Mathematical Applications to Reliability and Maintenance Problems in Engineering Systems

Mathematical Problems in Engineering

Guest Editors: Wenbin Wang, Philip Scarf, Shaomin Wu, and Enrico Zio Mathematical Applications to Reliability and Maintenance Problems in Engineering Systems Mathematical Problems in Engineering

Mathematical Applications to Reliability and Maintenance Problems in Engineering Systems

This is a special issue published in “Mathematical Problems in Engineering.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Mohamed Abd El Aziz, Egypt Abdel-Ouahab Boudraa, France Ahmed El Hajjaji, France Farid Abed-Meraim, France Francesco Braghin, Italy Fouad Erchiqui, Canada Silvia Abrahao,˜ Spain Michael J. Brennan, UK Anders Eriksson, Sweden Paolo Addesso, Italy Maurizio Brocchini, Italy Giovanni Falsone, Italy Claudia Adduce, Italy Julien Bruchon, France Hua Fan, China Ramesh Agarwal, USA Javier Buldufh, Spain Yann Favennec, France Juan C. Aguero,¨ Australia Tito Busani, USA Roberto Fedele, Italy Ricardo Aguilar-Lopez,´ Mexico Pierfrancesco Cacciola, UK Giuseppe Fedele, Italy Tarek Ahmed-Ali, France Salvatore Caddemi, Italy Jacques Ferland, Canada Hamid Akbarzadeh, Canada Jose E. Capilla, Spain Jose R. Fernandez, Spain Muhammad N. Akram, Norway Ana Carpio, Spain S. D. Flapper, Netherlands Mohammad-Reza Alam, USA Miguel E. Cerrolaza, Spain Thierry Floquet, France Salvatore Alfonzetti, Italy Mohammed Chadli, France Eric Florentin, France Francisco Alhama, Spain Gregory Chagnon, France Francesco Franco, Italy Juan A. Almendral, Spain Ching-Ter Chang, Taiwan Tomonari Furukawa, USA Saiied Aminossadati, Australia Michael J. Chappell, UK Mohamed Gadala, Canada Lionel Amodeo, France Kacem Chehdi, France Matteo Gaeta, Italy Igor Andrianov, Germany Xinkai Chen, Japan Zoran Gajic, USA Sebastian Anita, Romania Chunlin Chen, China CiprianG.Gal,USA Renata Archetti, Italy Francisco Chicano, Spain Ugo Galvanetto, Italy Felice Arena, Italy Hung-Yuan Chung, Taiwan Akemi Galvez,´ Spain Sabri Arik, Turkey Joaquim Ciurana, Spain Rita Gamberini, Italy Fumihiro Ashida, Japan John D. Clayton, USA Maria Gandarias, Spain Hassan Askari, Canada Carlo Cosentino, Italy Arman Ganji, Canada Mohsen Asle Zaeem, USA Paolo Crippa, Italy Zhong-Ke Gao, China Francesco Aymerich, Italy Erik Cuevas, Mexico Xin-Lin Gao, USA Seungik Baek, USA Peter Dabnichki, Australia Giovanni Garcea, Italy Khaled Bahlali, France Luca D’Acierno, Italy Fernando Garca, Spain Laurent Bako, France Weizhong Dai, USA Laura Gardini, Italy Stefan Balint, Romania Purushothaman Damodaran, USA A. Gasparetto, Italy Alfonso Banos, Spain Farhang Daneshmand, Canada Vincenzo Gattulli, Italy Roberto Baratti, Italy FabioDeAngelis,Italy J. Geiser, Germany Martino Bardi, Italy Stefano de Miranda, Italy Oleg V. Gendelman, Israel Azeddine Beghdadi, France Filippo de Monte, Italy Mergen H. Ghayesh, Australia Abdel-Hakim Bendada, Canada Xavier Delorme, France Anna M. Gil-Lafuente, Spain Ivano Benedetti, Italy Luca Deseri, USA Hector Gomez,´ Spain Elena Benvenuti, Italy Yannis Dimakopoulos, Greece Rama S. R. Gorla, USA Jamal Berakdar, Germany Zhengtao Ding, UK Oded Gottlieb, Israel Enrique Berjano, Spain Ralph B. Dinwiddie, USA Antoine Grall, France Jean-Charles Beugnot, France Mohamed Djemai, France Jason Gu, Canada Simone Bianco, Italy Alexandre B. Dolgui, France Quang Phuc Ha, Australia David Bigaud, France George S. Dulikravich, USA Ofer Hadar, Israel Jonathan N. Blakely, USA Bogdan Dumitrescu, Finland Masoud Hajarian, Iran Paul Bogdan, USA Horst Ecker, Austria F. Hamelin, France Daniela Boso, Italy Karen Egiazarian, Finland Zhen-Lai Han, China Thomas Hanne, Switzerland Panos Liatsis, UK Eva Onaindia, Spain Takashi Hasuike, Japan Wanquan Liu, Australia J. Ortega-Garcia, Spain Xiao-Qiao He, China Yan-Jun Liu, China A. Ortega-Monux,˜ Spain Mar´ıa I. Herreros, Spain Peide Liu, China Naohisa Otsuka, Japan Vincent Hilaire, France Peter Liu, Taiwan Erika Ottaviano, Italy Eckhard Hitzer, Japan Jean J. Loiseau, France Alkiviadis Paipetis, Greece Jaromir Horacek, Czech Republic Paolo Lonetti, Italy Alessandro Palmeri, UK Muneo Hori, Japan L. M. Lopez-Ochoa,´ Spain Anna Pandolfi, Italy A. Horvth, Italy V. C. Loukopoulos, Greece Elena Panteley, France Gordon Huang, Canada Valentin Lychagin, Norway Manuel Pastor, Spain Sajid Hussain, Canada Fazal Mahomed, South Africa P. N. Pathirana, Australia Asier Ibeas, Spain Yassir T. Makkawi, UK F. Pellicano, Italy Giacomo Innocenti, Italy Noureddine Manamanni, France Haipeng Peng, China Emilio Insfran, Spain Didier Maquin, France Mingshu Peng, China Nazrul Islam, USA Paolo Maria Mariano, Italy Zhike Peng, China Payman Jalali, Finland Benoit Marx, France Marzio Pennisi, Italy Reza Jazar, Australia G. A. Maugin, France Matjaz Perc, Slovenia Khalide Jbilou, France Driss Mehdi, France Francesco Pesavento, Italy Linni Jian, China Roderick Melnik, Canada Maria do R. Pinho, Portugal Zhongping Jiang, USA Pasquale Memmolo, Italy Antonina Pirrotta, Italy Bin Jiang, China Xiangyu Meng, Canada Vicent Pla, Spain Ningde Jin, China Jose Merodio, Spain Javier Plaza, Spain Grand R. Joldes, Australia Luciano Mescia, Italy J. Ponsart, France Joaquim J. Judice, Portugal Laurent Mevel, France Mauro Pontani, Italy Tadeusz Kaczorek, Poland Yuri V. Mikhlin, Ukraine Stanislav Potapenko, Canada Tamas Kalmar-Nagy, Hungary Aki Mikkola, Finland Sergio Preidikman, USA Tomasz Kapitaniak, Poland Hiroyuki Mino, Japan C. Pretty, New Zealand Haranath Kar, India Pablo Mira, Spain Carsten Proppe, Germany K. Karamanos, Belgium Vito Mocella, Italy Luca Pugi, Italy Chaudry Khalique, South Africa Roberto Montanini, Italy Yuming Qin, China Nam-Il Kim, Korea Gisele Mophou, France Dane Quinn, USA Do Wan Kim, Korea Rafael Morales, Spain Jose Ragot, France Oleg Kirillov, Germany Aziz Moukrim, France K. R. Rajagopal, USA Manfred Krafczyk, Germany Emiliano Mucchi, Italy Gianluca Ranzi, Australia Frederic Kratz, France Domenico Mundo, Italy Sivaguru Ravindran, USA Jurgen Kurths, Germany Jose J. Munoz,˜ Spain Alessandro Reali, Italy K. Kyamakya, Austria Giuseppe Muscolino, Italy Giuseppe Rega, Italy Davide La Torre, Italy Marco Mussetta, Italy Oscar Reinoso, Spain Risto Lahdelma, Finland Hakim Naceur, France Nidhal Rezg, France Hak-Keung Lam, UK Hassane Naji, France Ricardo Riaza, Spain Antonino Laudani, Italy Dong Ngoduy, UK Gerasimos Rigatos, Greece Aime’ Lay-Ekuakille, Italy Tatsushi Nishi, Japan JoseRodellar,Spain´ Marek Lefik, Poland Ben T. Nohara, Japan R. Rodriguez-Lopez, Spain Yaguo Lei, China Mohammed Nouari, France Ignacio Rojas, Spain Thibault Lemaire, France Mustapha Nourelfath, Canada Carla Roque, Portugal Stefano Lenci, Italy Sotiris K. Ntouyas, Greece Aline Roumy, France Roman Lewandowski, Poland Roger Ohayon, France Debasish Roy, India QingQ.Liang,Australia Mitsuhiro Okayasu, Japan R. Ruiz Garc´ıa, Spain Antonio Ruiz-Cortes, Spain Francesco Soldovieri, Italy M. E. Vazquez-M´ endez,´ Spain Ivan D. Rukhlenko, Australia Raffaele Solimene, Italy Josep Vehi, Spain Mazen Saad, France Ruben Specogna, Italy Kalyana C. Veluvolu, Korea Kishin Sadarangani, Spain Sri Sridharan, USA F. J. Verbeek, Netherlands Mehrdad Saif, Canada Ivanka Stamova, USA Franck J. Vernerey, USA Miguel A. Salido, Spain Yakov Strelniker, Israel Georgios Veronis, USA Roque J. Saltaren,´ Spain S. A. Suslov, Australia Anna Vila, Spain F. J. Salvador, Spain Thomas Svensson, Sweden R.-J. Villanueva, Spain A. Salvini, Italy Andrzej Swierniak, Poland U. E. Vincent, UK Maura Sandri, Italy Yang Tang, Germany Mirko Viroli, Italy Miguel A. F. Sanjuan, Spain Sergio Teggi, Italy Michael Vynnycky, Sweden Juan F. San-Juan, Spain Roger Temam, USA Junwu Wang, China Roberta Santoro, Italy Alexander Timokha, Norway Shuming Wang, Singapore Ilmar F. Santos, Denmark Rafael Toledo, Spain Yan-Wu Wang , China J. A. Sanz-Herrera, Spain Gisella Tomasini, Italy Yongqi Wang, Germany Nickolas S. Sapidis, Greece F. Tornabene, Italy J. Witteveen, Netherlands E. J. Sapountzakis, Greece Antonio Tornambe, Italy Yuqiang Wu, China Themistoklis P. Sapsis, USA Fernando Torres, Spain Dash Desheng Wu, Canada Andrey V. Savkin, Australia Fabio Tramontana, Italy Xuejun Xie, China Valery Sbitnev, Russia S. Tremblay, Canada Guangming Xie, China Thomas Schuster, Germany I. N. Trendafilova, UK Gen Qi Xu, China Mohammed Seaid, UK George Tsiatas, Greece Hang Xu, China Lotfi Senhadji, France Antonios Tsourdos, UK Xinggang Yan, UK J. Serra-Sagrista, Spain Vladimir Turetsky, Israel Luis J. Yebra, Spain Leonid Shaikhet, Ukraine Mustafa Tutar, Spain Peng-Yeng Yin, Taiwan Hassan M. Shanechi, USA E. Tzirtzilakis, Greece Ibrahim Zeid, USA Sanjay K. Sharma, India Filippo Ubertini, Italy Qingling Zhang, China Bo Shen, Germany Francesco Ubertini, Italy Huaguang Zhang, China Babak Shotorban, USA Hassan Ugail, UK Jian Guo Zhou, UK Zhan Shu, UK Giuseppe Vairo, Italy Quanxin Zhu, China Dan Simon, USA K. Vajravelu, USA Mustapha Zidi, France Luciano Simoni, Italy R. A. Valente, Portugal Alessandro Zona, Italy C. H. Skiadas, Greece Raoul van Loon, UK Michael Small, Australia Pandian Vasant, Malaysia Contents

Mathematical Applications to Reliability and Maintenance Problems in Engineering Systems, WenbinWang,PhilipScarf,ShaominWu,andEnricoZio Volume 2015, Article ID 629497, 2 pages

Parameter Estimation of a Delay Time Model of Wearing Parts Based on Objective Data,Y.Tang, J. J. Jing, Y. Yang, and C. Xie Volume 2015, Article ID 419280, 8 pages

A Decision Optimization Model for Leased Manufacturing Equipment with Warranty under Forecasting Production/Maintenance Problem, Zied Hajej, Nidhal Rezg, and Ali Gharbi Volume 2015, Article ID 274530, 14 pages

Prognostics and Health Management: A Review on Data Driven Approaches,KwokL.Tsui,NanChen, Qiang Zhou, Yizhen Hai, and Wenbin Wang Volume 2015, Article ID 793161, 17 pages

Parametric Sensitivity Analysis for Importance Measure on Failure Probability and Its Efficient Kriging Solution, Yishang Zhang, Yongshou Liu, and Xufeng Yang Volume 2015, Article ID 685826, 13 pages

Gear Crack Level Classification Based on EMD and EDT,HaipingLi,JianminZhao,XinghuiZhang, and Hongzhi Teng Volume 2015, Article ID 137274, 10 pages

Weibull Failure Probability Estimation Based on Zero-Failure Data,PingJiang,YunyanXing,XiangJia, and Bo Guo Volume 2015, Article ID 681232, 8 pages

Fast Prediction with Sparse Multikernel LS-SVR Using Multiple Relevant Time Series and Its Application in Avionics System, Yang M. Guo, Pei He, Xiang T. Wang, Ya F. Zheng, Chong Liu, and Xiao B. Cai Volume 2015, Article ID 460514, 10 pages

Fuzzy Dynamic Reliability Models of Parallel Mechanical Systems Considering Strength Degradation Path Dependence and Failure Dependence, Peng Gao and Liyang Xie Volume 2015, Article ID 649726, 9 pages

Delayed Age Replacement Policy with Uncertain Lifetime, Xueyan Li and Chunxiao Zhang Volume 2015, Article ID 528726, 6 pages

Minimizing the Discrepancy between Simulated and Historical Failures in Turbine Engines: A Simulation-Based Optimization Method, Ahmed Kibria, Krystel K. Castillo-Villar, and Harry Millwater Volume 2015, Article ID 813565, 11 pages

Accelerated Testing with Multiple Failure Modes under Several Temperature Conditions,ZongyueYu, Zhiqian Ren, Junyong Tao, and Xun Chen Volume2014,ArticleID839042,8pages Reliability Analysis of the Proportional Mean Residual Life Order,M.Kayid,S.Izadkhah,andH.Alhalees Volume 2014, Article ID 142169, 8 pages

Reliability Analysis of a Cold Standby System with Imperfect Repair and under Poisson Shocks, Yutian Chen, Xianyun Meng, and Shengqiang Chen Volume 2014, Article ID 507846, 11 pages Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 629497, 2 pages http://dx.doi.org/10.1155/2015/629497

Editorial Mathematical Applications to Reliability and Maintenance Problems in Engineering Systems

Wenbin Wang,1,2 Philip Scarf,3 Shaomin Wu,4 and Enrico Zio5,6

1 Donlinks School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, China 2Faculty of Business and Law, Manchester Metropolitan University, Manchester M15 6BH, UK 3Salford Business School, University of Salford, Salford M5 4WT, UK 4Kent Business School, University of Kent, Canterbury CT2 7PE, UK 5CentraleSupelec, 92295 Chatenay-Malabry, France 6Department of Energy, Polytechnic University of Milan, 20133 Milano, Italy

Correspondence should be addressed to Wenbin Wang; [email protected]

Received 8 March 2015; Accepted 8 March 2015

Copyright © 2015 Wenbin Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reliability and maintenance are well known concepts, which illustrates an accelerated testing procedure, in which both contribute to retaining engineering systems in their func- high temperatures and low temperatures are applied to a tioning states. Reliability is one of the fundamental criteria product. This allows building the reliability function of the in engineering systems design and maintenance serves to product by statistical analysis, accounting for multiple failure support it throughout the systems life. As such, maintenance modes and variable working conditions. acts in parallel to production and can have a great impact on The work by M. Kayid et al. introduces and studies a new the availability and capacity of production and on the quality stochastic order called proportional mean residual life order. of the products. For this reason, it deserves great attention, Several characterizations and preservation properties of the careful planning, and continuous improvement. new order under some reliability operations are discussed. Toinform the strategic decision-making on reliability and The paper by Y. Chen et al. addresses the reliability maintenance of engineering systems, mathematical models analysis of a two-component, cold-standby system with a and optimization techniques have long been used. These single repairman, who may have vacations. The paper derives models and techniques can help in achieving the desired a number of classical reliability indices under such condition: target of system reliability and retain it with cost-effective system availability, system reliability, the rate of occurrence maintenance. of system failures, and the mean time to the first failure of the The interest in the mathematical models and optimization system. techniques for system reliability and maintenance is demon- In the paper entitled “Gear Crack Level Classification strated by the over 90 papers submitted to this special issue. BasedonEMDandEDT,”theauthorsuseEmpiricalMode Finally after an intense and rigorous reviewing process, 13 Decomposition to process vibration signals and, then, the papers were selected for publication. Euclidean Distance Technique to measure the (Euclidean) The paper entitled “Prognostics and Health Management: similarity between the test sample and samples from four A Review on Data Driven Approaches” presents an extensive classes, for gear crack fault classification. The results obtained review on the stochastic processes and regression-based show that the proposed method has high accuracy rates in models for Prognostics and Health Management (PHM), classifying different crack levels and in adaptive to different based on available monitored data. Some practical examples conditions. and applications are also illustrated. X. Li and C. Zhang present a paper titled “Delayed Age The paper entitled “Accelerated Testing with Multi- Replacement Policy with Uncertain Lifetime.” The authors ple Failure Modes under Several Temperature Conditions” consider the delayed age replacement policy with uncertain 2 Mathematical Problems in Engineering lifetimes and find that the optimal replacement time is irrelevant to the uncertain distribution of lifetime of the first unit, over the infinite time span. Y. Gao et al. propose a new scheme of health index prediction, which utilizes multiple relevant time series to enhance the completeness of the information and adopt a prediction model based on least squares support vector regression to perform the health trend prediction. Z. Hajej et al. in their paper develop a mathematical model to study the lease contract with basic and extended warranty, based on a win-win relationship between the lessee andthelessor.Theinfluenceoftheproductionratesis considered to determine a theoretical condition under which a compromise-pricing zone exists, under different schemes of maintenance policies. P. Gao and L. Xie develop a fuzzy dynamic reliability model for parallel mechanical systems, with respect to stress and strength parameters. A practical example is chosen to demonstrate the proposed model. Y. Tang et al. develop a delay time-based model for optimization of inspection intervals, which is completely based on maintenance data for estimating the model parameters. Then, they illustrate the method on a filter and a blowout preventer rubber core. The availability of sufficient data for reliability and main- tenancemodelingisalwaysaprobleminapplications.In this regard, Y. Peng et al. present a method to estimate the uncertainty intervals of the failure probability estimate by Weibull distributions, in the case of no available failure data. Some engineering experience or hypothesis testing is required for the set-up of the shape parameter. Y. Zhang et al. present an active learning Kriging solution to calculate moment-independent importance measures based on the failure probability. Two numerical examples and two engineering examples are analyzed to demonstrate the significance of the proposed parametric sensitivity index, as well as the efficiency and precision of the calculation method. A. Kibria et al. address the problem of estimating the failure rate of a component and provide a simulation-based optimization method for the minimization of the discrepancy between the simulated and the historical percentages of failures for turbine engine components. The method can be considered as a decision-making tool for maintenance, repair, and overhaul.

Acknowledgments First, we would like to thank all authors for their excellent contributions to this special issue. Of course, sincere thanks go also to all reviewers for their careful (and voluntary) reviewwork,whichhashelpedtoimprovethequalityofthe papers published and thus the significance of the special issue. Wenbin Wang Philip Scarf Shaomin Wu Enrico Zio Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 419280, 8 pages http://dx.doi.org/10.1155/2015/419280

Research Article Parameter Estimation of a Delay Time Model of Wearing Parts Based on Objective Data

Y. Tang,1 J. J. Jing,2 Y. Yang,3 and C. Xie1

1 School of Mechatronic Engineering, Southwest Petroleum University, Chengdu 610500, China 2 Safety, Environment, Quality Supervision & Testing Research Institute, CCDE, Guanghan 618000, China 3 School of Science, Southwest Petroleum University, Chengdu 610500, China

Correspondence should be addressed to Y. Tang; [email protected]

Received 10 June 2014; Revised 22 September 2014; Accepted 27 October 2014

Academic Editor: Wenbin Wang

Copyright © 2015 Y. Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The wearing parts of a system have a very high failure frequency, making it necessary to carry out continual functional inspections and maintenance to protect the system from unscheduled downtime. This allows for the collection of a large amount of maintenance data. Taking the unique characteristics of the wearing parts into consideration, we establish their respective delay time models in ideal inspection cases and nonideal inspection cases. The model parameters are estimated entirely using the collected maintenance data. Then, a likelihood function of all renewal events is derived based on their occurring probability functions, and the model parameters are calculated with the maximum likelihood function method, which is solved by the CRM. Finally, using two wearing parts from the oil and gas drilling industry as examples—the filter element and the blowout preventer rubber core—the parameters of the distribution function of the initial failure time and the delay time for each example are estimated, and their distribution functions are obtained. Such parameter estimation based on objective data will contribute to the optimization of the reasonable function inspection interval and will also provide some theoretical models to support the integrity management of equipment or systems.

1. Introduction functional inspections and repairs/replacements based on the experience and subjective judgment of personnel. Since For the reliability maintenance and security management of the maintenance personnel do not identify their failure equipment, the establishment of the failure time distribution time distribution rule by the quantitative method, they will model (FTDM) of parts or systems is one of the key steps develop irrational function inspection intervals, which may toward making scientific maintenance decisions [1]. Parame- lead to excess maintenance, a reduction in the reliability of ter estimation is the first step in the application of the FTDM, the system, lower production efficiency, and so on. Before the and the accuracy and precision of these estimations directly occurrence of the functional failure of the wearing parts, we affects whether the optimized predetermined maintenance foundthatsomesymptomswereoftenpresent,suchasfatigue interval is reasonable. Therefore, parameter estimation is cracks, wear, and corrosion. These symptoms are a potential a crucial part in the process of the establishment of the failure process of the wearing parts. Though they present no FTDM. When a substantial amount of maintenance data immediate threat to the equipment, if, allowed to accumulate can be collected, the parameter estimation method can for a period of time, such defects will gradually increase use completely objective data, which is more accurate than and reveal themselves and cause the functional failure of the subjective data [2]. The failure frequency of wearing parts is equipment. This failure process is consistent with the delay relatively higher than other parts used in the operation pro- time theory. cess of the equipment. Traditional solutions to ensure the safe In1973,Christerputforwardtheconceptofthedelaytime and reliable operation of the equipment include initiating model (DTM) in the context of building maintenance [3]. 2 Mathematical Problems in Engineering

A number of research papers followed, applying the DTM Inspect Inspect Inspect to industrial asset inspection problems. Christer and Waller [4]usedsubjectivedatatosolvetheDTMparameters,and t Christer and Redmond [5] studied the maximum likelihood 1󳰀 2󳰀 3󳰀 4󳰀 5󳰀 functionmethodofthesubjectiveestimateoftheDTM.Cor- 1234AB 5 C responding with the subjective estimation methods, Baker Figure 1: Function inspection schematic diagram, where “I” and Wang [6, 7] put forward a parameter estimation method e founded on objective data. Christer and Wang [8]then denotes the initial defect point of a part; “ ” denotes failure points caused by a defect; and A, B, and C denote the times of the three proposedamethodthatusessubjectiveandobjectivedata function inspections. to solve the DTM for multicomponents in complex systems. Recent research on the DTM includes the work by Wang et al. [9], who proposed a parameter estimation method of the Normal Failure delay time maximum likelihood of the DTM, and Hu et al. [10], who t established a nonideal DTM solved by a genetic algorithm. o Wang [11, 12]alsopresentedatwo-stageprognosismodelin uu+ Potential defect Function failure condition-based maintenance and an inspection model based occurring time occurring time on a three-stage failure process. However, in these previous studies, the parameter estima- Figure 2: The DTM schematic diagram. tion of the DTM was solved by using simulation data, subjective data, or a combination of subjective data and objective 󸀠 󸀠 󸀠 data, mainly because there was rarely sufficient maintenance thetime“PointB”,thefailuresattimes1,2 and 3 can be data to allow the use of fully objective data to solve it [13– avoided.Iftheinspectioniscarriedoutatthetime“PointC”, then “Defect 5” can be identified prior to failure, but failures at 16]. Moreover, the formula of the likelihood function of the 󸀠 󸀠 󸀠 󸀠 DTM is quite complex because of both the failure density times 1 ,2,3 and 4 will have already occurred. In summary, function and the failure probability function with the integral from the function inspection of the three different times term, and the maximum could not be solved by traditional above, we know that a fault could be averted only if the derivation methods [17, 18]. However, we found that, from the inspection is set between the defect’s appearance time and process of frequent functional inspections and maintenance the failure occurrence time. Therefore, it is necessary to deter- renewals of the wearing parts, a large amount of maintenance mine a reasonable regular inspection to lower the failure rate data is collected in a short period of time. The parameters of of parts. the established DTM of the wearing parts could therefore be estimated based on sufficient objective data. This effectively 2.2. Delay Time Concept. The delay time concept is similar removes the subjective factors and reduces the estimation in definition to the well-known potential failure interval in error of the parameters. It is necessary to explore a novel reliability-centered maintenance (RCM), defining the failure algorithm to effectively look for the complicated maximum process of an asset as a two-stage process, as shown in likelihood function of the DTM and to get the parameter Figure 2. The first stage is the normal operating stage ofan estimation values. Thus, we will obtain the FTDM of the asset, from new to the point that a potential defect is wearing parts, which can help to optimize the maintenance identified. The second stage is defined as the delay time, interval and make scientific maintenance decisions in the which is from the point of the potential defect to the point of next study. function failure [2]. Through research and analysis, the most The remainder of this paper is organized as follows. In wearing parts (e.g., a filter core, bop rubber core, mud pump Section 2,weintroducetheconceptofthefunctioninspection bearings, etc.) comply with the DTM. It is the existence of and the DTM. In Section 3, we establish the ideal inspection such a delay time that provides the opportunity for preventive model and the nonideal inspection model of the wearing maintenance (PM) to be carried out, to remove the identified parts. Section 4 is devoted to estimating the parameters of the defects before the parts fail. With appropriate modeling of DTM by maximizing the log-likelihood, which is solved with the durations of these two stages, optimal function inspection the CRM. In Section 5,wetaketwoexamplesfromtheoiland intervals can be identified to optimize a criterion function of gas drilling industry to verify the DTM of the wearing parts interest. and the solving method. We provide a brief discussion of our results in Section 6. 3. The DTM of Wearing Parts

2. The Delay Time Concept 3.1. Notation and Assumptions 2.1. Function Inspection. A schematic diagram of function 3.1.1. Notation. In this paper, we use the following notation. inspectionforthepartsorcomponentsisillustratedin 𝑢: The initial time of the defect Figure 1 [19]. The function inspection is carried out at the 𝑔(𝑢) 𝑢 time “Point A” and the inspection is perfect, so “Defect 1” : Probability density function of the initial time can be identified and the parts will be repaired and replaced. 𝐺(𝑢): Cumulative distribution function of the initial Similarly, if a perfect function inspection is carried out at time 𝑢 Mathematical Problems in Engineering 3

V t :Thedelaytimeofthedefect Last renewal time Defect was not detected at n time 𝑓(V): Probability density function of the delay time V Function failure time 𝐹(V): Cumulative distribution function of the delay time V 0 t1 ··· tn tk 𝑁𝐼: The total number of inspection renewals Initial defect time

𝑁𝐹: The total number of failure renewals (a) Failure renewal event

𝑃𝑚(𝑡𝑖):Theprobabilityofthe𝑖th inspection renewal Defect was not detected at tm time Last renewal time happening at time 𝑡𝑖 Defect was detected

𝑝𝑏(𝑡𝑗): The density function of the 𝑗th failure renewal happening at time 𝑡𝑗 0 t1 ··· tm tk 𝑃𝑛(𝑡): The probability of no failure happening during the whole of observation period 𝑡 Initial defect time (b) Inspection renewal event 𝑡𝑖:Theoccurrencetimeofthe𝑖th inspection renewal The part has worked normally in 𝑡𝑚: The last inspection time before the 𝑖th inspection Last renewal time the observation period renewal

𝑡𝑗:Theoccurrencetimeofthe𝑗th inspection renewal 𝑡 𝑗 𝑛: The last inspection time before the th inspection 0 t1 ··· tn tk renewal (c) Nonrenewal event 𝑡𝑘:Thelastinspectiontimebeforetheendofthe observation Figure 3: Function inspection process. 𝑡:Thetimeofendoftheobservation 𝑚:Thetotalinspectionnumberbefore𝑖th times inspection renewal inspection renewal event, and a nonrenewal event, as shown 𝑛:Thetotalinspectionnumberbefore𝑗th times in Figure 3 [12]. failure renewal We combine the function inspection process of the 𝑘:Thetotalinspectionnumberbeforetheendofthe wearing parts above with the theory of the DTM [2]tosetup observation the failure density function or the failure probability function 𝑡 𝑐 of the three kinds of events [20]. 𝑐:The th opportunity renewal time Let 𝑋 be a random variable of the time to failure. Then, the 𝑁𝑐: The total number of opportunity renewals. probability of the 𝑖th inspection renewal happening at time 𝑡𝑖 is given by 3.1.2. Assumptions. The following assumptions are used in this paper. 𝑃𝑚 (𝑡𝑖)=𝑃(𝑋<𝑡𝑖) (1) The wearing part is a single component, inspected at 𝑡𝑖 (1) 𝑡 (𝑡 = 𝑖𝑇, 𝑖 = 1, 2, 3, ..) 𝑇 time point 𝑖 𝑖 ,where is a = ∫ 𝑔 (𝑢) [1 − 𝐹 𝑖(𝑡 −𝑢)]𝑑𝑢, 𝑡 regular inspection interval. 𝑚 (2) When defects are identified at the inspection point, they need to be repaired immediately. Otherwise, the where 𝑃(𝑋<𝑡𝑖) is the probability that the time to failure 𝑋 wearing part would continue to work until either a is less than the inspection renewal time 𝑡𝑖. failure occurs or the next inspection is reached. The density function of the 𝑗th failure renewal happens at 𝑡 (3) If defects are identified or function failures occur time 𝑗,whichisgivenby between two continual inspection points, the wearing part needs be immediately repaired or replaced. 𝑑 𝑝 (𝑡 )= 𝑃(𝑋≤𝑡) 𝑏 𝑗 𝑑𝑡 𝑗 3.2. The Ideal Inspection Model. The defects can easily be (2) 𝑡𝑗 found for some wearing parts due to their simple structures. = ∫ 𝑔 (𝑢) 𝑓(𝑡 −𝑢)𝑑 , We have assumed that the process of the function inspection 𝑗 𝑢 𝑡𝑛 forthesekindsofwearingpartsisanidealcase.Wecan detect any defects for such wearing parts perfectly at the next inspection time point. There are three kinds of events in the where 𝑃(𝑋<𝑡𝑗) is the probability that the time to failure 𝑋 failure process for the wearing parts: a failure renewal event, is less than the failure renewal time 𝑡𝑗. 4 Mathematical Problems in Engineering

The probability of no failure happening during the whole the parameters. This computing process can be simplified by of observation period 𝑡 is given by taking natural logarithms to get the log-likelihood function:

𝑃𝑛 (𝑡) =1−𝑃(𝑋<𝑡) 𝑁𝐼 𝑡 𝑡 (3) log 𝐿 (Θ) = ∑ log [𝑃𝑚 (𝑡𝑖)] =1−∫ 𝑔 (𝑢) 𝑑𝑢 + ∫ 𝑔 (𝑢)(1−𝐹(𝑡−𝑢)) 𝑑𝑢, 𝑖=1 0 𝑡 𝑘 (6) 𝑁 where 𝑃(𝑋 < 𝑡) is the probability that nonfailure time 𝑋 is 𝐹 + ∑ [𝑝 (𝑡 )] + [𝑃 (𝑡)]. less than the whole of observation period 𝑡. log 𝑏 𝑗 log 𝑛 𝑗=1 3.3. The Nonideal Inspection Model. Some wearing parts have a complex structure and shape and therefore need greater Finally, the parameters of the DTM can be estimated by inspection conditions and techniques in the inspection pro- maximizing the log-likelihood. cess. A few such defects cannot be identified perfectly, so we have assumed that the process of the function inspection for 4.2. Method of Solving Maximum Likelihood Function. The these kinds of wearing parts is a nonideal inspection case. traditional maximum likelihood estimation (MLE) method Assuming that the probability of the defect being identified used to adopt the derivative of likelihood function, solving is 𝑟 and then referring to the ideal inspection model in its maximum value by setting the derivative equal to zero Section 3.2 (including (1)–(3)), the density function or the and finally solving the estimated parameters of the function. probability function of the three kinds of events is established However, (6) is a more complicated log-likelihood function, fortheprocessofthenonidealinspection[20]andisgivenby containing the failure density function and the failure proba- 𝑚 𝑡 bility function with integral terms; taking the derivative of it 𝑚−𝑗+1 𝑗 is quite complex and cannot be solved by the traditional MLE 𝑃𝑚 (𝑡𝑖)=∑ (1−𝑟) 𝑟 ∫ 𝑔 (𝑢) [1 − 𝐹 𝑖(𝑡 −𝑢)]𝑑𝑢 𝑗=1 𝑡𝑗−1 method. Hence, for the parameter estimation, we need to find a suitable calculation method to cope with this problem. By 𝑡𝑖 +𝑟∫ 𝑔 (𝑢) [1 − 𝐹 (𝑡 −𝑢)]𝑑 , analyzing the characteristics of (6) and the value range of the 𝑖 𝑢 parameters in this study, we can convert it to an extremum 𝑡𝑚 problem of no constraints, which can be solved by sophisti- 𝑛 𝑡𝑘 𝑝 (𝑡 )=∑ (1−𝑟)𝑛−𝑘+1 ∫ 𝑔 (𝑢) 𝑓(𝑡 −𝑢)𝑑 cated optimization algorithms. There are several nonderiva- 𝑏 𝑗 𝑗 𝑢 tive optimization algorithms: simplex substitution method 𝑘=1 𝑡𝑘−1 (4) (small-scale, rapid convergence), CRM (small-scale, noncou- 𝑡 𝑗 pling problem), grid point method (nonconvex problems), + ∫ 𝑔 (𝑢) 𝑓(𝑡𝑗 −𝑢)𝑑𝑢, 𝑡 andMonteCarlomethod(nonconvexproblems)[21]. 𝑛 Since the log-likelihood function of (6) belongs to a 𝑡 𝑡 small scale and noncoupling problem, we choose the CRM 𝑃 (𝑡) =1−∫ 𝑔 (𝑢) 𝑑𝑢 + ∫ 𝑔 (𝑢)(1−𝐹(𝑡−𝑢)) 𝑑 𝑛 𝑢 to estimate the parameters in the distribution function of the 0 𝑡𝑘 initial defect time and the defect delay time of wearing parts. 𝑘 𝑡 𝑗 𝑘−𝑗+1 The CRM is a direct search method without derivation: taking + ∑ ∫ (1−𝑟) 𝑔 (𝑢)(1−𝐹(𝑡−𝑢)) 𝑑𝑢. 𝑛 coordinate directions of 𝑥 as the search direction (namely, 𝑗=1 𝑡𝑗−1 0 starting from 𝑥 ), the first search along the 𝑥1 direction, the second search along the 𝑥2 direction,...,the𝑛th search along 4. Delay Time Model Algorithm the 𝑥𝑛 direction, the 𝑛+1th search along the 𝑥1 direction, and so on, continuously search along the direction of rotation 4.1. Establishing the Likelihood Function. In view of the until the convergence condition is met [22]. DTM characteristics of the two cases (ideal and nonideal The remaining calculation steps of the CRM are as inspection) and the solving method, we decided to use follows. the maximum likelihood function method to estimate the 0 parameters of the models. We take the density function or the Step 1. Select the initial point 𝑥 (also known as the ini- probability function of the three kinds of renewal events for tial base point), the initial step Δ𝑥𝑖,andtheconvergence the wearing parts to establish the likelihood function of the precision 𝜀𝑖.The initial step is generally a desirable variable DTM for a period of observed time [20], which is given by estimation range (𝑢𝑖 −𝑙𝑖)of1/100, the convergence precision 𝜀𝑖 can take the initial step Δ𝑥𝑖 of 1/100,andlet𝑘=0. 𝑁𝐼 𝑁𝐹 𝐿 (Θ) = ∏𝑃𝑚 (𝑡𝑖) ⋅ ∏𝑝𝑏 (𝑡𝑗) ⋅𝑃𝑛 (𝑡) , (5) 𝑖=1 𝑗=1 Step 2. Begin a round of exploratory searching along itself direction one by one. To give 𝑥1 an incremental Δ𝑥1,other where Θ is the set of parameters of the probability density variables remain unchanged. Then we get a test point: function and the cumulative distribution function for the initial time and delay time. The likelihood function is taken 𝑇 𝑥𝑠 =(𝑥(𝑘+1) +Δ𝑥 ,𝑥(𝑘+1),...,𝑥(𝑘+1)) . (7) to look for the maximum, to obtain the estimated values of 1 1 2 𝑛 Mathematical Problems in Engineering 5

𝑠 𝑠 𝑘−1 To examine the feasibility of 𝑥 ,if𝑓(𝑥 )<𝑓(𝑥 ),then Table 1: Failure renewal time of the filter element. 𝑠 𝑥 begins as a coordinate direction search. Otherwise, search Renewal time Renewal time 𝑥𝑘−1 S/N Time from point along the opposite direction: (weeks) (weeks) 𝑇 1 21 16 30 𝑥𝑠 = (𝑥(𝑘+1) −Δ𝑥 ,𝑥(𝑘+1),...,𝑥(𝑘+1)) . (8) 1 1 2 𝑛 2 5 17 14 3 22 18 33 If the two search directions fail, stopping at the original 4 11 19 27 point and not moving, then take a step Δ𝑥2 to search along the 𝑥2 direction until 𝑥𝑛. The end point of this round of search 5 15 20 8 𝑘 was called base point 𝑥𝐵. 6 37 21 7 7 26 22 25 𝑥𝑘 Step 3. Carry through the Pattern Move to point 𝑀.The 8 19 23 13 𝑥𝑘−1 Pattern Move direction is from a base point 𝐵 to the 9 18 24 29 𝑥𝑘 current base point 𝐵, and the amount of move is the distance 10 3 25 9 between two points: 11 23 26 11 𝑘 𝑘 𝑘 𝑘−1 12 16 27 17 𝑥𝑀 =𝑥𝐵 + (𝑥𝐵 −𝑥𝐵 ) . (9) 13 10 28 15 Then, examine the feasibility of the Pattern Move. If 14 9 29 6 𝑘 𝑘 𝑠 𝑓(𝑥𝑀)<𝑓(𝑥𝐵) is established, then take the point 𝑥𝑀 as the 15 18 startingpointforthenextroundsearch;otherwise,cancelthis 𝑘 Pattern Move and take the point 𝑥𝐵 as the starting point for the next round search. Table 2: Inspection renewal time of the filter element.

Step 4. Repeat Steps 2 and 3 to search. The Pattern Move S/N Renewal time (weeks) Time Renewal time (weeks) is a loop execution until the exploratory searches of each 1161028 coordinate direction have all failed and still remain unmoving 241112 in the original base point. Then, test whether it satisfies the 3161220 Δ𝑥 <𝜀 convergence criteria 𝑖 𝑖 or not. If it is not satisfied, the 4201316 variable step size Δ𝑥𝑖 (𝑖=1,2,3,...,𝑛)is reduced by half, and 516148 a new round of exploratory search is started. If it is satisfied, ∗ 𝑘 ∗ 𝑘 6321528 put out an optimal point 𝑥 =𝑥𝐵 and 𝑓(𝑥 )=𝑓(𝑥𝐵)→ min 𝑓(𝑥). 7241612 8321736 9161824 5. Numerical Example 5.1. The Ideal Inspection Model Numerical Example. The filter element that plays a key role in the hydraulic system of the remote control station is a wearing part in the oil and gas the regular inspection interval for the filter element was drilling process. It has the advantages of a simple structure approximatelyonceeveryfourweeks.Between2003and2013, and low inspection requirements and is therefore considered the filter element in a hydraulic system of a remote control as suitable for the ideal inspection model. Learning from station (FKQ-400-5-B) in the Tarim Oilfield had 29 failure the oil field maintenance records, the maintenance personnel renewals and 18 inspection renewals. Detailed maintenance renew the filter element for the hydraulic system in the records for the filter element were processed and collated, as following two cases. shown in Tables 1 and 2. Through analysis of the time distribution regularities of (i) Failure renewal: if the filter element is clogged or failure renewal and inspection renewal in Tables 1 and 2,we leaking or has exploded and so forth, resulting in the assumed that the initial time and the delay time follow the well control equipment being unable to be operated Weibull distribution or the exponential distribution. By using initsnormalstate,weregarditasafailureanditwill the data in Tables 1 and 2 with (1)–(3),wetookthoseobtained be maintained or replaced immediately. equations into (5) to get the likelihood function of the DTM (ii) Inspection renewal: within the regular inspection, the in an ideal inspection case. We then took the logarithm for it, filter element needs to be cleaned or replaced when a to easily seek for the parameter estimation of the model. We large amount of grease and deposit has been detected considered that the maximum of the likelihood function was in it. a small scale in the absence of a constraint range and took the CRM. The parameters of the initial time and the delay time of Combined with the actual environments and conditions the filter element were estimated with different distribution in the oilfield exploitation and construction, we found that types, as shown in Table 3. 6 Mathematical Problems in Engineering

Table 3: The parameters estimation of the DTM for the filter element.

Model The initial time 𝑢 The delay time V

Parameter 𝛼1 𝛽1 𝛼2 𝛽2 AIC = 2 log 𝐿+2𝑘 𝐸-𝐸 — 17.98 — 2.36 56.24 𝐸-𝑊 — 40.42 1.23 5.94 62.08 𝑊-𝑊 1.23 18.24 0.27 2.09 60.16 𝑊-𝐸 1.12 18.06 — 2.21 58.84 Notes: 𝑊: Weibull distribution, 𝐸: exponential distribution. 𝑊/𝐸 indicates that the initial time obeys a Weibull distribution, and the delay time is an exponential distribution. “—” means the parameters do not need to be estimated in the exponential distribution.

By comparing and analyzing the minimum of the Akaike Table 4: Failure renewal time of rubber core. information criterion (AIC), we find that the distribution Early inspection time S/N Failure renewal time (months) regularities of the initial failure time and the delay time (months) about the filter element obeying the exponential distribution aremoreaccurateintheidealinspectioncase[20]. From 13 — the results of Table 3,wehaveobtainedtheirparameter 27 — estimation values: the average initial time 𝑢 is 17.98weeks, and 3124 the average delay time V is 2.36 weeks. 44 — Based on the functional expression of the exponential 5157 distribution, we obtain the failure time distribution functions 6233,12 for the filter element: the probability density function of −17.98𝑢 7195,13 the initial failure time 𝑔(𝑢) = 17.98𝑒 ,thecumulative distribution function of the initial failure time 𝐺(𝑢) = 1− 85 — −17.98𝑢 𝑒 , the probability density function of the delay time 9158 −2.36V 𝑓(V) = 2.36𝑒 , and the cumulative distribution function 10 1 — −2.36V of the delay time 𝐹(V)=1−𝑒 .

5.2. The Nonideal Inspection Model Numerical Example. The Ramblowoutpreventer(RamBOP)isoneofthekeysafety Table 5: Inspection renewal time of the rubber core. devicesinthewellcontrolsystem,whosemainroleisto ensure the safety of the drilling operation. Therefore, the field Early inspection time S/N Failure renewal time (months) staff used to carry out a strict function inspection of it. The (months) rubber seal on the ram end of the Ram BOP is called the 12 — rubber core, and it is an important component part that is 2168 the key to effectively closing the well in the process of well control. The rubber core is a wearing part with a very high 31810 failure frequency, and it used to be renewed in the following 45 — circumstances [23]. 510— 6122,8 (i) Failure renewal: in the oil and gas well, the high 79 6 pressure and high speed multiphase fluid that flows 88 2 throughtheRamBOPcancorrodetherubbercore, causing shedding or serious deformation in the process of closing the well, resulting in well liquid leakage and ineffective closing of the wellhead. In drilling operation, it is necessary to carry out a function (ii) Inspection renewal: through regular inspection, if the inspection for the Ram BOP approximately every 1-2 months. rubber core is found to be forming an axial scratch, After the completion of the drilling, the Ram BOP is sent back aging, corroding, cracking, or peeling, we need to to the equipment factory approximately every 5–10 months replace it immediately. for maintenance. Based on the actual situation, the rubber core of the Ram BOP meets a nonideal process of function (iii) Opportunity renewal: no matter how well the rubber inspection, where the probability of perfectly inspecting the core works, when we maintain the other parts of the defect (through the consultation of on-site experts) generally Ram BOP, we also need to inspect the rubber core. If ranges from 80% to 90%. Through research into the Tarim problems are found or potential failure is realized, it Oilfield,wehavetherepairrecordoftherubbercoreinthe must be replaced immediately. Ram BOP (FZ35-70), as shown in Tables 4, 5,and6. Mathematical Problems in Engineering 7

Table 6: Opportunity renewal time of the rubber core. The probability density function of the delay time is

1.21 Opportunity renewal time Early inspection time 𝑢 2.21 S/N 𝑓 (V) = 0.68 ( ) 𝑒−(𝑢/3.27) . (13) (months) (months) 3.27 16 — 2114The cumulative distribution function of the delay time is 312— (V/3.27)2.21 𝐹 (V) =1−𝑒 . (14) 44 — 5127As we all know, PM includes a regular inspection, repair, 69 —andreplacementroutinedesignedtoreducetheriskofsystem 7158failure. The probability density function and the cumulative distribution function of failure time, including the initial time anddelaytime,playaveryimportantroleintheapplication of the DTM, and the accurate parameter estimation for the model also directly affects the optimization of the reasonable PM interval. Therefore, we have obtained an optimal value of parameter estimation for the initial time model and delay Because of the existence of the opportunistic repair of the time model based on the objective data, and these models rubber core in the function inspection process of the Ram can be applied to the function test model of the potential BOP, the maximum likelihood function was changed [20]: unmeasured failures to optimize the PM interval on account of the cost, availability, and risk guideline.

𝑁𝐼 𝑁𝐹 𝑁𝑐 𝐿=∏𝑃𝑚 (𝑡𝑖)⋅∏𝑝𝑏 (𝑡𝑗)⋅∏𝑃𝑛 (𝑡𝑐). (10) 6. Conclusion 𝑖=1 𝑗=1 𝑐=1 In the ideal inspection and nonideal inspection cases, theDTM(includingthefailurerenewalmodel,inspection We can assume that the initial time and the delay time follow renewal model, and nonfailure renewal model) was estab- the Weibull distribution or the exponential distribution. By lished for the wearing parts, which can be extended to using the data in Tables 4–6 with (4),wetookthoseobtained other single components and simple systems obtaining a equations into (10) to get the likelihood function of the DTM large number of objective maintenance data. Moreover, the in a nonideal inspection case. maximum likelihood for DTM was solved by the CRM in Wethen took the logarithm for it to get the log-likelihood, thenoconstraintrange.Theexamplesofthefilterelement to seek for reasonable parameters of the DTM by the maxi- forthehydraulicsystemofaremotecontrolstationandthe mum likelihood in the no constraint range. Finally, through rubber core of Ram BOP were verified for the ideal inspection calculation with the CRM, the initial time and delay time model and nonideal inspection model, respectively. We have parameters of the model of the rubber core can be estimated, estimated the parameters of the DTM and determined more as shown in Table 7. accurate distribution types for their failure function, and From the parameter estimation results in Table 7,wecan both the average initial defect time and the average failure delay time were determined to provide data for the further see that both the initial time and the delay time of rubber core function to optimize the inspection interval, which verifies obeying the Weibull distribution are more accurate because of the feasibility of the models and the method. We then the value of AIC minimum, and the most reasonable values 𝛼 = 1.25 𝛽 = 10.28 𝛼 = 2.21 obtained the probability density function and the cumulative of parameter estimation are 1 , 1 ; 2 , distribution function of failure time, including the initial time 𝛽2 = 3.27. Therefore, the average initial time 𝑢 is 10.28 months V and the delay time. This study provides a specific DTM for and the average delay time is 3.27 months. the wearing parts, which will help to provide an optimal BasedonthefunctionalexpressionoftheWeibulldis- function inspection interval, lower repair costs, and establish tribution, we obtain the probability density function of the a maintenance decision model to reduce the risk of system initial failure time as failure.

0.25 𝑢 −(𝑢/10.28)1.25 Conflict of Interests 𝑔 (𝑢) = 0.12 ( ) 𝑒 . (11) 10.28 The authors declare that there is no conflict of interests regarding the publication of this paper. The cumulative distribution function of the initial failure time is Acknowledgments TheworkreceivedthesupportoftheNaturalScienceFounda- (𝑢/10.28)1.25 𝐺 (𝑢) =1−𝑒 . (12) tion of China (Grant no. 51274171) and the Graduate Student 8 Mathematical Problems in Engineering

Table 7: The parameter estimation of the DTM for the rubber core.

Model The initial time 𝑢 The delay time V

Parameter 𝛼1 𝛽1 𝛼2 𝛽2 𝑟 AIC = 2 log 𝐿+2𝑘 𝐸-𝐸 — 12.56 — 5.21 0.80 39.04 𝐸-𝑊 — 10.83 0.72 5.42 0.80 40.48 𝑊-𝑊 1.25 10.28 2.21 3.27 0.80 38.72 𝑊-𝐸 1.50 10.02 — 5.05 0.80 40.16 Notes: 𝑊: Weibull distribution, 𝐸: exponential distribution. 𝑊/𝐸 indicates that the initial time obeys a Weibull distribution, and the delay time is an exponential distribution. “—” means the parameters do not need to be estimated in the exponential distribution.

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Research Article A Decision Optimization Model for Leased Manufacturing Equipment with Warranty under Forecasting Production/Maintenance Problem

Zied Hajej,1,2 Nidhal Rezg,1,2 and Ali Gharbi3

1 LaboratoiredeGenie´ Industriel, de Production et de Maintenance, UniversitedeLorraine,57045Metz,France´ 2ICN Business School, 13 rue Michel Ney, 54000 Metz-Nancy, France 3AutomatedProductionEngineeringDepartment,ProductionSystemDesignandControlLaboratory, Eecole´ de Technologie Superieure,´ University of Quebec,1100NotreDameStreetWest,Montreal,QC,CanadaH3C1K3´

Correspondence should be addressed to Zied Hajej; [email protected]

Received 2 July 2014; Revised 24 September 2014; Accepted 24 September 2014

Academic Editor: Shaomin Wu

Copyright © 2015 Zied Hajej et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Due to the expensive production equipment, many manufacturers usually lease production equipment with a warranty period during a finite leasing horizon, rather than purchasing them. The lease contract contains the possibility of obtaining an extended warranty for a given additional cost. In this paper, based on the forecasting production/maintenance optimization problem, we develop a mathematical model to study the lease contract with basic and extended warranty based on win-win relationship between the lessee and the lessor. The influence of the production rates in the equipment degradation consequently on the total cost byeach side during the finite leasing horizon is stated in order to determine a theoretical condition under which a compromise-pricing zone exists under different possibilities of maintenance policies.

1. Introduction concept since the leasing contract contains the warranty for the maintenance service. In this case, we can consider the Duetotherapidadvancesintechnology,thetechnological warranty as a selling argument to attract and win more obsolescence appeared in the market at a quick rate with customers concerning the point of view of the lessor. From thenewandbetterequipment.Ontheotherhand,the the customers’ point of view, warranty means reducing the owning cost of this new equipment became very high. Due cost of repairs or replacement of the defective equipment to these different reasons, more and more industries started during the warranty period. leasing equipment rather than owning them. The maintenance actions in leasing contract are considered the most The maintenance of leased equipment offered by the important element negotiable between the manufacturer and equipment owner is generally quantified in a lease con- the consumer. Prior to the onset of leasing aspect, most tract provided by the lessor to the lessee (Murthy and businesses owned the equipment and the different preventive Asgharizadeh [1]). Concerning the maintenance strategy for and corrective maintenance actions were approved inter- leased equipment, some research works treated this type of nally. This began to change with the progress complexity of problem with numerous preventive maintenance policies, equipment with specialist services and their uneconomical which have been proposed and studied under various situ- maintenance actions made in-house. On the other hand, ations, such as perfect or imperfect maintenance. Yeh and most manufacturers considered the maintenance as a no Chang [2] determine the optimal number of lease periods and basic activity which is why they focus only on the principal define the maintenance strategy for leased equipment that is activities, which are considered as the basic of the business. In basedonaminimalrepairtorestoretheequipmenttoan this context, the notion of warranty is attached to the leasing operating condition when the leased equipment fails and an 2 Mathematical Problems in Engineering imperfect preventive maintenance is done to avoid failures, same maintenance service of well-known equipment. In this whentheageoftheequipmentreachesacertainthreshold context, Bouguerra et al. [9]developedamathematicalmodel value.Inthesamecontext,Jaturonnateeetal.[3]proposed to study the opportunity provided by the extended warranty amethodoffailure-ratereduction,inwhichthefailurerate for the consumer and manufacturer and proposed a strategy of equipment is reduced after each preventive maintenance of a long guarantee plan of the preventive maintenance action, to solve the optimal maintenance policy of leased for the systems subjected to the random troubleshooting. equipment under periodical preventive maintenance actions. This strategy considers diverse options for maintenance Concerning the warranty periods of the production policies during the following periods: basic guarantee period, machines, in numerous cases, due to the complexity of main- extended guarantee period, and postguarantee period. tenance actions for various machines, the consumers prefer to Wu and Longhurst [10] showed the influence of both buy a supplementary period of warranty by covering an extra the length of warranty period and replacement time on cost in order to avoid problems of production/maintenance thelifecyclecostofequipment.Theyformulatedthe cost increase and the production system perturbation. In this expected life cycle cost considering the opportunity-based context, Berke and Zaino [4] treated two types of warranties age-replacement policy with minimal repair for an extended contracts that were intended to assure the consumer that the warranty and maintenance. They also proved the conditions product would perform its planned functions under specific for the existence of optimal solutions for both the length of conditionsandperiodsoftime.Thefirsttypedefineda the extended warranty period and the design life for special combination policy that proposed an initial free-replacement cases. warranty and from a certain period the replacement item’s Recently, another type of problem that deals with leas- cost is calculated on a sliding scale. The other was the fleet ing/warranty problem is treated by Hajej et al. [11]; they warranty, which guarantees a purchaser of a large quantity of handled the optimization problem of production and main- wanted items an average field performance. Remaining with tenance policies for leased equipment under a lease con- warranty and maintenance aspect, Kim et al. [5] defined the tract with warranty periods. A mathematical model of the relation between the warranty and preventive maintenance total production and maintenance cost is developed and an by showing the impact of PM over warranty period on optimal production planning as well as the corresponding thecostofwarrantyservice.Yunetal.[6]proposedtwo optimal maintenance strategy is derived by choosing the new warranty servicing strategies, concerning imperfect and optimal warranty periods for the lessee in order to minimize minimal repairs. In the first strategy, they involve a func- the total cost. tional optimization to determine the optimal improvement MotivatedbyourworkinHajejetal.[11], we can consider in reliability, when an imperfect repair is carried over the this work is a continuation of our work in Hajej et al. [11] warranty period and depends on the age of the item. In the where we determine the most optimal basic warranty periods second strategy, they include only a parameter optimization for the lessee. This study shows that it has novelty and to determine the optimal reliability improvement that does originality relative to this type of problem, which considers not depend on the age. a mathematical model to study the opportunity provided to On the other hand, there are other types of warranties extend the warranty for the lessee. Based on a forecasting applied to nonrepairable products; we can cite the renewing production and maintenance problem for leasing machine, free-replacement warranty (RFRW) in which in the case of we will determine the total cost of leasing machine for product failure under warranty period it is replaced by a new each side in order to determine, for any given situation, one with a full warranty. Chien [7] investigated analytically zone of possible compromise yielding a win-win relationship the impacts of the RFRW on the optimal age-replacement with respect to the extended warranty cost. The zone of policy for a repairable product with a general failure model. compromise is characterized by the maximum extra cost They presented a general model that contains two types of the lessee should pay for the extended warranty and the failurewhentheproductfails.Intype1,thefailure(minor minimum price at which the lessor should sell it. Indeed, we failure) is removed by a minimal repair, whereas in type have shown the influence of production rates as well as the 2 of failure (catastrophic failure) it is removed only by a maintenance actions of the manufacturing machine over the replacement. Chien [8]presentedanewwarrantystrategy warranty and the extended warranty period on the warranty basedonanage-replacementpolicyforproducts,which servicing cost. combines a fully renewable free-replacement with a pro This study proposes a new idea of production and mainte- rata warranty policy (RFRW/RPRW policy). They developed nance coupling in the leasing aspect with warranty. This study a cost model from the user/buyer and discussed special showsthatithasnoveltyandoriginalityrelativetothistypeof cases of the model, in order to determine the corresponding problem, which considers and proposes a new maintenance local optimal replacement age by minimizing the long run strategy for leasing contract with extended warranty based on expected cost rate. win-win relationship between the lessee and the lessor. This The majority of the researches concerning the war- originality is characterised by the influence of the production ranty problems consider a fixed warranty period while the variation rates on the machine degradation degree that is dynamic warranty period or otherwise the extended warranty new in the literature charactering by analytical study that period, especially in the lease contract, helps the lessor to shows the evolution of the machine failure rate according keep contact with clients after the end of warranty period. toitsuserespectingatthesametimethecontinuityofthe Extended warranty helps the customer to continue with the equipment reliability from a period to another. Secondly, in Mathematical Problems in Engineering 3 ouropinion,noanalyticalornumericalwayhasbeenstated Basic warranty period Extended warranty period Δt in the literature, which leads to a decision framework to the ... L lessee and/or to the lessor making the identification of pricing k=0k=1k=2 ... X=a·Δ zones of the extended warranty acceptable for both sides. Xe =b·Δt The remainder part of this paper is organized as fol- Figure1:Lifecycleofleasingmachinewithwarrantyandextended lows. Section 2 states the problem. Section 3 presents and warranty periods. develops the mathematical model concerning the forecasting production problem and the different policies of maintenance considering the influence of production rates on the leasing instant 𝑋𝑒 for an additional cost 𝐶𝑋 paid by the lessee when machine degradation. Section 4 presents a numerical exam- leasing the machine. Namely, all maintenance actions during ple illustrating our approach followed by a variability study the basic and extended warranty periods are supported by showing the impact of variation of preventive maintenance thelessoratnocosttothelessee.Fortherestofthe costsonourmodel.Finally,theconclusionisgivenin leasing periods, the equipment is not under warranty and the Section 5. maintenanceactionsareundertheresponsibilityofthelessee. The maintenance actions are considered the well-known preventive maintenance policy with minimal repair at failure 2. Problem Description with negligible duration keeping the system failure rate nearly the same. The role of maintenance is to increase the 2.1. Problem Statement. In this work, we are considering the availability of machine reducing the maintenance costs in problem of forecasting production and maintenance problem order to ensure the production plan on the leasing horizon for leasing machine with warranty periods. The idea of our L. problem is to define a new aspect in the leasing contract. According to the forecasting problem as well as the Generally, several pieces of equipment are leased with a optimal production plan of leasing machine obtained, our warranty period but there are leasing contracts that propose objective is to develop a mathematical model to study the to the lessee (who leases the equipment) the possibility of opportunity provided by the extended warranty from the purchasing an additional period of warranty which will start lessee and the lessor perspectives. We will express the total at the end of the basic warranty period by adding additional expected cost incurred by each side during the product’s costs. Hence, the lessee has to decide whether to buy or life cycle in order to determine, for any given situation, not the extended warranty period and what the price of the the maximum extra cost the lessee should pay for the extended warranty is. It is a difficult decision for each side. extended warranty and the minimum price at which the The lessee does not know if the extra cost (the price ofthe lessor should sell it. Taking into account the influence of extended warranty) of the leasing equipment would exceed preventive maintenance actions performing on the leasing the potential repairs cost that would be borne by him in case machine during the basic and extended warranty periods, he does not take the extended warranty. On the other hand, we are considering different cases of maintenance strategies for the lessor not to lose, the price of extended warranty approvedduringthelifecycleofleasedmachine. should be higher than the cost of claims servicing borne (maintenance actions) by him during the additional warranty period. 3. Mathematical Model We will answer all these questions by proposing a forecasting model in whichthe lessee leases a manufacturing 3.1. Forecast Production Plan machine. The equipment is leased for a multihorizon 𝐿⋅Δ𝑡 (we (i) Stochastic Production Model. Basedontheapproach assumed that the production horizon is portioned equally to proposed by Zied et al. [12]andHajejetal.[11], the periods with a length equal to Δ𝑡 ) with a warranty period 𝑋⋅ production planning problem is formulated as a quadratic Δ𝑡. We suppose that leasing production machine is designed model whose decision variables include production rates and in order to produce only one type of product in a manufactur- inventory levels. The purpose of this section is to develop ing system composed also by a manufacturing store, where a mathematical model that will allow us to determine the the customer receives his demand over the leasing finite ∗ ∗ ∗ optimal production plan 𝑈 (𝑈 = 𝑢(𝑘) and 𝑘=1,...,𝐿−1) horizon L. Moreover, for the forecasting problem, we assume during the leasing horizon 𝐿. that the satisfaction of the demand is under a given inventory Formally, the stochastic production model is defined as service level 𝛼 and the fluctuation of the demand is a normal ̂ follows: distribution with mean and variance given, respectively, by 𝑑 and 𝜎𝑑 (Figure 1). 𝐿 2 The considered leased machine is subject to the random Minimize 𝑍=∑𝑓𝑘 (𝑈𝑘,𝑆𝑘)=𝐶𝑠 ⋅𝐸{𝑆𝐿} failures. Its failure rate 𝜆(𝑡) increases with both time and 𝑘=0 (1) production rate. An influence of the production rate variation 𝐿 on the equipment degradation and hence on the average + ∑𝐶 ⋅𝐸{𝑆2}+𝐶 ⋅𝑈2 𝑠 𝑘 pr 𝑘 number of failures is considered. 𝑘=0 The leasing contract includes the machine under warranty period X with the possibility of being extended until subject to 4 Mathematical Problems in Engineering

(i) inventory balance equation constraints 3.2. Maintenance Policy. Based on the work of Wu and Longhurst [10], the maintenance strategy considers the man- 𝑆𝑘+1 =𝑆𝑘 +𝑈𝑘 −𝑑𝑘 𝑘∈{0,1,...,𝐿−1} ; (2) ufacturing system’s degradation according to the production rate during the leasing horizon L. The correlation of the degradation of the machine production rates is manifested (ii) service level requirement for each period by an increased failure rate according to both time and production rate. [𝑆 ≥0] ≥𝛼 𝑘∈{0,1,...,𝐿−1} ; Prob 𝑘+1 (3) We assume that, during the machine life cycle, perfect preventive maintenance or replacement is performed peri- 𝑖⋅𝑇 𝑖 = 0,1,...,𝑁 𝑁 (iii) capacity constraints odically at times , 𝑗 (with 𝑗 number of preventive maintenance over each interval during the leasing max periods: basic warranty, extended warranty, and postwarranty 0≤𝑈𝑘 ≤𝑈 𝑘∈{0,1,...,𝐿−1} . (4) and 𝑇 preventive maintenance action interval) following which the unit is as good as new. (ii) Deterministic Production Model. An approach that trans- The evolution of the machine failure rate according to forms the stochastic problem into a deterministic equivalent its use (which is in our case the production rate for each is necessary. This deterministic problem maintains the main period) respecting at the same time the continuity of the properties of the original problem. equipment reliability from a period to another is presented The quadratic total expected cost of production and by an analytical equation. inventory over the leasing periods 𝐿 can be expressed then The failure rate in the interval k is expressed as follows: as follows: 𝑈𝑘 𝜆𝑘 (𝑡) =𝜆𝑘−1 (Δ𝑡) + ⋅𝜆𝑛 (𝑡) ∀𝑡 ∈ [0, Δ𝑡] (9) 𝐿−1 𝑈max 𝑍 (𝑢) =𝐶 ×(𝑆̂2 )+∑𝐶 ⋅ 𝑆̂2 +𝐶 ×𝑢2 𝑠 𝐿 𝑠 𝑘 pr 𝑘 𝑘=0 with (5) 𝑈 𝐿 (𝐿+1) 𝜆 =𝜆 ,Δ𝜆(𝑡) = 𝑘 ⋅𝜆 (𝑡) , +𝐶 ×𝜎2 × 𝑘=0 0 𝑘 𝑈 𝑛 (10) 𝑠 𝑑 2 max

where 𝜆𝑛(𝑡) is the nominal failure rate corresponding to the with the following. maximal production rate. Allowing maintenance strategy, we can define the differ- (i) It has mean variables ent numbers of preventive maintenance over each interval ̂ during the leasing periods given by the following: 𝐸{𝑆𝑘}=𝑆𝑘,𝐸{𝑢𝑘}=𝑢𝑘 (6) 𝑁1: number of PM actions during the basic warranty 𝑉 =0 𝑢 periods [0, X)withavalueequaltoIn(𝑋/𝑇); and variance variables 𝑢𝑘 .(Variable 𝑘 is deterministic.) 𝑁2: number of PM actions during the leasing periods [0, 𝐿)withavalueequaltoIn(𝐿/𝑇); (ii) The inventory balance (2) can be reformulated as 𝑁3: number of PM actions between the end of basic ̂ ̂ ̂ warranty and the end of the leasing periods [𝑋, 𝐿) 𝑆𝑘+1 = 𝑆𝑘 +𝑢𝑘 − 𝑑𝑘 𝑘=0,1,...,𝐿−1. (7) with a value equal to In((𝐿 − 𝑋)/𝑇);

𝑁4: number of PM actions during the basic and Proof. See the Appendix. extended warranty periods [0, 𝑋𝑒)withavalueequal to In(𝑋𝑒/𝑇);

(iii) Service Level Constraint. As another step to transform 𝑁5: number of PM actions during the extended the stochastic problem into an equivalent deterministic one, warranty [𝑋,𝑒 𝑋 ) with a value equal to In((𝑋𝑒 −𝑋)/𝑇); we consider a service level constraint in a deterministic form 𝑁 by determining a minimum cumulative production quantity 6: number of PM actions between the end of depending on the service level requirements. extended warranty periods and the end of leasing periods [𝑋𝑒,𝐿)with a value equal to In((𝐿 −𝑒 𝑋 )/𝑇), For 𝑘∈{0,1,...,ℎ𝑖 −1}we have with In: integer part of a real number. −1 ̂ ̂ Prob (𝑆𝑘+1 ≥0)≥𝛼󳨐⇒(𝑈𝑘 ≥𝑉𝑑,𝑘 ⋅𝜑𝑑,𝑘 (𝛼) + 𝑑𝑘 − 𝑆𝑘), (8) We express below the analytic expression of the total maintenance cost incurred by each side during the leasing 𝜑 (𝑈, 𝑁 ) where 𝜑𝑑,𝑘: cumulative Gaussian distribution function with period of machine where 𝑀 𝑖 corresponds to the ̂ expected number of failures that occur during the different mean 𝑑𝑘 and finite variance Var(𝑑𝑘)=𝑉𝑑,𝑘 ≥0and 𝑉𝑑,𝑘: variance of demand 𝑑 at period 𝑘. intervals defined above, considering the production rate in each production period Δ𝑡: 𝜉 (𝑈, 𝑁 ) =𝐶 × (𝑁 −1) +𝐶 ×𝜑 (𝑈, 𝑁 ) . Proof. See the Appendix. 𝑖 pm 𝑖 cm 𝑀 𝑖 (11) Mathematical Problems in Engineering 5

X=a·Δt Let In denote the integer part of (⋅). Then the average number Xe =b·Δt according to failure rate defined above is L 𝜑𝑀 (𝑈,𝑖 𝑁 ) ... 0 T 2·TN5 ·T 0 T 𝑁 −1 In((𝑗+1)×(𝑇/Δ𝑡)) 𝑖 Δ𝑡 N5 ·T N6 ·T = ∑ [ ∑ ∫ 𝜆 (𝑡) 𝑖 N N6 PM actions 0 5 PM actions 𝑗=0 [𝑖=In(𝑗×(𝑇/Δ𝑡))+1 supported by the supported by the lessor lessee (𝑗+1)×𝑇−In((𝑗+1)×(𝑇/Δ𝑡))×Δ𝑡 + ∫ 𝜆In((𝑗+1)×(𝑇/Δ𝑡))+1 (𝑡) 𝑑𝑡 0 (12) Figure 2: Evolution of failure rate for Policy I-1. (In((𝑗+1)×(𝑇/Δ𝑡))+1)×Δ𝑡 (( ((𝑗+1)×(𝑇/Δ𝑡))+1)) + ∫ In 𝑈 B C (𝑗+1)×𝑇 max X=a·Δt

] ×𝜆𝑛 (𝑡) 𝑑𝑡 . L ] ... 0 T 2·T N3T Using the total cost, we can determine the zone of possible N3 ·T A N compromise yielding a win-win relationship between the 3 PM actions supported by the lessee lessor and the lessee characterized by the maximum addi- Figure 3: Average number of failures for the case in which he does tional cost for the lessee who should pay for the extended not take the extended warranty period. warranty and the minimum price at which the lessor should sell it. There are different cases of maintenance strategy adopted during the leasing horizon taking into account the impact of preventive maintenance on the warranty servicing and the case in which he takes it is necessary to determine the cost. cost of the extended warranty period paid by the lessee. In this The following maintenance policies will be considered for case, side of lessee, we state for each maintenance policy that lessor and lessee sides. the best situation for buying the extended warranty period would be least cost for the lessee. This is the best situation (i) Policy I. Periodic PM actions during the post–basic obtained, where the total maintenance cost incurred to him warranty period: for this policy we consider the in the case of purchase of extended warranty would be lower following possibility. than in the case he does not take it. We assume that 𝜉𝑐𝑃𝑛 and 𝜉𝑐𝑃𝑦 are the total maintenance (a) Policy I-1. PM actions are performed during the costs acquired to the lessee for maintenance policy (𝑃), extended warranty [𝑋, 𝑋𝑒)attimes𝑖⋅𝑇, 𝑖= respectively,forthecaseinwhichhedoesnottakethe 0,1,...,𝑁5,supportedbythelessorandPM extended warranty period (𝑛) and the case in which he takes actions are performed from the end of extended it (𝑦). warranty [𝑋𝑒,𝐿)at times 𝑖⋅𝑇, 𝑖 = 0,1,...,𝑁6, We recall that 𝑁1 =𝑋/𝑇; 𝑁3 =(𝐿−𝑋)/𝑇; 𝑁5 =(𝑋𝑒 − supported by the lessee (Figure 2). 𝑋)/𝑇; 𝑁6 =(𝐿−𝑋𝑒)/𝑇; 𝑋=𝑎⋅Δ𝑡; 𝑋𝑒 =𝑎1 ⋅Δ𝑡; Δ𝑡is length of production period. (ii) Policy II. Periodic PM actions during the warranty period: for this case we consider two different possi- (i) Policy I (see Figures 3 and 4). bilities. (a) Policy I-1. Consider (a) Policy II-1. PM actions are performed only [0, 𝑋) during the basic warranty period at times 𝜉𝑐𝐼−1𝑦 +𝜉𝑐𝑋 ≤𝜉𝑐𝐼−1𝑛 𝑖⋅𝑇, 𝑖=0,1,...,𝑁1,supportedbythelessor. (13) (b) Policy II-2. PM actions are performed during 󳨐⇒ 𝜉 𝑐𝑋 ≤𝜉𝑐𝐼−1𝑛 −𝜉𝑐𝐼−1𝑦 󳨐⇒ 𝜉 𝑐𝑋 ≤𝐴𝑐𝐼, both the basic and the extended warranty periods [0, 𝑋𝑒) at times 𝑖⋅𝑇, 𝑖 = 0,1,...,𝑁4, where supported by the lessor. 𝜉𝑐𝐼−1𝑛

In(𝑋/Δ𝑡) Δ𝑡 3.3. Maximum Additional Cost Paid for Extended Warranty: =𝐶 ×[ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 Lessee Side. The subsection determines the maximum addi- 𝑖=1 0 tional cost that the lessee should pay for the extended 𝑁 −1 In((𝑗+1)⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡) warranty during the leasing periods. The comparison of the 3 Δ𝑡 total maintenance costs acquired to the lessee between the + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 case in which he does not take the extended warranty period 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 6 Mathematical Problems in Engineering

X=a·Δt X =b·Δt In(𝐿/Δ𝑡) Δ𝑡 e ] D − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁6⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 ] +𝐶 × (𝑁 −𝑁) . ... pm 3 6 0 T 2·T N5T 0 T L (14) N ·T N ·T A 5 6 B C

Figure 4: Average number of failures for the case in which he takes (ii) Policy II. the extended warranty period.

(a) Policy II-1. Consider (𝐿/Δ𝑡) Δ𝑡 ] + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝜉𝑐𝐼𝐼−1𝑦 +𝜉𝑐𝑤 ≤𝜉𝑐𝐼𝐼−1𝑛 𝑖=In(𝑁3⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 ] 󳨀→ 𝜉 ≤𝜉 −𝜉 󳨀→ 𝜉 ≤𝐵 +𝑁 ×𝐶 , 𝑐𝑋 𝑐𝐼𝐼−1𝑛 𝑐𝐼𝐼−1𝑦 𝑐𝑋 𝑐1 3 pm 𝜉𝑐𝐼𝐼−1𝑛 𝜉𝑐𝐼−1𝑦 In(𝑋/Δ𝑡) Δ𝑡 =𝐶 × [ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 In(𝑋/Δ𝑡) Δ𝑡 cm 𝑖 0 =𝐶 × [ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 [𝑖=In(𝑁1⋅(𝑇/Δ𝑡))+1 cm 𝑖 𝑖=1 0 [ In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋/Δ𝑡 𝑁3−1 (( ) ( )) ( ) Δ𝑡 + ∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 In 𝑋 /Δ𝑡 𝑖 ( 𝑒 ) Δ𝑡 0 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁 ⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 5 In(𝐿/Δ𝑡) Δ𝑡 ] + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋 /Δ𝑡 𝑁6−1 (( ) ( )) ( 𝑒 ) Δ𝑡 0 𝑖=In(𝑁3⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 ] + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑗=0 𝑖=In 𝑗⋅(𝑇/Δ𝑡) + 𝑋 /Δ𝑡 +1 +𝑁 ×𝐶 , ( ) ( 𝑒 ) 3 pm 𝜉 (𝐿/Δ𝑡) Δ𝑡 𝑐𝐼𝐼−1𝑦 ] + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 In(𝑋/Δ𝑡) Δ𝑡 𝑖=In(𝑁 ⋅(𝑇/Δ𝑡))+(𝑋 /Δ𝑡)+1 6 𝑒 ] =𝐶 × [ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 0 𝑖=In 𝑁 ⋅(𝑇/Δ𝑡) +1 +𝑁 ×𝐶 , [ ( 1 ) 6 pm

(𝑋𝑒/Δ𝑡) Δ𝑡 𝐴𝑐𝐼 + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑖=(𝑤/Δ𝑡) 0 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋/Δ𝑡 𝑁3−1 (( ) ( )) ( ) Δ𝑡 [ In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋 /Δ𝑡 =𝐶 × ∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 𝑁6−1 (( ) ( )) ( 𝑒 ) Δ𝑡 cm 𝑖 0 [ 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑗=0 0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 In(𝐿/Δ𝑡) Δ𝑡 In(𝐿/Δ𝑡) + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 Δ𝑡 0 ] 𝑖=In(𝑁 ⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 3 0 𝑖=In(𝑁6⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 ] In(𝑋 /Δ𝑡) 𝑒 Δ𝑡 +𝑁 ×𝐶 , 6 pm − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁 ⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 5 𝐵𝑐1

In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋 /Δ𝑡 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋/Δ𝑡 𝑁6−1 (( ) ( )) ( 𝑒 ) Δ𝑡 𝑁3−1 (( ) ( )) ( ) Δ𝑡 − ∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 =𝐶 × [ ∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 𝑖 cm 𝑖 𝑗=0 0 𝑗=0 0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 [ 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 Mathematical Problems in Engineering 7

In(𝐿/Δ𝑡) Δ𝑡 𝐵𝑐2 + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 In(𝑋/Δ𝑡) 𝑖=In(𝑁3⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 Δ𝑡 =𝐶 × [ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 In(𝑋 /Δ𝑡) 0 𝑒 Δ𝑡 [𝑖=In(𝑁1⋅(𝑇/Δ𝑡))+1 − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋/Δ𝑡 𝑖=In(𝑋/Δ𝑡) 𝑁3−1 (( ) ( )) ( ) Δ𝑡 + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋 /Δ𝑡 𝑗=0 0 𝑁6−1 (( ) ( )) ( 𝑒 ) Δ𝑡 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 − ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑗=0 0 In(𝐿/Δ𝑡) Δ𝑡 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁3⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 In(𝐿/Δ𝑡) Δ𝑡 − ∑ ∫ 𝜆 (𝑡) 𝑑𝑡] 𝑖 In(𝑋 /Δ𝑡) 0 𝑒 Δ𝑡 𝑖=In(𝑁6⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 ] − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁 ⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 +𝐶 ×(𝑁 −𝑁). 5 pm 3 6 𝑁 −1 In((𝑗+1)⋅(𝑇/Δ𝑡))+(𝑋 /Δ𝑡) (15) 6 𝑒 Δ𝑡 − ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑗=0 0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 (b) Policy II-2. Consider In(𝐿/Δ𝑡) Δ𝑡 ] 𝜉 +𝜉 ≤𝜉 − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑐𝐼𝐼−2𝑦 𝑐𝑋 𝑐𝐼𝐼−2𝑛 0 𝑖=In(𝑁6⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 ] 󳨀→ 𝜉 ≤𝜉 −𝜉 󳨀→ 𝜉 ≤𝐵 𝑐𝑋 𝑐𝐼𝐼−2𝑛 𝑐𝐼𝐼−2𝑦 𝑐𝑋 𝑐2 +𝐶 ×(𝑁 −𝑁). pm 3 6

𝜉𝑐𝐼𝐼−2𝑛 (16)

In(𝑋/Δ𝑡) Δ𝑡 3.4. Minimum Price for Selling the Extended Warranty: Lessor =𝐶 × [ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 Side. The same as lessee side, this subsection determines the 0 [𝑖=In(𝑁1⋅(𝑇/Δ𝑡))+1 minimum price at which the lessor can sell the extended warranty during the leasing periods. We establish, for each In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋/Δ𝑡 maintenance policy, the best situation so that selling the 𝑁3−1 (( ) ( )) ( ) Δ𝑡 extended warranty period would be winning for the lessor. + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 This best situation is acquired where the total maintenance cost incurred to him during the leasing horizon in the case of selling the extended warranty would be less than what it In(𝐿/Δ𝑡) Δ𝑡 ] would cost him in the case in which he does not sell it. + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 We assume that 𝜉𝑀𝑃𝑛 and 𝜉𝑀𝑃𝑦 are the total maintenance 𝑖=In(𝑁 ⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 3 ] costs acquired to the lessor for maintenance policy (𝑃), +𝑁 ×𝐶 , respectively, for the case without the extended warranty 3 pm period (𝑛) and the case with the extended warranty period (𝑦). 𝜉𝑐𝐼𝐼−2𝑦 (i) Policy I.

In 𝑋 /Δ𝑡 ( 𝑒 ) Δ𝑡 (a) Policy I-1. Consider =𝐶 × [ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 0 𝜉𝑀𝐼−1𝑦 −𝜉𝑀𝑋 ≤𝜉𝑀𝐼−1𝑛 [𝑖=In(𝑁5⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1

󳨀→ 𝜉 𝑀𝑋 ≥𝜉𝑀𝐼−1𝑦 −𝜉𝑀𝐼−1𝑛 󳨀→ 𝜉 𝑀𝑋 ≥𝐴𝑀1, In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋 /Δ𝑡 𝑁6−1 (( ) ( )) ( 𝑒 ) Δ𝑡 In(𝑋/Δ𝑡) + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 Δ𝑡 𝑗=0 0 𝜉 =𝐶 ×[ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡] , 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 𝑀𝐼−1𝑛 cm 𝑖 𝑖=1 0

In(𝐿/Δ𝑡) Δ𝑡 𝜉 ] 𝑀𝐼−1𝑦 + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁6⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 ] In(𝑋/Δ𝑡) Δ𝑡 =𝐶 × [ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 +𝑁 ×𝐶 , 𝑖=1 0 6 pm [ 8 Mathematical Problems in Engineering

In 𝑗+1 ⋅(𝑇/Δ𝑡) +(𝑋/Δ𝑡) 𝑁5−1 (( ) ) Δ𝑡 𝜉 ] 𝑀𝐼𝐼−2𝑛 + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑁 In((𝑗+1)⋅(𝑇/Δ𝑡)) 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 1 Δ𝑡 ] =𝐶 × [∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 0 +𝐶 ×𝑁, 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+1 pm 5 [ 𝐴 =𝐶 In(𝑋/Δ𝑡) Δ𝑡 𝑀1 cm ] + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋/Δ𝑡 𝑖=In 𝑁 ⋅(𝑇/Δ𝑡) +1 𝑁5−1 (( ) ( )) ( ) Δ𝑡 ( 1 ) ] [ ] × ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 +𝐶 ×𝑁, 0 pm 1 [ 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 ] 𝜉𝑀𝐼𝐼−2𝑦 +𝐶 ×𝑁5. pm In 𝑗+1 ⋅(𝑇/Δ𝑡) 𝑁4 (( ) ) Δ𝑡 (17) =𝐶 × [∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 0 [𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+1

(ii) Policy II. In(𝑋𝑒/Δ𝑡) Δ𝑡 ] + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁4⋅(𝑇/Δ𝑡))+1 ] (a) Policy II-1. Consider +𝐶 ×𝑁, pm 4 𝜉 −𝜉 ≤𝜉 𝑀𝐼𝐼−1𝑦 𝑀𝑋 𝑀𝐼𝐼−1𝑛 𝐵𝑀2

In 𝑗+1 ⋅ 𝑇/Δ𝑡 󳨀→ 𝜉 𝑀𝑋 ≥𝜉𝑀𝐼𝐼−1𝑦 −𝜉𝑀𝐼𝐼−1𝑛 󳨀→ 𝜉 𝑀𝑋 ≥𝐵𝑀1, 𝑁4 (( ) ( )) Δ𝑡 =𝐶 × [∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 𝜉 0 𝑀𝐼𝐼−1𝑛 [𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+1 In 𝑗+1 ⋅(𝑇/Δ𝑡) In(𝑋 /Δ𝑡) 𝑁1 (( ) ) Δ𝑡 𝑒 Δ𝑡 [ + ∑ ∫ 𝜆 𝑡 𝑑𝑡 =𝐶 × ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑖 ( ) cm 0 𝑗=0 0 𝑖=In 𝑁 ⋅(𝑇/Δ𝑡) +1 [ 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+1 ( 4 ) In 𝑗+1 ⋅(𝑇/Δ𝑡) 𝑁1 (( ) ) Δ𝑡 In(𝑋/Δ𝑡) Δ𝑡 + ∑ ∫ 𝜆 (𝑡) 𝑑𝑡] − ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑖 0 0 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+1 𝑖=In(𝑁1⋅(𝑇/Δ𝑡))+1 ] +𝐶 ×𝑁, In(𝑋/Δ𝑡) Δ𝑡 pm 1 ] − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝜉𝑀𝐼𝐼−1𝑦 (18) 𝑖=In(𝑁1⋅(𝑇/Δ𝑡))+1 ]

In 𝑗+1 ⋅(𝑇/Δ𝑡) +𝐶 ×(𝑁 −𝑁). 𝑁1 (( ) ) Δ𝑡 pm 4 1 =𝐶 × [∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 cm 𝑖 0 (19) [𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+1

In(𝑋/Δ𝑡) Δ𝑡 3.5. Win-Win Interval for the Extended Warranty Cost. The + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 existence of win-win interval of the extended warranty cost is 0 determined according to the previous subsections, where we 𝑖=In(𝑁1⋅(𝑇/Δ𝑡))+1 have determined, for each maintenance policy, the maximum

In(𝑋𝑒/Δ𝑡) Δ𝑡 additional cost that the lessee must pay and the minimum + ∑ ∫ 𝜆 (𝑡) 𝑑𝑡] +𝐶 ×𝑁, price at which the lessor can sell the extended warranty. We 𝑖 pm 1 𝑖=In(𝑋/Δ𝑡) 0 ] determine a theoretical sufficient condition under which a win-win interval will exist and with which the maximum In(𝑋 /Δ𝑡) 𝑒 Δ𝑡 buying additional cost for the lessee is greater than the 𝐵 =𝐶 ×[ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡] . 𝑀1 cm 𝑖 minimumsellingpriceforthelessor. 𝑖=In(𝑋/Δ𝑡) 0 (i) Policy I-1. Using (14) and (17),thecompromiseintervalfor the extended warranty cost exists if (b) Policy II-2. Consider 𝐴 ≤𝐴 󳨐⇒ 𝐶 𝑀1 𝑐𝐼 cm

𝜉𝑀𝐼𝐼−2𝑦 −𝜉𝑀𝑋 ≤𝜉𝑀𝐼𝐼−2𝑛 In(𝑋/Δ𝑡) Δ𝑡 ×[ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 󳨀→ 𝜉 𝑀𝑋 ≥𝜉𝑀𝐼𝐼−2𝑦 −𝜉𝑀𝐼𝐼−2𝑛 󳨀→ 𝜉 𝑀𝑤 ≥𝐵𝑀2, 𝑖=1 0 Mathematical Problems in Engineering 9

𝑁 −1 In((𝑗+1)⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡) 3 Δ𝑡 (iii) Policy II-2. Using (16) and (19),thecompromiseinterval + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 for the extended warranty cost exists if 0 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1

In(𝐿/Δ𝑡) 𝐵 ≥𝐵 󳨐⇒ 𝐶 Δ𝑡 𝑐2 𝑀2 cm + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 In(𝑋/Δ𝑡) 𝑖=In(𝑁 ⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 Δ𝑡 3 [ × ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 [𝑖=In(𝑁1⋅(𝑇/Δ𝑡))+1 In(𝑋𝑒/Δ𝑡) Δ𝑡 − ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 𝑖 In 𝑗+1 ⋅(𝑇/Δ𝑡) +(𝑋/Δ𝑡) 0 𝑁3−1 (( ) ) Δ𝑡 𝑖=In(𝑁5⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋 /Δ𝑡 𝑁6−1 (( ) ( )) ( 𝑒 ) Δ𝑡 − ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 In(𝐿/Δ𝑡) Δ𝑡 𝑗=0 0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁3⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 In(𝐿/Δ𝑡) Δ𝑡 In 𝑋 /Δ𝑡 − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 ( 𝑒 ) Δ𝑡 0 − ∑ ∫ 𝜆 𝑡 𝑑𝑡 𝑖=In(𝑁6⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 𝑖 ( ) 0 𝑖=In(𝑁5⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋/Δ𝑡 𝑁5−1 (( ) ( )) ( ) Δ𝑡 In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋 /Δ𝑡 ] 𝑁6−1 (( ) ( )) ( 𝑒 ) Δ𝑡 − ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 − ∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 ] 𝑖 𝑗=0 0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 +𝐶 ×(𝑁 −𝑁 −𝑁)≥0. (22) pm 3 6 5 In(𝐿/Δ𝑡) Δ𝑡 (20) − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 𝑖=In(𝑁6⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1

In 𝑗+1 ⋅ 𝑇/Δ𝑡 𝑁4 (( ) ( )) Δ𝑡 (ii) Policy II-1. Using (15) and (18),thecompromiseinterval − ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 0 for the extended warranty cost exists if 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+1

In(𝑋𝑒/Δ𝑡) Δ𝑡 − ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 𝐵 ≥𝐵 󳨐⇒ 𝐶 𝑖 𝑐1 𝑀1 cm 0 𝑖=In(𝑁4⋅(𝑇/Δ𝑡))+1

In 𝑗+1 ⋅(𝑇/Δ𝑡) +(𝑋/Δ𝑡) 𝑁3−1 (( ) ) Δ𝑡 In 𝑗+1 ⋅(𝑇/Δ𝑡) 𝑁1 (( ) ) Δ𝑡 × [ ∑ ∑ ∫ 𝜆 (𝑡) 𝑑𝑡 𝑖 + ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑗=0 0 0 [ 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 𝑗=0 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+1

In 𝐿/Δ𝑡 ( ) Δ𝑡 In(𝑋/Δ𝑡) Δ𝑡 + ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 + ∑ ∫ 𝜆 (𝑡) 𝑑𝑡] 0 𝑖 𝑖=In(𝑁 ⋅(𝑇/Δ𝑡))+(𝑋/Δ𝑡)+1 0 3 𝑖=In(𝑁1⋅(𝑇/Δ𝑡))+1 ]

In(𝑋𝑒/Δ𝑡) Δ𝑡 +𝐶 ×(𝑁 −𝑁 −𝑁 +𝑁)>0. pm 3 6 4 1 −2× ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 𝑖=In(𝑋/Δ𝑡) 0 4. Numerical Example In 𝑗+1 ⋅ 𝑇/Δ𝑡 + 𝑋 /Δ𝑡 𝑁6−1 (( ) ( )) ( 𝑒 ) Δ𝑡 − ∑ ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 In order to illustrate the model developed previously, we 𝑗=0 0 consider a forecasting production/maintenance problem for 𝑖=In(𝑗⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 a company represented by a leasing machine which has to In(𝐿/Δ𝑡) Δ𝑡 satisfy a stochastic demand assumed Gaussian, under service ] 𝐿 − ∑ ∫ 𝜆𝑖 (𝑡) 𝑑𝑡 level, over a finite leasing horizon. The number of leasing 0 Δ𝑡 Δ𝑡 𝑖=In(𝑁6⋅(𝑇/Δ𝑡))+(𝑋𝑒/Δ𝑡)+1 ] periods is equal to 24, with =1um.Theleasingmachine has a degradation law characterized by a Weibull distribution 𝛼 𝛽 +𝐶 ×(𝑁3 −𝑁6)>0. with shape parameter and scale parameter (with these two pm parameters, the degradation is linear 𝛾=2). From the failure (21) rate equation, we determined the average number of failures 10 Mathematical Problems in Engineering

Table 1 [0, 𝑋𝑒)), the optimal preventive maintenance interval for ∗ lessor is equal to 𝑇𝑀 =3andforthelesseeisequalto 𝑑1 𝑑2 𝑑3 𝑑4 𝑑5 ∗ 𝑇 =2, and the win-win interval for the extended warranty 15 17 15 15 15 𝐶 cost existed between 202.25 and 203.476 mn units (Figure 5) 𝑑 𝑑 𝑑 𝑑 𝑑 6 7 8 9 10 which characterizes the threshold values, respectively, for 14 16 14 16 13 the lessor and the lessee. In this case, the best compromise 𝑑11 𝑑12 𝑑13 𝑑14 𝑑15 corresponds to the middle of this interval with an extended 15 14 15 12 15 warranty cost of 202.863 mn units.

𝑑16 𝑑17 𝑑18 𝑑19 𝑑20 In fact, from the lessor side, as preventive maintenance 13 15 11 16 13 actions become more efficient, the average number of failures gets lower. Consequently, he would pay less for minimal 𝑑21 𝑑22 𝑑23 𝑑24 𝑑25 repairs and therefore his threshold value for the extended 15 12 14 16 14 warranty cost becomes lower. From the lessor side, taking the extended warranty will result in having the leasing machine Table 2 entering the postwarranty period with a higher reliability, ∗ ∗ ∗ ∗ ∗ 𝑢 (1) 𝑢 (2) 𝑢 (3) 𝑢 (4) 𝑢 (5) thanks to preventive maintenance actions performed during [𝑋, 𝑋 ] 9 14 8 12 12 𝑒 .Consequently,thelesseeiswillingtopaymoreforthe ∗ ∗ ∗ ∗ ∗ 𝑢 (6) 𝑢 (7) 𝑢 (8) 𝑢 (9) 𝑢 (10) extended warranty whereas preventive maintenance becomes more efficient giving a higher reliability and minimal average 15 9 13 14 11 number of failures and hence least minimal repairs during the 𝑢∗ 𝑢∗ 𝑢∗ 𝑢∗ 𝑢∗ (11) (12) (13) (14) (15) postwarranty period. 10 5 11 12 5 As another example, for Policy II-1, since the period ∗ ∗ ∗ ∗ ∗ 𝑢 (16) 𝑢 (17) 𝑢 (18) 𝑢 (19) 𝑢 (20) during which PM is performed is related only to the basic 15 16 12 10 6 warranty period, the win-win interval for the extended ∗ ∗ ∗ ∗ ∗ 𝑢 (21) 𝑢 (22) 𝑢 (23) 𝑢 (24) 𝑢 (25) warranty cost is found between 3.41471 and 200.238; since 2517314there is no preventive maintenance for lessor side, the expected number of minimal repairs during this period remains obviously the same with or without the extended assuming that after each preventive maintenance action the warranty whereas on the lessee side the optimal preventive 𝑇∗ =3 equipment is in state “as good as new.” maintenance interval is equal to c . The following arbitrarily chosen input data are also Also, from Figure 6, we can notice, for Policy I-1, that considered: choosing an extended warranty period would be interesting neither for the lessee nor for the lessor. The win-win interval =3 =10 𝛼 = 0.95 Cpr1 mu, Cpr2 mu, service level , does not exist because the minimum price at which the lessor 𝐶 =5 𝑆 =20 𝑠 mu, initial inventory 0 ,thevarianceof should sell the extended warranty (equal to 202.347) is greater 𝑉𝑑 = 1.21 𝑋=2 𝑋𝑒 =6 𝐶 = 1500 demand 𝑘 , , , cm ,and than the maximum additional cost that the lessee should pay 𝐶 = 200 pm . for the extended warranty (equal to 194,706). The extended Tocomputethefailurerate,weassumethatthenominal warranty would not be as advantageous for the lessor as for degradation follows a Weibull distribution given by the lessee due to the fact that since there is no preventive maintenance, the average number of failures during [𝑋,𝑒 𝑋 ] 𝛾 𝑡 𝛾−1 remains obviously the same with or without the extended 𝜆 (𝑡) = ⋅( ) . (23) 𝑛 𝛽 𝛽 warranty.

The average of forecasting demand is presented in Table 1. For 𝐶𝑝𝑚 =200and𝐶𝑐𝑚 =1500.For compromise intervals for Applying our analytical model, we used the numerical the extended warranty cost see Table 3. algorithms for constrained global optimization with MATH- EMATICA, in order to realize this optimization. Firstly, we Variation of PM and CM Costs. We identify the impact 𝐶 areinterestedtofindtheforecastingoptimalproductionplan, of varying the maintenance preventive cost ( pm)andthe 𝐶 which is presented in Table 2. According to the production maintenance corrective cost ( cm) during the leasing hori- 𝐶 𝐶 plan obtained, we have observed, for each maintenance zon. Besides the nominal values ( pm =200, cm =1500), ∗ (𝐶 ∈ {300, 400, 600}; 𝐶 ∈ policy, the optimal preventive maintenance interval 𝑇 and we consider a higher value pm cm {1700, 1900, 2000}) the existence of a win-win interval where lower and upper . Obviously, the effect of varying the main- boundaries are, respectively, the minimum price at which the tenance costs can be observed for different policies. Certainly, lessor should sell the extended warranty and the maximum for these different policies, we note that the period over which additional cost that the lessee should pay for the extended preventive maintenance is performed has a direct impact on warranty. theaveragenumberoffailuresandonthenumberofminimal 𝑋, 𝑋 The forecasting production plan is presented in Table 2. repairs during the extended warranty period [ 𝑒]. From Table 3, we canunderstand, for example, that, We can look at Tables 4 and 5 that for Policy I-1 (preven- for Policy II-2 (preventive maintenance performed during tive maintenance actions are performed during [𝑋𝑒,𝐿)the Mathematical Problems in Engineering 11

Table 3

𝑇 Policy I-1 Policy II-1 Policy II-2 Lessor Lessee Lessor Lessee Lessor Lessee 1 803.724 804.288 3.41471 800.309 803.194 803.194 2 403.72 404.288 3.41471 400.309 403.194 203.476 3 202.347 0 194.706 3.41471 200.238 202.25 403.759 4 203.724 204.288 3.41471 200.309 203.759 403.194

Lessor (Policy II-2) Lessee (Policy II-2)

𝜉mX 𝜉CX T T 7 7 6 6 5 5 203.476 4 202.25 4 3 3 2 2 ∗ Tc 1 ∗ 1 TM

Extended warranty for the lessor

Extended warranty for the lessee 𝜉 202.25 202.863 203.476 X

Figure 5: Win-win interval for the extended warranty for option II-2.

𝐶 𝐶 for any values of pm and cm;sincethereisnopreventive Extended warranty for the lessee Extended warranty for the lessor maintenance for the lessor side, the expected number of 𝜉 194.706 202.347 X minimal repairs during this period remains obviously the same with or without the extended warranty. Figure 6: Absence of a win-win interval for the extended warranty 𝐶 cost for Policy I-1. 𝑐𝑚 =1500.See Table 4.

𝐶𝑝𝑚 =200.See Table 5.

Δ𝑡 preventive maintenance interval is increased if the preventive Effects of the Variation of Production Period Length . In this and corrective costs are increased but it is beneficial for section, we investigate the effects of varying the periodicity Δ𝑡 neither the lessee nor the lessor to adopt the extended length ( ) of production during the product’s life cycle. Δ𝑡 warranty period whatever the maintenance costs are. This can Beside the nominal value ( = 1 um), we consider a higher Δ𝑡 be explained by the fact that, at the end of warranty period value ( =2). 𝑋𝑒,itwouldbetoolatetostartpreventivemaintenance Obviously, the effect of varying production period can action and the degradation of leasing machine increased only be observed for Policy II-1 and Policy II-2. The optimal and the preventive maintenance action cannot improve the preventive maintenance interval for lessor is decreased for ∗ ∗ reliability of leasing machine even if they are performed more production period Δ𝑡 = 2 (Policy II-1: 𝑇𝑀 =2and 𝑇𝐶 =2) frequently during [𝑋𝑒,𝐿]. (Table 6)withahighercostrelativetoΔ𝑡 = 1 (Policy II-1: ∗ ∗ For Policies II-1 (preventive maintenance performed 𝑇𝑀 =3and 𝑇𝐶 =3)(Table 3). The compromise interval during [0, 𝑋)) and II-2 (preventive maintenance carried out for the extended warranty cost gets larger as the number during [0, 𝑋𝑒)), we can remark differing trend compared to of PM actions increases (production period increasing). In Policy I-1, since the win-win intervals exist for any values of fact,iftheproductionperiodlengthorthedemandincreases, 𝐶 𝐶 pm and cm. These intervals become larger as the mainte- the principal machine produces more to meet the customers’ nance costs increase but for Policy II-1, the minimum price demands; thus the machine will undergo more failures and at which the lessor should sell the extended warranty is fixed the preventive maintenance interval increases. According to 12 Mathematical Problems in Engineering

Table 4

𝐶 Policy I-1 Policy II-1 Policy II-2 pm Lessor Lessee Lessor Lessee Lessor Lessee 200 202,347 0 194,706 3.41471 200.238 202.25 203.476 300 302,34 0 294,706 3,41471 300,238 300,282 303,759 400 402,34 0 394,706 3,41471 400,238 400,282 403,759 600 602,347 0 594,706 3,41471 600,238 600,282 603,759

the previous results presented through the variability of Δ𝑡, Balance equation (2) 𝑆𝑘+1 =𝑆𝑘 +𝑈𝑘 −𝑑𝑘 𝑘 ∈ {0,1,...,𝐿− the production period length is really impacted visibly. 1} can be converted into an equivalent inventory balance equation, as follows: For 𝐶𝑝𝑚 =200and𝐶𝑐𝑚 =1500andΔ𝑡 =2.For compromise intervals for the extended warranty cost, see Table 6. (2) 󳨐⇒ 𝐸 {𝑆 𝑘+1}=𝐸{𝑆𝑘}+𝑈𝑘 −𝑑𝑘 (A.3) ̂ ̂ ̂ 5. Conclusion 󳨐⇒ 𝑆𝑘+1 = 𝑆𝑘 +𝑈𝑘 − 𝑑𝑘. This paper treats a forecasting production/maintenance problem correlated to the adoption of an extended warranty Equation (A.3) represents the mean variation of inventory period for a leasing machine during a finite leasing horizon. at each period k, 𝑘 ∈ {1,2,...,𝑁−.Furthermore, 1} 𝑢𝑖,𝑘 Firstly, we have developed a mathematical model for a is deterministic, since it does not depend on the random forecasting problem in order to determine a forecasting variables 𝑑𝑘 and 𝑆𝑘.Thatis,𝐸{𝑈}𝑘 =𝑈 with 𝑉(𝑈𝑘)= production plan. Secondly, an analytical model has been 0 for all 𝑘. Taking the difference between (2) and (A.3) proposed in order to present a study of the opportunity provided by the extended warranty in leasing contract from both the lessee and the lessor. We proposed different main- 𝑆 − 𝑆̂ =𝑆 − 𝑆̂ −(𝑑 − 𝑑̂ ) tenance policies during the finite leasing horizon, which we 𝑘+1 𝑘+1 𝑘 𝑘 𝑘 𝑘 2 2 have considered to be the influence of production rates on 󳨐⇒ ( 𝑆 − 𝑆̂ ) =((𝑆 − 𝑆̂ )−(𝑑 − 𝑑̂ )) the degradation degree of leasing machine and including 𝑘+1 𝑘+1 𝑘 𝑘 𝑘 𝑘 periodic preventive maintenance actions with different costs. ̂ 2 󳨐⇒ 𝐸 ( (𝑆 𝑘+1 − 𝑆𝑘+1) ) For each maintenance policy, we expressed the total cost 2 incurred by the lessee and by the lessor in order to determine =𝐸((𝑆 − 𝑆̂ )−(𝑑 − 𝑑̂ ) ) the maximum additional cost the lessee should pay for the 𝑘 𝑘 𝑘 𝑘 extended warranty and the minimum price at which the ̂ 2 󳨐⇒ 𝐸 ( (𝑆 𝑘+1 − 𝑆𝑘+1) ) (A.4) lessor should sell it. For each policy and for any given situ- 2 2 ation, conditions of existence of a win-win interval between =𝐸((𝑆 − 𝑆̂ ) +(𝑑 − 𝑑̂ ) −2 ⋅ (𝑆 − 𝑆̂ )⋅(𝑑 − 𝑑̂ )) the lessee and the lessor have resulted. 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 𝑘 2 For future research, we will consider a more complex 󳨐⇒ 𝐸 ( (𝑆 − 𝑆̂ ) ) system with other types of warranty policies (including the 𝑘+1 𝑘+1 2 2 number of warranty dimensions, the renewability of a war- =𝐸((𝑆 − 𝑆̂ ) )+𝐸((𝑑 − 𝑑̂ ) ) ranty, and the warranty compensation methods). Concerning 𝑘 𝑘 𝑘 𝑘 the maintenance strategy, we will consider new hypotheses: −2⋅𝐸((𝑆 − 𝑆̂ )⋅(𝑑 − 𝑑̂ )) . the corrective and preventive times are not negligible. 𝑘 𝑘 𝑘 𝑘

𝑆 𝑑 Appendix Since 𝑘 and 𝑘 are independent random variables we can deduce that Proof of (5). The inventory variable 𝑆𝑘 is statistically ̂ described by its mean 𝐸{𝑆𝑘}=𝑆𝑘 and variance ̂ ̂ ̂ ̂ 𝐸((𝑆𝑘 − 𝑆𝑘)⋅(𝑑𝑘 − 𝑑𝑘)) = 𝐸 ((𝑆𝑘 − 𝑆𝑘))⋅𝐸((𝑑𝑘 − 𝑑𝑘)) . (A.5) ̂ 2 𝐸{(𝑆𝑘 − 𝑆𝑘) }=Var (𝑆𝑘). (A.1) Also, it is easy to see that

The expected inventory cost is ̂ ̂ 𝐸((𝑆𝑘 − 𝑆𝑘)) = 𝐸 𝑘(𝑆 )−𝐸(𝑆𝑘)=0, 2 ̂2 (A.6) 𝐶𝑠 ⋅𝐸{𝑆 }=𝐶𝑠 ⋅ 𝑆 . (A.2) ̂ ̂ 𝑘 𝑘 𝐸((𝑑𝑘 − 𝑑𝑘)) = 𝐸 𝑘(𝑑 )−𝐸(𝑑𝑘)=0. Mathematical Problems in Engineering 13

Table 5

𝐶 Policy I-1 Policy II-1 Policy II-2 cm Lessor Lessee Lessor Lessee Lessor Lessee 1500 202.34 0 192.915 4.55294 200.238 200.282 203.759 1700 202.66 0 191.97 4.55294 200.27 200.32 204.26 1900 202,973 0 191,025 4.55294 200.302 200.358 204.761 2000 203.129 0 190,553 4.55294 200.318 200.376 205.012

Table 6

𝑇 Policy I-1 Policy II-1 Policy II-2 Lessor Lessee Lessor Lessee Lessor Lessee 1 903,830 904,180 3.41471 800.309 803.194 803.194 2 303,27 0 305,28 4.7141 500.39 504.194 504.76 3 345,347 304,706 4.91471 501.238 522.25 503.759 4 360,724 306,288 4.94417 511.309 533.759 513.140

Consequently, Proof of (8). Consider

𝑆 (𝑘+1) =𝑆(𝑘) +𝑈(𝑘) −𝑑(𝑘) ̂ 2 ̂ 2 ̂ 2 𝐸((𝑆𝑘+1 − 𝑆𝑘+1) )=𝐸((𝑆𝑘 − 𝑆𝑘) )+𝐸((𝑑𝑘 − 𝑑𝑘) ) 󳨐⇒ Prob (𝑆 (𝑘+1) ≥0) ≥𝛼 (A.7) 2 2 2 󳨐⇒ (𝑆 (𝑘) +𝑈(𝑘) −𝑑(𝑘) ≥0) ≥𝛼 󳨐⇒ ( 𝜎 ) =(𝜎 ) +(𝜎 ) . Prob 𝑠𝑘+1 𝑠𝑘 𝑑𝑘 󳨐⇒ Prob (𝑆 (𝑘) +𝑈(𝑘) ≥𝑑(𝑘)) ≥𝛼 (A.10) ̂ ̂ 󳨐⇒ Prob (𝑆 (𝑘) +𝑈(𝑘) − 𝑑 (𝑘) ≥𝑑(𝑘) − 𝑑 (𝑘))≥𝛼 If we assume that 𝜎𝑠(0) = 0 and 𝜎𝑑𝑘 is constant and equal to ̂ ̂ 𝜎𝑑 for all 𝑘’s, we can deduce that 𝑆 (𝑘) +𝑈(𝑘) − 𝑑 (𝑘) 𝑑 (𝑘) − 𝑑 (𝑘) 󳨐⇒ Prob ( ≥ )≥𝛼 𝑉𝑑,𝑘 𝑉𝑑,𝑘

2 (𝜎 ) =𝑘⋅(𝜎)2 𝑑(𝑘)̂ 𝑘 (𝑑(𝑘)) =𝑉 ≥ 𝑠𝑘 𝑑 with average demand at period and Var 𝑑,𝑘 0 variance of demand 𝑑 at period 𝑘. 2 ̂2 2 [𝑌≥𝑋]≥𝛼 󳨐⇒ 𝐸 (𝑆 𝑘)−𝑆𝑘 =𝑘⋅(𝜎𝑑) (A.8) ThisequationisintheformofProb , 𝑋 = ((𝑑 − 𝑑̂ )/𝑉 ) with 𝑘 𝑘 𝑑𝑘 being a Gaussian random 󳨐⇒ 𝐸 (𝑆 2)=𝑘⋅(𝜎)2 + 𝑆̂2. 𝑑 𝜑 𝑘 𝑑 𝑘 variable representing the demand 𝑘,and 𝑑𝑘 is a cumulative Gaussian distribution function of the form 𝐹(𝑌) ≥𝛼 such as

̂ Substituting (A.3) in the expected cost (1) 𝑆 (𝑘) +𝑈(𝑘) − 𝑑 (𝑘) 󳨐⇒ 𝜑 𝑑,𝑘 ( )≥𝛼. (A.11) 𝑉𝑑,𝑘

𝐿 𝜑 =0 𝜑 =1 2 2 2 Since lim𝑑𝑘 →−∞ 𝑑𝑘 and lim𝑑𝑘 →+∞ 𝑑𝑘 ,thefunction 𝑍=𝐶𝑠 ⋅𝐸(𝑆𝐿)+∑ 𝐶𝑠 ⋅𝐸(𝑆𝑘)+𝐶 ⋅𝑈𝑘 , pr 𝜑𝑑 is strictly increasing, and we note that it is indefinitely 𝑖=0 𝑘 𝜑 differentiable. That is why we conclude that 𝑑𝑘 is invertible. 𝐿−1 𝐿 𝑍 (𝑢) =𝐶 ×(𝑆̂2 )+∑𝐶 ⋅ 𝑆̂2 +𝐶 ×𝑢2 +𝐶 ×(𝜎 )2 × ∑𝑘, Thus, 𝑠 𝐿 𝑠 𝑘 pr 𝑘 𝑠 𝑑 𝑖=0 𝑘=0 (A.9) 𝑆 (𝑘) +𝑈(𝑘) − 𝑑̂(𝑘) 𝐿−1 ≥𝜑−1 (𝛼) 𝑍 (𝑢) =𝐶 ×(𝑆̂2 )+∑𝐶 ⋅ 𝑆̂2 +𝐶 ×𝑢2 +𝐶 ×𝜎2 𝑑,𝑘 𝑠 𝐿 𝑠 𝑘 pr 𝑘 𝑠 𝑑 𝑉𝑑,𝑘 𝑘=0 ̂ −1 (A.12) 𝐿 (𝐿+1) 󳨐⇒ 𝑆 (𝑘) +𝑈(𝑘) − 𝑑 (𝑘) ≥𝑉 ⋅𝜑 (𝛼) × . 𝑑,𝑘 𝑑,𝑘 2 −1 ̂ 󳨐⇒ 𝑈 (𝑘) ≥𝑉𝑑,𝑘 ⋅𝜑𝑑,𝑘 (𝛼) + 𝑑 (𝑘) −𝑆(𝑘) . 14 Mathematical Problems in Engineering

Notation [7] Y.-H. Chien, “Optimal age-replacement policy under an imper- Δ𝑡 fect renewing free-replacement warranty,” IEEE Transactions on : Length of a production period Reliability,vol.57,no.1,pp.125–133,2008. 𝐿:Numberofleasingperiods 𝑋 [8] Y.-H. Chien, “Optimal age for preventive replacement under : Warranty periods a combined fully renewable free replacement with a pro-rata 𝑋 𝑒: Warranty period including both basic warranty,” International Journal of Production Economics,vol. period 𝑋 and extension 124, no. 1, pp. 198–205, 2010. 𝑈 𝑀 𝑘:Productionratebymachineduring [9]S.Bouguerra,A.Chelbi,andN.Rezg,“Adecisionmodelfor period 𝑘 (𝑘=0,1,...,𝐿) adopting an extended warranty under different maintenance ̂ 𝑑(𝑘): Average demand during period 𝑘 policies,” International Journal of Production Economics,vol.135, (𝑘=0,1,...,𝐿) no. 2, pp. 840–849, 2012. 𝑉𝑑(𝑘): Variance of demand during period 𝑘 [10] S. Wu and P. Longhurst, “Optimising age-replacement and (𝑘=0,1,...,𝐿) extended non-renewing warranty policies in lifecycle costing,” 𝑆𝑘:Inventorylevelof𝑆 at the end of period 𝑘 International Journal of Production Economics,vol.130,no.2, (𝑘=0,1,...,𝐿) pp. 262–267, 2011. ̂ 𝑆𝑘: Average inventory level of 𝑆 during period [11] Z. Hajej, N. Rezg, and A. Gharbi, “Integrated maintenance pol- 𝑘 (𝑘=0,1,...,𝐿) icy optimization under lease/warranty contract,” in Proceedings 𝐶 : Unitproductioncostofleasingmachine of the International Conference on Industrial Engineering (ICIE pr ’13),Dubai,UAE,October2013. 𝐶𝑠: Holding cost of product unit during one [12] H. Zied, D. Sofiene, and R. Nidhal, “Optimal integrated period maintenance/production policy for randomly failing systems mu: Monetary unit 𝑈max with variable failure rate,” International Journal of Production : Maximal production rate of leasing Research, vol. 49, no. 19, pp. 5695–5712, 2011. machine min 𝑈 : Minimal production rate of leasing machine 𝛼: Probability index related to customer satisfaction and expressing the service level 𝑆0:Initialinventory 𝐶 pm: Preventive maintenance cost 𝐶 cm: Corrective maintenance cost.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] D. N. P. Murthy and E. Asgharizadeh, “Optimal decision making in a maintenance service operation,” European Journal of Operational Research,vol.116,no.2,pp.259–273,1999. [2]R.H.YehandW.L.Chang,“Optimalthresholdvalueoffailure- rate for leased products with preventive maintenance actions,” Mathematical and Computer Modelling,vol.46,no.5-6,pp.730– 737, 2007. [3] J. Jaturonnatee, D. N. P. Murthy, and R. Boondiskulchok, “Optimal preventive maintenance of leased equipment with corrective minimal repairs,” European Journal of Operational Research,vol.174,no.1,pp.201–215,2006. [4]T.M.BerkeandN.Zaino,“Warranties:Whatarethey?What do they really cost?” in Proceedings of the Annual Reliability and Maintainability Symposium, pp. 326–330, January 1991. [5] C. S. Kim, I. Djamaludin, and D. N. P. Murthy, “Warranty and discrete preventive maintenance,” Reliability Engineering and System Safety,vol.84,no.3,pp.301–309,2004. [6] W. Y. Yun, D. N. P. Murthy, and N. Jack, “Warranty servicing with imperfect repair,” International Journal of Production Economics, vol. 111, no. 1, pp. 159–169, 2008. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 793161, 17 pages http://dx.doi.org/10.1155/2015/793161

Review Article Prognostics and Health Management: A Review on Data Driven Approaches

Kwok L. Tsui,1 Nan Chen,2 Qiang Zhou,1 Yizhen Hai,1 and Wenbin Wang3,4

1 Department of System Engineering and Engineering Management, City University of Hong Kong, Hong Kong 2Department of Industrial & Systems Engineering, National University of Singapore, Singapore 117576 3Donlinks School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, China 4Faculty of Business and Law, Manchester Metropolitan University, Manchester M15 6BH, UK

Correspondence should be addressed to Wenbin Wang; [email protected]

Received 1 July 2014; Revised 25 October 2014; Accepted 31 October 2014

Academic Editor: Shaomin Wu

Copyright © 2015 Kwok L. Tsui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Prognostics and health management (PHM) is a framework that offers comprehensive yet individualized solutions for managing system health. In recent years, PHM has emerged as an essential approach for achieving competitive advantages in the global market by improving reliability, maintainability, safety, and affordability. Concepts and components in PHM have been developed separately in many areas such as mechanical engineering, electrical engineering, and statistical science, under varied names. In this paper, we provide a concise review of mainstream methods in major aspects of the PHM framework, including the updated research from both statistical science and engineering, with a focus on data-driven approaches. Real world examples have been provided to illustrate the implementation of PHM in practice.

1. Introduction failure of LED lighting system in Xiamen, China, two months after installation although the manufacturer promised five To fulfill the increasing demand on functionality and quality, years’ lifespan of their products, not to mention the infamous modern systems are often built with overwhelming complex- sudden acceleration failures of Toyota automobiles which ities. These systems are often featured rich electronics and have significantly damaged the company’s profit and reputa- intricate interactions among subsystems/components. For tion [4]. example, a typical car consists of about 2,000 functional com- In view of the high impact and extreme costs usually ponents, 30,000 parts, and 10 million lines of software code associated with system failures, methods that can predict and [1]. prevent such catastrophes have long been investigated. Appli- Additionally, extremely high requirements of system cationsofdevelopedmethodsarenotrareindomainssuchas reliability are essential since a single failure can result in catas- electronics-rich systems, aerospace industries, or even public trophicconsequences.Despiteeveryeffortmadeinthepast, health environment [5, 6]. In general, these methodologies disasters keep occurring with profound implications. In June can all be grouped under the framework of prognostics and 2009, the Metro rail crash in Washington D.C. killed nine health management (PHM). Particularly, prognostics is the people and injured dozens more, suspiciously due to sensor process of predicting the future reliability of a product by circuit “anomalies” under the rail track [2]. Brazil blackouts assessing the extent of deviation or degradation of the prod- in November 2009 affected more than 60 million people uct from its expected normal operating conditions; health andshutdowneverythingfromsubwaytolightbulbs[3]. management is the process of real time measuring, recording, Despite the explanation attributed to lightning, wind, and and monitoring the extent of deviation and degradation from rain, it was still believed that “there was obviously some fail- normal operation condition [7, 8]. Different from traditional ure, either technical or human.” Other examples include the handbook based reliability prediction methods (e.g., U.S. 2 Mathematical Problems in Engineering

Department of Defense Mil-Hdbk-217 and Telcordia SR-332 increasing accuracy of prognostics for individual items. For (formerly [9])), which assumed that constant hazard rate of instance, many different types of data during the whole life each component can be tailored by independent “modifiers” cycle of the products can be easily retrieved, especially in crit- to account for various quality, operating, and environmental ical applications. These data could include production pro- conditions, PHM methodologies instead monitor the health cess information, quality records, operation logs, and sensor stateinrealtimeanddynamicallyupdatethereliability measurements. Moreover, unlike manually entered data used function (hazard rates) based on in situ measurements and before which are slow, costly, and error-prone, most of tailored evolution models obtained from historical data. Due current records are automatic, accurate, and timely entries to the success of existing PHM methodologies, there is no due to the advancements of technology. The use of Radio Fre- need to exhibit the growing interests of studying new PHM quency Identification (RFID) technology, for example, is not techniques and applying PHM to underdeveloped domains. rare in supply chain distribution network, healthcare, and Nevertheless, the increasingly complex modern systems even military applications, to provide reliable and timely pose new challenges on PHM. One of the most prominent tracking or surveillance of products/components. Advanced problems is called No Fault Found (NFF) problem (related sensor technologies also enable abundant measurements at terminologies include “cannot duplicate,” “re-test OK,” “trou- both macro and micro scale, such as vibration, frequency ble not identified,” and “intermittent malfunctions”) [10–12], response,magneticfields,andthecurrent/voltage,toname particularly in electronic-rich systems. As the name sug- afew. gests, it refers to the situation that no failure/fault can be Inresponsetotheemergingchallengesaswellasopportu- detected/replicated during laboratory tests even when the nities, this paper tries to review recent advancements of PHM failure has been reported in the field13 [ ]. NFF issues not only methodologies, with a focus on data-driven approaches, and make the prognosis and diagnosis extremely difficult, but their applications in practice, and identify research problems also can cause skyrocketing maintenance cost. As reported that may lead to further improvement of PHM in both theory by Williams et al. [13], NFF failures account for more than and practice. Before we move on to the next section, we would 85% of all filed failures and 90% of overall maintenance costs like to use an example from real practice to better illustrate the in avionics; it is estimated that NFF related activities cost the ideas in PHM. U.S. Department of Defense 2∼10 billion U.S. dollars per year [14]. Evidently, NFF contributes a lot in operational cost in AMotivatingExample.Bearingsarefoundinmostmechan- many different application areas. On top of the maintenance ical systems with rotational components. They provide nec- costs, potential safety hazards related to NFFs are even more essary support as well as constraining the moving parts striking. For example, both Toyota and National Highway to desired motion mode. It is hard to overemphasize the Traffic Safety Administration (NHTSA) spent quite a long importance of keeping bearings under normal working con- time to investigate the root causes of sudden acceleration ditions in engineering applications. Breaking down of a single failures in some car models, a problem that might be linked bearing may cause failure of an entire system. For example, to 89 deaths in 71 crashes since 2000 according to NHTSA on August 30th 2010, a Qantas Boeing 747 aircraft departing [15]. Unfortunately, no conclusive finding has been reached from San Francisco International Airport encountered an despite the efforts in trying to repeat the failures in a variety accidental engine shutdown, which has later been confirmed of laboratory conditions. These intermittent faults are also to be caused by a fractured turbine blade and a failed bearing suspected to be the main reason for other catastrophes, like [16]. Washington Metro crashes and Brazil blackouts. As a bearing tends to exhibit larger vibration as it Intermittent faults or NFF problems pose significant bar- degrades, its health condition can thus be assessed via the riers to apply traditional methods for reliability prediction, vibration signal collected by sensors. Such signal is often which are often empirical and population based. From the referred to as degradation signal intheliteratureofPHM. aforementioned examples, we can find that intermittent faults When the amplitude of a bearing’s vibration exceeds certain are often tightly related to the environmental conditions and threshold, the bearing can be considered as no longer suitable operation histories of the particular individual system. They for further operations. Figure 1 demonstrates the degradation can hardly be repeated due to unknown random disturbances paths of three different bearings, where the 𝑥-axisisthework- involved. Therefore, laboratory testing and assessment can ing time of bearings (in minutes), the 𝑦-axis is the average only provide a reference on the “average” characteristics of amplitude of the vibration at different harmonic frequencies, the whole population and are insufficient to provide accurate and the horizontal line is the vibration threshold considered modeling and prediction for each individual. To reduce the as indicator of bearing failures [17, 18]. maintenance cost and eliminate safety hazards caused by NFF, Unlike conventional reliability analysis which mostly the paradigm of PHM needs to shift from empirical to data provides population-based assessment, individualized pre- supported and from population based to individual based. diction results are possible by taking advantage of the degra- Companioned with the challenges, the fast development dation signal. Based on the vibration data collected up to the of information and sensing technology has enabled the current time, we can build models to predict the evolution collection of many in situ measurements during operations path of vibration signal in the future and consequently andprovidedthecapabilityofrealtimedatamanagementand predict the remaining useful life (RUL) of an in-service processing for each individual. These advancements provide bearing by assuming its failure as the vibration signal hits us great opportunity to develop sophisticated models with the threshold for the first time. Unfortunately, many factors Mathematical Problems in Engineering 3

0.035 Other than these three major tasks, there are also some other important components listed in Figure 2. Nevertheless, 0.03 they are often prepared offline and only timely updating may be needed during the system operations. For example, signal 0.025 processing/feature extraction is the procedure to preprocess the signals using rules or methods developed according to 0.02 engineering knowledge, expert experience, or statistical find- ings from historical data. They serve the purpose to eliminate 0.015 noise, reduce data dimensions (complexities), and transform Degradation the data into proper space for future analysis. Similarly, 0.01 prognosis and diagnosis algorithms can also be developed offlinetocaterthespecialcharactersofthesignalsandsystem 0.005 properties. Upon new arrival of sensing signals, appropriate algorithms can be selected to compute distribution of RUL, 0 0 100 200 300 400 500 600 determine maintenance actions, or find root causes of abnor- Time malities. Duetoitsundeniableimportance,recentyearshaveseen Figure 1: Evolution of vibration signals of three bearings. prosperous development in different aspects of PHM. The reviews on statistical data-driven approaches by Si et al. [19] and Sikorska et al. [20]havecoveredmostofthemodels used in RUL estimation with a statistical orientation, but in the degradation make the exciting task very challenging. our work focuses on a wide range of the varied models Figure 1 demonstrates several important features of the in PHM methodologies from diagnosis to prognosis with degradation signals. For example, the degradation of bearings their motivations. The subsequent sections are devoted to exhibits two distinctive phases. At the initial stage, the vibra- discussion of research progress and open issues of these tasks/ tion signals emitted are small and stable. However, after a components in PHM to provide us with an overview for change point (often a crack appears), the magnitude of vibra- further advancement. tions increases dramatically and features large variability. Despite similar shapes of degradation paths, the location of 2.1. Signal Processing and Feature Extraction. In current changing point, increasing rate of vibration magnitude, and data rich environment, huge amounts of data are often soforthvaryfromonebearingtoanother.Wewillreturnto automatically collected in a short time period. Different from this example in Section 3 for more technical details. the problem that very limited data was available decades The rest of the paper is organized as follows. Section 2 ago, this overwhelming data poses new challenges in data reviews advancements in PHM. Section 3 uses three exam- management, analysis, and interpretation. Consequently, ples to illustrate the procedures and strategies of PHM. data preprocessing and feature extraction procedures become Section 4 concludes the paper with summing up comments standard in many complex systems to improve data quality, and future works that drive PHM further in academia and reduce data redundancy, and boost efficiency of analysis. Due industries. to its importance, many researchers have investigated this problem in the literature, as summarized in some of the 2. Overview of Data-Driven PHM Approaches review papers in different application areas (e.g., [21–23]). Instead of giving a comprehensive review on different In general, typical workflows in a PHM system can be con- techniques in the literature, in this section we will list some ceptually illustrated, as shown in Figure 2.Threemajortasks of the commonly used methods in the context of PHM. These can be identified in the flowchart: fault diagnostics, prognos- techniques can be roughly classified as statistical methods tics, and condition-based maintenance. The first task isto and engineering knowledge based methods. In the first diagnose and identify the root causes of system failures. The category, data can be transformed to optimize certain prede- root causes identified can provide useful information for termined criteria without the input of the domain knowledge. prognostic models as well as feedback for system design For example, principle component analysis (PCA) and inde- improvement. The second task takes the processed data and pendent component analysis (ICA) have been used widely existing system models or failure mode analysis as inputs to reduce data dimensions. Similar to analysis of variance and employs the developed library of prognosis algorithms to (ANOVA), distance evaluation technique (DET) [24]isalso online update degradation models and predict failure times of preferred and its varied versions are applied, such as a two- the system. The third task makes use of the prognosis results stage feature selection and weighting technique (TFSWT) (e.g., the distribution of remaining useful life) and considers via Euclidean distance evaluation technique (EDET) [25], a the cost versus benefits for different maintenance actions to modified version of ANOVA, which takes both the difference determine when and how the preventive maintenance will be between the variances in each group and the maximum conducted to achieve minimal operating costs and risks. All versus the minimum differences between the mean of each of these three tasks need to be executed dynamically and in group into consideration. However, certain useful informa- real time. tion cannot be retained when dealing with highly nonlinear 4 Mathematical Problems in Engineering

Condition monitoring and data collection

Data preprocessing/ signal processing

Feature selection

Statistical modeling

Fault Condition-based Prognostics diagnostics maintenance

Figure 2: An illustration of typical flows of PHM systems. data, as reported in [26]. Other techniques include mutual Table 1: Commonly used time-domain features. information (MI) based method [27], self-organizing map (SOM) [28], and density based methods [29]. These tech- Feature Definition Peak value max = max 𝑛𝑗 niques work well on nonlinear data and hence are employed 𝑗=1,...,𝑁 broadly in many applications. 1 𝑁 Mean 𝑢= ∑ 𝑛𝑗 Methods in the second category, on the other hand, 𝑁 𝑗=1 utilize some of the domain knowledge in the process of 𝑁 1 2 feature extractions. Based on the procedure of conditional- Standard deviation 𝜎=√ ∑ (𝑛𝑗 −𝑢) 𝑁−1𝑗=1 based maintenance, the data type could be summarized into 𝑁 1 2 three: value type (e.g., temperature, pressure, humidity, etc.), Root mean square RMS = √ ∑ (𝑛𝑗) waveform type (e.g., vibration data), and multidimensional 𝑁 𝑗=1 𝑁 3 type (e.g., image data, X-ray images, etc.) [30]. ∑𝑗=1(𝑛𝑗 −𝑢) = Waveform data analysis is among the most common Skewness SK (𝑁 − 1)𝜎3 methods in diagnostics of mechanical systems, due to the ∑𝑁 (𝑛 −𝑢)4 = 𝑗=1 𝑗 popularity of waveform data collected from sensors, particu- Kurtosis KU (𝑁 − 1)𝜎4 larly in vibration signal analysis of rotating elements [31, 32]. max |𝑛| Crest indicator CI = Different kinds of techniques and algorithms are developed 𝑁 2 √(1/𝑁) ∑𝑗=1(𝑛𝑗) in this field. They can be categorized into time-domain |𝑛| = max analysis, frequency-domain analysis, and the combination Clearance indicator CLI 󵄨 󵄨 2 ((1/𝑁) ∑𝑁 √󵄨𝑛 󵄨) of both. These methods often create features that have clear 𝑗=1 󵄨 𝑗󵄨 physical meanings or interpretations. For example, Table 1 𝑁 2 √(1/𝑁) ∑𝑗=1(𝑛𝑗) summarizes ten commonly used time-domain features [25]. Shape indicator SI = 󵄨 󵄨 (1/𝑁) ∑𝑁 󵄨𝑛 󵄨 Instead of using the raw waveform data, these ten summary 𝑗=1 󵄨 𝑗󵄨 statistics provide us with extracted information of the sig- max |𝑛| Impulse indicator MI = 󵄨 󵄨 nals. These ten features can be generally applied to many (1/𝑁) ∑𝑁 󵄨𝑛 󵄨 𝑗=1 󵄨 𝑗󵄨 applications. However, if the mechanism of how the abnormalities influence the measured signal is known, extracted featuresbasedondomainknowledgemaybemoreeffective. For example, Lei and Zuo [25]summarized11statistical Meanwhile, it is believed that some faults will show features specifically developed for gear damage detection. certain characters in frequency domain.Fouriertransformis Another time-domain analysis approach is time synchronous the most common form of further signal processing, which average (TSA), popularly used in fault detection of rotating decomposes a time waveform into its constituent frequencies. equipment [33, 34]. The idea is to use the average over a Fast Fourier transform (FFT) is usually used to generate the number of evolutions of raw signal in order to remove/reduce frequency spectrum from time series signals. A high vibra- noise. Time series models are naturally applied here as well tion level at a particular frequency may be the signature of [35], in an attempt to extract features based on parametric a particular fault type. Besides FFT spectrum, other methods models. For example, coefficients in a fitted autoregressive such as cepstrum [37], high-order spectra [38], and holospec- moving average model (ARMA) can be indicative of the trum analysis [39] are also developed for fault diagnostics in health condition [36]. the frequency domain. Mathematical Problems in Engineering 5

One limitation of frequency domain analysis is its inabil- network has been preferred by many engineers and widely ity to handle nonstationary waveform signals, commonly applied to fault diagnostics of various engineering systems observed during machine faults. A combination of both time [28, 66–69]. Unsupervised machine learning algorithms such and frequency domains, time-frequency analysis, has been as fuzzy 𝑐-means and self-organizing maps have been applied developed to solve the problem [40, 41]. A typical method when no response is provided [70–72]. It is worth mentioning is called short-time Fourier transform (STFT) [42], which that the sensitivity to a given fault is often a function of oper- divides the whole waveform signal into segments with short- ating conditions and the nature of the anomaly. Therefore, time window and then applies Fourier transform to each the environmental conditions need careful consideration. For segment. Wavelet transform is another popular method with example, self-organizing maps are used for regionalization of a similar idea. Wavelet analysis has been successfully applied the system operating conditions [60]. Excellent reference for to feature extraction and fault diagnostics in various applica- fundamentals of these methods can be found in Bishop [73], tions (e.g., [43–47]). A review on the application of wavelet Hastie et al. [74], and Kotsiantis et al. [75]. analysis in machine fault diagnosis and fault feature extraction can be found in Peng and Chu [48]. Other methods of 2.3. Data-Driven Prognostics Method. As mentioned in time-frequency analysis include spectrogram [49], Wigner- the introduction, prognostics algorithms predict the future Ville distribution [50–52], and Choi-Williams distribution reliability of a product considering current and past health [53]. information collected. Through constant inspection, the observed health information is often referred to as condition 2.2. Fault Diagnostics and Classification. Fault diagnostics monitoring (CM) data. CM data may be directly or indirectly is designed to efficiently and accurately identify the root related with the system health status and hence can be cause of the faults. Effective diagnosis can not only reduce viewed as system health indicators. Examples of CM data downtime and repair cost, but also provide useful informa- are amount of tire wear, chemical concentration, size of a tion for prognostics to improve its accuracy. Fault detection fatigue crack, power output of an amplifier, and the light is defined to be the task of determining if a system is intensity of LED. As a system degrades inevitably through experiencing problems. Fault diagnostics, then, is the task usage, its health status deteriorates and is manifested through oflocatingthesourceofafaultonceitisdetected.Because the observed CM data (e.g., the light intensity decreases as of its importance, researchers from different fields have the LED degrades). Hence, CM is normally viewed as the investigated the issue of fault diagnostics extensively, such as system degradation signal. Failures are often defined as in manufacturing processes [54, 55], discrete event systems the degradation reaches a predetermined threshold set by [56, 57], and communication systems and networks [58, 59]. experts. Thus, by modeling the evolution of degradation and We do not attempt to give a comprehensive review but focus calculating the time it first hits the failure threshold, we will on the approaches that are commonly used in PHM practices. be able to predict the system remaining useful life (RUL). In general, methodologies in fault diagnostics can be clas- Due to randomness in the evolution paths of the degradation, sified into two categories: model based approaches and model thecalculatedRULwillbeintheformofsomeprobability free approaches. In the model based category, some forms of distribution. Two excellent comprehensive review papers in underlying models linking failure modes and observations RUL research can be found in Si et al. [19] and Sikorska et al. are proposed. These models are often derived according [20]. The main difference between our paper and Si etal. to first principles and physical mechanisms. Based on the [19] could be understood through the illustration in Figure 2. model structure and parameters, observations can be used Our paper describes the entire process of data-driven to infer the root causes or the failure modes using different approach related to PHM, addresses the three PHM algorithms. In contrast, model free methods often do not objectives (fault diagnostics, prognostics, and conditional- assume the knowledge of underlying processes. Although based maintenance), and discusses additional prognostics in many cases implicit or statistical/surrogate models are approaches developed from other areas and relationships used in the fault diagnostics, we use the name to emphasize among different approaches. On the other hand, Si et al.19 [ ] that approaches in this category are purely data-driven focus mainly on various modeling approaches for prognostics without additional assumptions on the systems’ operation and do not address the tasks before the three PHM mechanisms. objectives. Sikorska et al. [20] focus on evaluating the various In fault diagnostics, we would like to know the exact time prognostics approaches from the industry point of view when a fault appears, its location, and its severity. Therefore, without details on methodologies. the diagnostics consists of three aspects: (1) anomaly detec- As a core ingredient of PHM, we summarize the major tion, as we first identify any potential performance deviation categories of data-driven prognostics in this section. from normal operation; (2) fault localization, which localizes the problem to the specific component or subsystem; (3) 2.3.1. Independent Increment Process Based Model. Generally fault classification, which discriminates known and unknown speaking, the stochastic process models (Table 2) consist of faults and identifies the type of the fault if it is previously two basic components: (1) astochasticprocess{𝑋(𝑡), 𝑡∈ known [60]. Many popular methods from machine learning T,𝑋 ∈ 𝜒} with initial value 𝑋(0) =0 𝑥 ,whereT is the and artificial intelligence are applied in this context, such as time space and 𝜒 is the state space of the process, and (2) a support vector machine [61, 62], 𝑘-nearest neighbors [25, 63], boundary set 𝐵,where𝐵⊂𝜒.Taking𝑋(0) =0 𝑥 outside the anddecisiontrees[64, 65]. Among them, artificial neural boundary set 𝐵, the first hitting time (FHT) is the random 6 Mathematical Problems in Engineering

Table 2: Summary of three stochastic processes.

Models FHT distributions Model parameters Shapes of hazard functions 2 Wiener process Inverse Gaussian distribution Drift 𝜐 and variance 𝜎 Upside down bathtub Gamma process Approximate inverse Gaussian or B-S distributions Shape 𝛼 and scale 𝛽 Upside down bathtub Inverse Gaussian process Approximate B-S distribution Mean 𝜇 and shape 𝜆 Upside down bathtub

2 variable 𝑇, defined as 𝑇=inf{𝑡 : 𝑋(𝑡) ∈𝐵}.Inmostcases,𝐵 an inverse Gaussian distribution IG(Δ𝑡, 𝜂(Δ𝑡) ).Likethe ⋆ is simplified as a threshold 𝑥 (or 𝐷) and the FHT is the first Gamma process, the inverse Gaussian process also has a ⋆ time when 𝑋(𝑡) reaches 𝑥 . monotone path, and the failure time distribution approx- In stochastic process model, it is supposed that the imated a Birnbaum-Saunders type distribution, which has degradation signal {𝑋(𝑡), 𝑡∈ T,𝑋 ∈ 𝜒} has stationary excellent properties for future computation. The IG process is independent increment, which means for any time 𝑡,and relatively new and has not been widely applied in degradation Δ𝑡 >, 0 the increment Δ𝑋(𝑡) = 𝑋(𝑡 + Δ𝑡) −𝑋(𝑡) only modeling,eventhoughitismoreflexibleinincorporating depends on Δ𝑡 and some other parameters denoted as 𝜑. random effects and covariates. It was introduced in Wang Usually, Δ𝑋(𝑡) follows a distribution that possesses the and Xu [88] to incorporate random effects. The random property of additivity. For example, if Δ𝑋(𝑡) follows a Normal drift model, random volatility model, random drift-volatility 2 distribution 𝑁(𝜇Δ𝑡, 𝜎 Δ𝑡) and 𝑋(0) =,then 0 𝑋(𝑡) will follow model, and incorporating covariates are thoroughly studied 2 aNormaldistribution𝑁(𝜇𝑡, 𝜎 𝑡) and {𝑋(𝑡)} is the Wiener byYeandChen[89]. process. Other typical choices of the Δ𝑋(𝑡) distribution are Gamma distribution and inverse Gaussian distribution, in {𝑋(𝑡)} 2.3.2. Markovian Process-Based Models. Another set of meth- which cases will be correspondingly called Gamma ods are built based on the memoryless Markov processes. processes and inverse Gaussian processes. AlthoughMarkovprocessstillbelongstostochasticpro- cesses, these methods are different from the previously Wiener Process. A Wiener process {𝑋(𝑡), 𝑡 ⩾0} can be repre- mentioned models in the sense that they assume a finite state sented as 𝑋(𝑡) = 𝜆𝑡+𝜎𝐵(𝑡) where 𝜆 is a drift parameter, 𝜎>0 of the degradation and focus on the transition probability is a diffusion coefficient, and 𝐵(𝑡) is the standard Brownian among those states. The methods in this category have the motion. The Probability Density Function (PDF) of the first ⋆ following major variations. hitting time 𝑇 is the inverse Gaussian distribution IG((𝑥 − 𝑥 )/𝜆, (𝑥⋆ −𝑥 )2/𝜎2) 0 0 . The process varies bidirectionally over Markov Chain Model. In general, it is assumed that the time with Gaussian noises. It only uses information contained degradation process {𝑋𝑛,𝑛⩾0} evolves on a finite state space in the current degradation status. Please refer to [76–81]for Φ={0,1,...,𝑁}, with 0 corresponding to the perfect healthy more information. Recently, there are some new researches state and 𝑁 representing the failed state of the monitored under this framework and use the history information given system. The RUL at time instant 𝑛 canbedefinedas𝑇= by the entire sequence of observations. These models update inf{𝑡 :𝑛+𝑡 𝑋 =𝑁|𝑋𝑛 =𝑁}̸ . The probability transition matrix the parameters recursively so the prognostics is history and the number of the states can be estimated from historical dependent [82, 83]. data. By dividing the health status into discrete states such as “Good,” “OK,” “Minor defects,” “Maintenance required,” and Gamma Process. One disadvantage of the Wiener process is “Unserviceable,” the method can provide meaningful results that it is not monotone with the Brownian motion embedded. that are easier to be understood by field engineers. For modeling monotonically increasing/decreasing degradation signals, the Gamma process is a better choice. Here, Semi-Markov Processes.Asemi-Markovprocess{𝑋(𝑡), 𝑡 ⩾0} Δ𝑋(𝑡) = 𝑋(𝑡 + Δ𝑡) −𝑋(𝑡) the increment for a given time extends the Markov chain model by including the random Δ𝑡 (𝛼Δ𝑡, 𝜎) interval has a Gamma distribution Ga with shape time that the process resides in each state. Although the 𝛼Δ𝑡 >0 𝜎>0 parameter and scale parameter .AGamma Markov property is generally lost by this extension, the model process has monotonic sample paths and can be viewed as remains of great practical value. In a semi-Markov model, the the limit of a compound Poisson process whose rate goes first hitting time represents the time that the process resides to infinity while the jump size tends to zero in proportion. in the initial and subsequent states before it first enters one of The first hitting time has an inverse Gamma distribution, 𝐵 ∗ the states that define set . defined by the identity 𝑃(𝑇 > 𝑡) = 𝑃(𝑋(𝑡) <𝑥 ).Details of modeling degradation with Gamma process are given by Hidden Markov Model (HMM).HMMconsistsoftwo Singpurwalla [84], Lawless and Crowder [85],andYeetal. stochastic processes, a hidden Markov chain {𝑍𝑛,𝑛 ⩾0} , [86], and maintenance related issues are considered by van which is unobservable and represents the real state of the Noortwijk [87]. degradation, and an observable process {𝑌𝑛,𝑛 ⩾0},whichis the observed signal from monitoring. Similar to Markovian- Inverse Gaussian Process. An inverse Gaussian process based models, it is assumed that the degradation process {𝑋(𝑡), 𝑡 ⩾0} with mean function V(𝑡) and scale parameter evolves according to a Markov chain on a finite state space. 𝜂 has the following properties. The increment Δ𝑋(𝑡) has Generally, a conditional probability measure 𝑃(𝑌𝑛 |𝑍𝑛 =𝑖), Mathematical Problems in Engineering 7

𝑖∈Φ,isusedtolink{𝑌𝑛,𝑛 ⩾0} and {𝑍𝑛,𝑛 ⩾0}.Assuchthe random effects for item 𝑖.Basedon(3), the distribution of fail- RUL at time instant 𝑛 canbedefinedas𝑇=inf{𝑡 :𝑛+𝑡 𝑍 = uretime,definedas𝑇=inf{𝑡 : 𝑓(𝑡;𝑖 𝜙,𝜃 )+𝜖𝑖𝑗 >𝐷}where 𝐷 is 𝑁|𝑍𝑛 =𝑁,𝑌̸ 𝑗,0≤𝑗≤𝑛}. The model is preferred when only the predetermined threshold, can be computed analytically or indirect observations are available [90]. numerically. Different examples were illustrated for different degradation models and distributions of random effects 𝜃𝑖 in 2.3.3. Filtering-Based Models. Similar as the HMM, the thepaper.K.YangandG.Yang[95] extended the idea and uti- Kalman filtering model does not use the CM directly as the lized both the life time data of failed devices and degradation true degradation signal. It assumes that the true state of the information from unfailed ones to improve the model esti- degradation is unobservable but related with CM data. The mation. Along this line with applications in a variety of fields, Kalman filtering model considers the unobserved condition, other representative works include Yang and Jeang [96], 𝑥𝑡,andtheobservedCMdata,𝑦𝑡,suchthat𝑥𝑡 =𝛼𝑥𝑡−1 +𝜖𝑡 Tseng et al. [97],andGoodeetal.[98]. Although in these and 𝑦𝑡 =𝛽𝑥𝑡 +𝜂𝑡,where𝜖𝑡 and 𝜂𝑡 are Gaussian noises and results the individual-to-individual variation has been con- 𝛼 and 𝛽 are the parameters of the state space model. The sidered using random effects, the data from individual item Kalman filtering model takes advantage of all historical data, areonlyusedtoassessthevariabilityamongitemsinthe unlike many methods that only depend on the last CM population and fit the random effects distribution 𝐺𝜃.The status. However, the linear assumption and Gaussian noise prediction of failure time is still population-wise although assumption limit its applications. Efforts have been made to it considers the variability within the population. In certain overcome these problems (e.g., [91, 92]). sense, the prediction interval is inflated to cover different degradation paths. 2.3.4. Regression Based Model. Methodsinthiscategory Gebraeel and his colleagues [17, 18, 99] instead developed mostly involve building parametric evolution path (linear or a Bayesian framework to model the degradation signals and nonlinear) of CM data with random effects. Most existing predict the residual life distribution. Different from previous methods in RUL estimation assume that products of the works, the residual life prediction is “customized” based on same type or from the same batch have exactly the same the data from each individual. For example, the degradation failure characteristics probabilistically. While the population signalcanbemodeledby behavior can provide some reference, they cannot accurately 𝐿 (𝑡 ) =𝜃+𝛽⋅𝑡 +𝜀(𝑡 ) , reflect the health evolution for each individual item since 𝑖 𝑖 𝑖 (4) individual products often experience different usage patterns, where 𝜃 and 𝛽 are random variables following certain dis- distinct environments, or even different quality due to process tribution, and 𝜀 is measurement error (independent normal variations. Consequently, it is crucial to adapt to the health or Brownian motion). From historical data, the joint distri- evolutions of each individual product rather than perceived bution of 𝜃 and 𝛽, denoted by 𝑃(𝜃, 𝛽),canbeestimatedas group averages for better reliability prediction. In recent prior knowledge regarding the population characteristics of years, some methods have been proposed to incorporate the degradation. For each operating new individual, the vibration population information with observations from individual signals are collected and used to update the degradation items to get better RUL estimation. model using Bayesian method: Meeker and Escobar [93] give an example of a linear degradation with Log-normal rate: 𝑃{𝜃,𝛽|𝐿(𝑡1),𝐿(𝑡2),...,𝐿(𝑡𝑘)} (5) 𝑋 (𝑡) =𝛽0 +𝛽 𝑡, 1 (1) ∝𝑃{𝐿(𝑡1),𝐿(𝑡2),...,𝐿(𝑡𝑘) | 𝜃, 𝛽} ⋅ 𝑃 (𝜃, 𝛽), where 𝛽0 is fixed, 𝛽1 ∼ LOGNOR(𝜇, 𝜎),and𝐷 is the where the left hand side of (5) represents the posterior predetermined threshold; thus the distribution of failure time distribution of parameters of the degradation model (4) given will be observations up to time 𝑡𝑘;thefirsttermontherighthand 𝐷−𝛽 𝐹(𝑡,𝛽 ,𝜇,𝜎)=𝑃(𝛽 +𝛽 𝑡>𝐷)=𝑃(𝛽 > 0 ) side corresponds to the likelihood function of the observed 0 0 1 1 𝑡 data implied by (4) with fixed 𝜃 and 𝛽;andthelasttermisthe (estimated) prior distribution of the model parameters within (𝑡) −[ (𝐷 − 𝛽 )−𝜇] =Φ (log log 0 ), (2) the population. In other words, the degradation model for nor 𝜎 each individual is self-updating when new observations are available. It is expected that the predicted failure time based 𝑡>0. on the posterior degradation model will be more accurate. Due to its simplicity and natural integration, Bayesian Lu and Meeker [94] proposed several random coefficients framework has continuously been investigated in the liter- model to describe individual health degradation by consider- ature to provide more accurate degradation modeling and ing both population trend and individual unit characteristics failure prognostics. For example, Xu and Zhou [100]studied through fixed and random effects, respectively: the modeling and prognostics using general nonlinear degra- 𝑦𝑖𝑗 =𝑓(𝑡𝑗;𝜙,𝜃𝑖)+𝜀𝑖𝑗 , (3) dation paths, where the Bayesian based model estimation and Monte Carlo based failure time prediction were presented. where 𝑦𝑖𝑗 is the degradation signal of item 𝑖 at time 𝑡𝑗; 𝜙 is the Chen and Tsui [101] extended this to a two-phase model, parameters for the fixed effects, and 𝜃𝑖 is the parameters of which allows for a change point of the linear degradation. 8 Mathematical Problems in Engineering

This model captures the deterioration of the bearing indicated underlying system, allowing maintenance activities to be by the vibration signal. Other references on this topic are performed only when necessary. The dominating objective Gebraeel and Lawley [102]andSietal.[82, 83]. of CBM in literature is to minimize the cost for maintenance activities. It is worth pointing out that, in some literature, the 2.3.5. Proportional Hazard Model. Proportional hazard term “condition based maintenance” has a broader definition model [103] has been extensively studied in various areas. that also involves the preceding steps of data manipulation, Proportional hazard model with time-dependent variable(s) diagnosis, and prognosis (e.g., [30]). In this section, we is able to incorporate both event data and CM data, which restrict our discussion to maintenance decision-making in can be particularly useful in cases of uncertain failure CBM. thresholds or hard failures [104–106]. The model assumes the The underpinning assumption of the PHM framework following form for hazard rate: is that systems are subject to stochastic deteriorations. The natural choice of maintenance strategy for stochastic deterio- ℎ (𝑡) =ℎ0 (𝑡) exp (𝛾z (𝑡)), (6) rating systems is called “control-limit policy,” or “failure limit where ℎ0(𝑡) isthebaselinehazardrate,𝛾 is a vector of coef- policy” [110, 112], where maintenance activities are conducted ficients, and z(𝑡) contains time-dependent variables. ℎ0(𝑡) when the system deterioration reaches a certain level. Under can be either parametric (e.g., Weibull) or nonparametric, such policy, prognostic results on system deterioration can and model parameters can be estimated by the maximum be used for maintenance decision-making. The control-limit likelihood method. In this model, the condition monitoring policy has been shown to be the optimal replacement rule data are viewed as time-dependent covariates in z(𝑡).System for systems with increasing deteriorations when considering failure distribution can be calculated based on (6). the average long-run cost per unit time [113]. Existing work on CBM can be classified in several ways depending on the 2.3.6. Threshold Regression Model. The parent process 𝑋(𝑡) nature of the system and the assumptions they make, which and boundary set 𝐵 of the FHT model will both generally are (1) whether the system health condition is completely have parameters 𝑧 that depend on covariates that vary across observable or partially observable; (2) whether the condition individuals: 𝑔𝜃(𝑧𝑖)=𝑧𝑖𝛽 [107]. Cox Proportional Hazard monitoring is continuous or intermittent; (3) whether the regression is, for most purposes, a special case of Thresh- maintenance program deals with single component or mul- old Regression [108].Anyfamilyofproportionalhazard tiple components. Note that these are not mutually exclusive functions can be generated by varying the time scales or and a single work usually falls into multiple categories. Below boundaries of a TR model, subject to only mild regulatory we discuss these topics in more detail. conditions. There is a connection between the shape of the In condition monitoring, the system health condition hazard function (HF) and the type of failure mode (cause of can be either completely or partially observed/identified. The failure). For example, an increasing HF corresponds to aging/ system health information obtained in the former case is wear-out and a decreasing hazard function generally suggests called direct information, while that in the latter case is called a mixture of defective or other weak units leading to infant indirect information. While it is a critical issue for the degra- mortality. dation modeling and prognosis previously reviewed in this paper, it has no major impact on the maintenance decision- 2.4. Condition Based Maintenance. Maintenance is defined as making.Forthisreason,wedonotdiscussthisissueagain a set of activities or tasks used to restore an item to a state in here; interested readers are referred to Jardine et al. [30], which it can perform its designated functions. Maintenance which summarized many works in this regard. strategiescanbebroadlyclassifiedintoCorrectiveMain- Depending on the budget constraint and/or technologies tenance and Preventive Maintenance strategies [109, 110]. used, condition monitoring can be either continuous or In corrective maintenance, maintenance activities are only intermittent (periodic or aperiodic), of which the latter is carried out after the failure happens and hence should only be also known as interval inspection. The case of CBM with used for noncritical systems. On the other hand, preventive intermittent condition monitoring has been studied exten- maintenance tries to prevent the failure from happening by sively in literature, primarily due to its wide implementation using either predetermined maintenance such as time-based in practice. The important decision variables are the control maintenance or condition-based maintenance (CBM). An limit/critical level and inspection interval; optimal critical example of predetermined maintenance is the commonly level and inspection intervals are found based on criteria suggested practice of changing engine oil every 3,000 miles or that are mostly cost-based [114–117]. In some works, critical three months (whichever comes first), regardless of the actual levels are assumed to be predetermined by expert knowledge oil condition. In recent decades, some companies such as GM and only optimal inspection strategies are studied [118–120]. have developed oil life monitoring systems which allows car To provide more refined maintenance policies to minimize owners to change oil only when necessary (e.g., [111]). Such cost, some researchers consider multiple control limits. For system is an excellent example of CBM implementation. In example, Castanier et al. [121] used different thresholds for recent years, CBM has become the most modern technique inspection scheduling, partial repair, preventive replacement, discussed in the literature and falls well within the framework and restarting for repairable systems. With the development of PHM. of sensing and information technology, continuous condition CBM is a maintenance program that makes maintenance monitoring has become available at reasonable costs in decisions based on the information collected about the many applications [122]. Comparing with interval inspection, Mathematical Problems in Engineering 9

Timing belt Gearbox

Siglab analyzer Speed controller #1 Shaft 1 Brake controller

Tested gear Gearbox system #2Shaft 2 #3 Motor Brake Laptop Shaft 3 #4

Accelerometers

(a) (b)

Figure 3: Experiment setup.

Figure 4: Different crack levels in the gears. the research in this area is relatively new but has become Diagnosisofgearfaultsiscrucialforpreventingsystemmal- increasingly popular. The fundamental difference of CBM function. In this example, we demonstrate the fault diagnosis with continuous monitoring is that the real-time system and classification for identifying different development stages information allows maintenance decisions to be made at any of cracks on gears in gearboxes [25, 133]. A gearbox test rig time and hence the greater chance to optimize the set criteria is shown in Figure 3,wheregear#3isthetestedgearwith [123–128]. potential cracks. Three different types of gears are tested: (1) While majority of the existing work deals only with single 0% crack level; (2) 25% crack level; (3) 50% crack level, as component, some researchers extend CBM to maintenance shown in Figure 4. The vibration signal is measured using decision-making for multiple components in the system. The accelerometers under various working conditions: 3 levels of rationale of developing multicomponent maintenance policy load from the magnetic brake (no load, half the maximum is that there are economic dependencies among multiple load, and maximum load), and 4 levels of motor speed components [129, 130]. High fixed maintenance cost, such (1200rpm,1400rpm,1600rpm,and1800rpm).Threesetsof as sending a maintenance team to a remote wind farm, data sample are obtained under each combination of the two can be mitigated by replacing/repairing multiple components factors.Henceweobtain36datasamplesforeachcracklevel. simultaneously [123, 126, 131, 132]. The data are then used for gear crack detection and crack level classification. 3. Illustrative Case Studies All the ten time-domain features listed in Table 1 are calculated as potential candidate features. Six features, peak, In this section, we use three examples to illustrate the mean, root mean square (RMS), skewness, kurtosis, and implementation of PHM. shapefactor,areselectedbyANOVAandTFSWT.After featureselection,threemethodsareappliedtoclassifying 3.1. Fault Diagnosis on Gear Crack Development. Gearboxes the three levels of gear cracks, namely, multinomial logit are one of the most commonly used parts in machinery. model (MLM), cumulative link model (CLM), and weighted 10 Mathematical Problems in Engineering

𝑘 nearest neighbor (WKNN). Interested readers are referred 0.035 to Lei and Zuo [25]andHaietal.[133] for technical details. To assess the performance of the methods, a leave-one-out cross- 0.03 validation approach is used; the classification accuracies for MLM, CLM, and WKNN are 98.1%, 94.4%, and near 100%, 0.025 respectively. The test results demonstrate that the proposed methods can accurately identify the crack development of 0.02 gears, which is very beneficial for early warning of potential gearbox malfunction. 0.015

0.01 3.2. Predicting RUL of Rotational Bearings. In this section, we distribution Probability return to the motivating example in Section 1,whichaimsto 0.005 predicttheRULofbearings.AsthepurposeofPHMismainly to provide individualized prediction results, it is necessary to 0 adapt the model to specific characteristics of each bearing, 0 100 200 300 400 500 revealed through past observations. A natural choice is to use Residual life Bayesian framework to integrate the prior information from other bearings with observations of the in-service unit. By 70% failure time selecting the conjugate prior distributions, model updating 80% failure time canbeefficientlydone: Figure 5: Predicted distribution of residual life at different observation times. 𝑃 (𝛽 |𝑌1,𝑌2,...,𝑌𝑛) ∝𝑃(𝑌1,𝑌2,...,𝑌𝑛 | 𝛽) ⋅𝜋(𝛽) , (7) where 𝛽 is the model parameters, 𝑌𝑖 is the observed vibration 𝑡 𝑃(𝑌 ,𝑌,...,𝑌 | 𝛽) magnitude at time 𝑖, 1 2 𝑛 is the likelihood 1600 function given the parameter, 𝜋(𝛽) is the prior distribution carrying information from historical bearing samples, and 1400 𝑃(𝛽 |𝑌1,𝑌2,...,𝑌𝑛) is the posterior distribution of 𝛽 integrating the prior information and current observations. As 1200 new observations are collected, model updating can be done 1000 repeatedly, and correspondingly the predicted failure time will also be updated: 800 Failure time Failure 600 𝑃(𝑌𝑘 |𝑌1,𝑌2,...,𝑌𝑛) (8) 400 = ∫ 𝑃(𝛽 |𝑌1,𝑌2,...,𝑌𝑛)⋅𝑃(𝑌𝑘 | 𝛽)𝑑𝛽, 200 where 𝑃(𝑌𝑘 |𝑌1,𝑌2,...,𝑌𝑛) is the prediction of the vibration 0 0 5 10 15 20 25 magnitude at some future time 𝑡𝑘.As𝑛 increases, the predic- Bearing index tion will be more and more accurate with smaller variance, as demonstrated in Figure 5. Figure 6: Prediction interval and the true values of failure time In this example, 25 bearings are tested and Figure 6 shows for each bearing (triangle means last observed life time of bearings the prediction interval of the failure time. The 𝑥-axisisthe which have not yet failed). index of the bearings used in the experiments. The circles with the same 𝑥-axis value represent the 0.05, 0.5, and 0.95 quantiles of the failure time, and the cross shows the true failure time. The results show that the prediction based on the (PF) based prognostic algorithm is used to predict RUL of above algorithm is acceptably accurate. In certain cases, the lithium-ion batteries based on accelerating testing data of six prediction interval is very tight, providing very informative batteries [134]. warnings on the potential failures. In the experiment, batteries were tested with full charging and discharging cycles, under the constant-current/constant- 3.3. Predicting RUL of Lithium-Ion Batteries Using Particle voltagemode.Thedischargeratewassetto1C,whichmeant Filter. Lithium-ion batteries are widely used in consumer the battery would be fully discharged in one hour. The experi- electronics as their sole power sources. The importance of ment was conducted under room temperature, and discharge batteries to those devices is arguably critical. As battery ages, capacity was calculated based on integrating current over its capacity degrades and is widely used as an indicator of the time for each cycle. Figure 7 shows the capacity degradation battery’s health. As a common rule, a battery is considered process of one testing battery. incapable of functioning as intended when its capacity drops As capacity is used as the default health indicator of to 80% of its initial value. In this example, a particle filtering batteries, there is no feature selection needed. We proceed Mathematical Problems in Engineering 11

observed capacity sequence very well. As expected, the prediction results are better and RUL PDF is narrower at the 1.1 later stage of the battery’s life.

1 4. Conclusions PHM is a framework that offers a complete set of tools for 0.9 managing system health with individualized solutions. In this Capacity (Ah) Capacity paper, we have reviewed methodologies in all major aspects of the PHM framework, namely, signal processing and feature 0.8 extraction, fault diagnosis and classification, fault prognosis, and condition based maintenance. As can be seen, PHM involves many subareas and hence a huge body of literature. 0.7 These areas are at very different stages of development. 0 200 400 600 800 While areas such as signal processing and feature extraction Cycle have long been studied, system failure prognosis based on Figure 7: Capacity degradation of a lithium-ion battery. condition monitoring is still at its infancy. Some subareas have already been extensively studied long before the concept of PHM. Excellent surveys have already been done in some subareas,forexample,infailureprognosis[19] and condition- directly to the degradation modeling and prognosis. The based maintenance [30]. Therefore, instead of focusing on an model of the degradation curve is assumed as follows: extensive literature review in all subareas, we have taken a 2 holistic view to summarize mainstream methods in them, the 𝐶𝑘 =𝛾1 ⋅ exp (𝛾2 ⋅𝑘)+𝛾3 ⋅𝑘 +𝛾4, (9) role of each area in PHM, and their relationships. where 𝐶𝑘 is the battery capacity at 𝑘th cycle and 𝛾𝑖’s are Data-driven methodologies in PHM are closely related the model parameters. For accurate estimation of 𝐶𝑘 and with those in some other major research directions, such as dynamically updating model parameters for better tracking, statistical quality control, reliability engineering, and design a PF approach is used. In the PF, the state-space model for of experiments. It is worthwhile to briefly discuss their trackinghasaprocessfunction𝑓𝑘 and a measurement relations with PHM. function ℎ𝑘: Γ =𝑓 (Γ ,𝜃 ) ;𝑦=ℎ (Γ ,𝜑 ) , 4.1. Statistical Quality Control. Statisticalqualitycontrolis 𝑘 𝑘 𝑘−1 𝑘−1 𝑘 𝑘 𝑘 𝑘 (10) an area that has been extensively studied for many decades. The main objective is to detect abnormalities or changes where 𝑦𝑘 istheobservedcapacity,Γ𝑘 is the collection of all in a process. It is generally applied to a large number of 𝛾𝑖’s estimated at the 𝑘th step, and 𝜃𝑘 and 𝜑𝑘 are two i.i.d. homogeneous units and focuses on identifying the abnormal noise sequences. In (10), 𝑓𝑘 and ℎ𝑘 actually define conditional distributions 𝑝(Γ𝑘 |Γ𝑘−1) and 𝑝(𝑦𝑘 |Γ𝑘),respectively.The ones which may be traced back to process faults. PHM, on the recursive Bayesian filtering is then carried out via the follow- other hand, focuses more on how faults happen and how to ing equations: predict future faults so that optimal maintenance policy can be made, rather than fault detection. Furthermore, research 𝑝(Γ𝑘 |𝑦1,...,𝑦𝑘−1) in PHM focuses more on individual behaviors along time instead of cross-sectional analysis on the population charac-

= ∫ 𝑝(Γ𝑘 |Γ𝑘−1)𝑝(Γ𝑘−1 |𝑦1,...,𝑦𝑘−1)𝑑Γ𝑘−1 teristics. (11) 𝑝(Γ |𝑦,...,𝑦 )𝑝(𝑦 |Γ) 4.2. Reliability Engineering. The research in PHM is closely 𝑝(Γ |𝑦,...,𝑦 )= 𝑘 1 𝑘−1 𝑘 𝑘 . related with those in reliability engineering, such as failure 𝑘 1 𝑘 𝑝(𝑦 |𝑦,...,𝑦 ) 𝑘 1 𝑘−1 prediction and maintenance. Many methods in PHM stem from those originally developed in reliability engineering. Equation (11) provides a recursive way to update the dis- However, they have different focuses of interests. Tradi- tribution of Γ𝑘 with newly observed values. For prognosis, tional reliability engineering focuses on the modeling and the battery capacity at 𝑗 step ahead, 𝐶𝑘+𝑗,canbeestimated prediction of the entire product population, without much by projecting Γ𝑘 to its all possible future paths based on emphasis on variability of the individuals and their respective 𝑝(Γ𝑘 |𝑦1,...,𝑦𝑘). Finally, the RUL distribution is calculated 𝑝(𝐶 ≤ 0.8𝐶 ) 𝐶 working conditions. Therefore, reliability engineering is most through 𝑘+𝑗 init where init is the initial capacity of the battery. valuable for manufacturer’s product design and warranty For demonstration, four batteries are used. Three of them policy making where population characteristics are crucial, are used for initializing model parameters and the last one while PHM is most valuable for end users who care more is used for testing. The predictions are made at 1/3, 2/3, and about the specific units they have on hand. 4/5 of the battery’s life by treating the data of subsequent cycles unknown to the algorithm. The results are shown in 4.3. Design of Experiments. Comparing with the PHM which Figure 8. It can be seen that the algorithm can track the emphasizes online monitoring and dynamic updating, design 12 Mathematical Problems in Engineering

Model C Model C

1.1 1.1 Prediction time 1.05 1.05

1 1

0.95 0.95 RUP pdf Capacity (Ah) Capacity Capacity (Ah) Capacity RUP pdf Prediction time 0.9 0.9 End of performance End of performance 0.85 0.85

0.8 0.8 0 200 400 600 800 0 200 400 600 800 Cycle number Cycle number

(a) (b) Model C

1.1

1.05

1 RUP pdf

0.95

Capacity (Ah) Capacity Prediction time 0.9

End of performance 0.85

0.8 0 200 400 600 800 Cycle number

Real data Estimated values Predicted values

(c)

Figure 8: RUL predictions for lithium-ion battery: (a) at 1/3 life; (b) at 1/2 life; (c) at 2/3 life.

of experiments (DOE) is an offline methodology. PHM and wide application. To achieve cost-effective, robust, and easy- DOE are implemented at different stages. DOE is applied to-implementsolutionssothatPHMcanbeappliedtomore mostly during the system planning and design phase, instead realworldapplications,therearemanychallengesaswellas ofitsoperatingphasewherePHMisapplied.ToolsinDOE research opportunities. 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Research Article Parametric Sensitivity Analysis for Importance Measure on Failure Probability and Its Efficient Kriging Solution

Yishang Zhang, Yongshou Liu, and Xufeng Yang

Institute of Aircraft Reliability Engineering, Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710129, China Correspondence should be addressed to Yishang Zhang; [email protected]

Received 27 May 2014; Revised 23 September 2014; Accepted 23 September 2014

Academic Editor: Shaomin Wu

Copyright © 2015 Yishang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The moment-independent importance measure (IM) on the failure probability is important in system reliability engineering, andit is always influenced by the distribution parameters of inputs. For the purpose of identifying the influential distribution parameters, the parametric sensitivity of IM on the failure probability based on local and global sensitivity analysis technology is proposed. Then the definitions of the parametric sensitivities of IM on the failure probability are given, and their computational formulae are derived. The parametric sensitivity finds out how the IM can be changed by varying the distribution parameters, which providesan important reference to improve or modify the reliability properties. When the sensitivity indicator is larger, the basic distribution parameter becomes more important to the IM. Meanwhile, for the issue that the computational effort of the IM and its parametric sensitivity is usually too expensive, an active learning Kriging (ALK) solution is established in this study. Two numerical examples and two engineering examples are examined to demonstrate the significance of the proposed parametric sensitivity index, as well as the efficiency and precision of the calculation method.

1. Introduction how small variation of the distribution parameters near a reference point changes the output value. The classical local Uncertainties existing in engineering analysis and design SA is defined as the partial derivative of the output with are inherently unavoidable in nature associated with the respect to the distribution parameters of inputs [9]. The manufacturing error, material property, loads, and so forth. global SA, also named as importance measure (IM), gives Fortunately, reliability analysis and sensitivity analysis are consideration to measure the effect of the output uncertainty now available to deal with the uncertainty existing in design on the uncertainty of the input parameters, covering their variables to improve the performance of a mechanical or variation range space as opposed to local SA using partial structural system [1–3].Reliabilityanalysisaimsatpredicting derivatives. the failure probability (or reliability) of the structure under Saltelli and Marivoet [10]andHeltonandDavis[11] the effects of random uncertainties. On the other hand, sensi- proposed the nonparametric techniques (input-output cor- tivity analysis focuses on the contribution of each uncertainty relation), but this method lacks model independence. With or distribution parameters of the input variables [4–7]. It the advantage of “global, quantitative and model free,” the is reasonable and practicable to obtain reliability sensitivity variance-based importance measures are gaining the increas- analysis for quantifying and ranking the effects of random ing attention of practitioners and have been used exten- uncertainties on the failure probability. In this paper, the sively for quantitative analysis [12–16]. However, Borgonovo reliability sensitivity analysis is concerned. addressed the following fact: “the premise of variance-based Sensitivity analysis (SA) is the study of how the output GSA technique that the variance is sufficient to identify response of a model (numerical or otherwise) is affected the variability of model output is not always true” [17]. by the input uncertainty, which can be classified into two The “moment-independent” importance measures have been groups: local SA and global SA [8]. The local SA investigates presented [6, 16–19]. They are also global, quantitative, 2 Mathematical Problems in Engineering model-free, and additionally moment-independent, thus The remainder of this work is organized as follows. attracting more and more attention of practitioners recently. Section 2 reviews the definition of the moment-independent Generally speaking, in reliability analysis, researchers often importance measure of the basic variable on the failure pay the most attention to the failure probability. With this probability. And the parametric sensitivity of IM on failure respect, Cui et al. [6] introduced a moment-independent probability is firstly presented. In Section 3, the established importance measure of the basic variable on the failure ALK solution can effectively solve the problem that the probability which was further developed by Li et al. [20]. computational cost of the parametric sensitivity of IM relies This moment-independent importance measure on fail- on small failure probability. The effectiveness of the proposed ureprobabilityisappliedtoquantifytheaverageeffectofthe parametric sensitivity of IMs and efficiency of the ALK basicvariablesonthereliabilityofthemodelandobtainthe method are demonstrated by several examples in Section 4. importance ranking. The IM on the failure probability can be The discussions and conclusions are given at the end ofthis used in the priordesign stage for variables screening when a paper. reliability design solution is yet identified and the postdesign stage for uncertainty reduction after an optimal design has 2. Definition of the Parametric Sensitivity of been determined. Uncertain inputs inherent in most engineering problems are assumed as random variables obeying IM on Failure Probability probabilistic distributions. Obviously, system reliability and 2.1.ReviewoftheImportanceMeasureontheFailureProb- reliability IM on failure probability are decided by distribu- ability. Consider a probabilistic reliability model 𝑌= tion parameters. One can directly change the input’s IMs by 𝑔(𝑋1,𝑋2 ⋅⋅⋅𝑋𝑛),where𝑔(𝑋1,𝑋2 ⋅⋅⋅𝑋𝑛) is the performance controlling or modifying some input’s distribution parame- function, 𝑌 is the model output, and X ={𝑋1,𝑋2 ⋅⋅⋅𝑋𝑛} is ters; namely, changing the input’s distribution parameters can the 𝑛-dimensional vector of random input variables with joint also influence the failure probability, which would facilitate probability density function (PDF) 𝑓X(x).Denote𝑃𝑓 by the its use under various scenarios of design under uncertainty, 𝑌 unconditional failure probability; that is, 𝑃𝑓 =𝑃{𝑔(𝑋)≤0}. for instance, in reliability-based design. It is necessary to fur- 𝑌 When the 𝑖th input variable 𝑋𝑖 is fixed at one given value, the ther recognize effects of the distribution parameters within conditional failure probability 𝑃𝑓 canbeobtained. system reliability on the importance ranking. At present, Cui 𝑌|X𝑖 Based on the idea of the moment-independent impor- et al. [7] defined the parametric sensitivities to illustrate the X influences of the distribution parameters on the importance tance analysis, the importance measure of basic variable 𝑖 measures. on the failure probability is defined by Cui et al. [6]as Combined with the local SA technique of input param- 󵄨 󵄨 1 󵄨 󵄨 eters, the effects of the distribution parameters on the IM 𝜂X = 𝐸X [󵄨𝑃𝑓 −𝑃𝑓 󵄨] 𝑖 2 𝑖 󵄨 𝑌 𝑌|X𝑖 󵄨 onfailureprobabilitycanbeintroduced,bywhichIMs +∞ 󵄨 󵄨 (1) of the inputs can be controlled or modified by changing 1 󵄨 󵄨 = ∫ 󵄨𝑃𝑓 −𝑃𝑓 󵄨 𝑓X (x𝑖)𝑑x𝑖, the distribution parameters. This can provide important 2 −∞ 󵄨 𝑌 𝑌|X𝑖 󵄨 𝑖 guidance for robust design, reliability-based design, and reliability-based optimization in engineering. However, its where X𝑖 represents a random basic variable 𝑋𝑖 or a set of solution still relies on the corresponding method for failure (𝑋 ,𝑋 ⋅⋅⋅𝑋 ) 1≤𝑖 ≤ random basic variables 𝑖1 𝑖2 𝑖𝑔 ,where 1 probability and the computation of the derivative operation 𝑖2 ⋅⋅⋅𝑖𝑔 ≤𝑛. 𝐸[⋅] is the operator of expectation. on failure probability existing in the parametric sensitivity of As the absolute value in (1) is difficult to compute, it IM. is transformed into square operation by Li et al. [20]. The The Monte Carlo simulation (MCS) procedure is easy modified version of importance measures on the failure to implement and is available for computing the parametric probability can be expressed as follows: sensitivity of IM based purely on model evaluation [15, 21], but it has to face the problem of “curse of computational 2 −3 𝑃 𝛿X =𝐸[𝑃𝑓 −𝑃𝑓 ] cost” for the problem with small failure probability (10 – 𝑖 𝑌 𝑌|X𝑖 −4 10 or smaller). Thus, to deal with this problem, the Kriging +∞ 2 (2) approach is widely used for deterministic optimization prob- = ∫ (𝑃𝑓 −𝑃𝑓 ) 𝑓X (x𝑖)𝑑x𝑖. lems [22]andreliabilityanalysis[23]hasbeenintensively −∞ 𝑌 𝑌|X𝑖 𝑖 investigated. Furthermore, an advanced Kriging method, namedactivelearningKriging(ALK),hasbeenprovedto In the reliability analysis, the failure domain of this be highly efficient in reliability analysis problems [24–26]. structure system is defined as This work would employ the ALK method to compute the 𝐹=(x :𝑔(x) <0) . parametric sensitivity of IM on failure probability. In the ALK (3) method, the Kriging model is updated by adding new training points to the design of experiment (DOE) in iterations by Suppose the indicator function of this failure domain is 𝐼 (x) active learning until the Kriging model satisfies necessary given as 𝐹 ;thatis, accuracy. The computational efficiency of the Kriging method can be validated by several numerical and engineering 1𝑔(x) <0, 𝐼𝐹 (x) ={ (4) examples. 0𝑔(x) >0. Mathematical Problems in Engineering 3

Then, the unconditional failure probability and condi- Xi tional failure probability on 𝑋𝑖 canbeexpressedas 𝜃j X𝑖 P f𝑌 𝑃 = ∫ ⋅⋅⋅∫ 𝐼 (x) 𝑓 (x) 𝑑x =𝐸(𝐼 (x)), 𝑓𝑌 𝐹 X 𝐹 (5)

𝜃j ∈ 𝜃j 𝑃 = ∫ ⋅⋅⋅∫ 𝐼 (x) 𝑓 (x |𝑥)𝑑x =𝐸(𝐼 (x |𝑥)) . X𝑘 −X𝑖 𝑓𝑌|𝑋 𝐹 X 𝑖 𝐹 𝑖 (6) Pf 𝑖 𝑌|X𝑖

X ∈ X k −X𝑖 Here, 𝑓X(x |𝑥𝑖) is the conditional joint PDF on 𝑥𝑖. According to the probability theory, 𝑓X(x |𝑥𝑖) is defined as 𝑃 Figure 1: Influence of the distribution parameter on the 𝑓𝑌 and 𝑃𝑓 . 𝑌|X𝑖 𝑓 (x) 𝑓 (x |𝑥)= X , X 𝑖 𝑓 (𝑥 ) (7) 𝑗 𝜃𝑗 ∈ 𝑋𝑖 𝑖 To analyze the effect of changing the th parameter 𝑋𝑘 𝜃𝑗 𝛿𝑃 −𝑋𝑖 of input on X𝑖 , the derivative of the IM can be defined as where 𝑓𝑋 (𝑥𝑖) is the PDF of 𝑥𝑖. 𝑖 𝜕𝛿𝑃 𝑋𝑖 𝑗 𝜕𝜃𝑋 2.2. The Parametric SA of IM on Failure Probability. For the 𝑘 influential distribution parameter, it is significant to identify +∞ 2 how it influences IM on failure probability. We suppose that 𝜕∫ (𝑃𝑓 −𝑃𝑓 ) 𝑓𝑋 (𝑥𝑖)𝑑𝑥𝑖 = −∞ 𝑌 𝑌|𝑋𝑖 𝑖 each input only depends on one distribution parameter in 𝑗 𝜕𝜃𝑋 order to simplify the notation in the following. 𝑘 𝜃𝑗 (𝑖= 1,2⋅⋅⋅𝑛) 𝑗 As stated above, 𝑋𝑖 is the th distribution +∞ 𝑃 𝜕𝑃𝑓 𝑓 𝑋 [ 𝑌 𝑌|𝑋𝑖 ] parameter of input 𝑖 which influences the unconditional = ∫ 2(𝑃𝑓 −𝑃𝑓 )⋅( − ) 𝑓𝑋 (𝑥𝑖)𝑑𝑥𝑖. 𝑌 𝑌|𝑋𝑖 𝑗 𝑗 𝑖 failure probability 𝑃𝑓 , but not the conditional failure prob- −∞ 𝜕𝜃 𝜕𝜃 𝑌 [ 𝑋𝑘 𝑋𝑘 ] 𝑗 𝑗 ability 𝑃𝑓 .Itisalsonoticedthat𝜃 ∈ 𝜃 is not the (9) 𝑌|X𝑖 𝑋𝑘 −𝑋𝑖 distribution parameter of input 𝑋𝑖,butitstillinfluences 𝑃 𝑃𝑓 and 𝑃𝑓 . 𝜃−𝑋 is a vector containing all distribution To compute the above formula, the derivatives of 𝑓𝑌 and 𝑌 𝑌|X𝑖 𝑖 𝑃𝑓 contributionwithrespecttothedistributionparameter parameters of the input variables but 𝑋𝑖.Thecontributions 𝑌|X𝑖 𝑗 of input can be given by of distribution parameter 𝜃 on 𝑃𝑓 and 𝑃𝑓 can be shown 𝑋𝑖 𝑌 𝑌|X𝑖 in Figure 1. 𝜕𝑃𝑓 𝜕𝑓 (x) 1 𝑗 𝑌 = ∫ ⋅⋅⋅∫ 𝐼 (x) X 𝑓 (x) 𝑑x To analyze the effect of changing the th parameter 𝑗 𝐹 𝑗 X 𝑗 𝜕𝜃 𝑅𝑛 𝜕𝜃 𝑓X (x) 𝜃 (𝑖=1,2⋅⋅⋅𝑛) 𝑋 𝑋𝑖 𝑋𝑖 𝑋𝑖 of input 𝑖, the sensitivity derivative of the 𝑃 IM on failure probability 𝛿X canbedefinedas 𝑖 𝐼 (x) 𝜕𝑓 (x) =𝐸( 𝐹 X ), 𝑗 𝑓X (x) 𝜕𝜃 𝑋𝑖 𝜕𝛿𝑃 (10) 𝑋𝑖 𝑗 𝜕𝑃𝑓 𝜕𝑓 (x) 1 𝜕𝜃 𝑌 = ∫ ⋅⋅⋅∫ 𝐼 (x) X 𝑓 (x) 𝑑x 𝑋𝑖 𝑗 𝐹 𝑗 X 𝑅𝑛 𝑓 (x) 𝜕𝜃𝑋 𝜕𝜃𝑋 X +∞ 2 𝑘 𝑘 𝜕∫ (𝑃𝑓 −𝑃𝑓 ) 𝑓𝑋 (𝑥𝑖)𝑑𝑥𝑖 = −∞ 𝑌 𝑌|𝑋𝑖 𝑖 𝐼 (x) 𝜕𝑓 (x) 𝑗 =𝐸( 𝐹 X ), 𝜕𝜃𝑋 𝑗 𝑖 𝑓X (x) 𝜕𝜃 𝑋𝑘 +∞ 𝜕𝑃𝑓 𝑃 [ 𝑌 𝑓𝑌|𝑋 𝜕𝑓X (x |𝑥𝑖) 1 = ∫ 2(𝑃𝑓 −𝑃𝑓 )⋅ 𝑖 = ∫ ⋅⋅⋅∫ 𝐼 (x) 𝑓 (x |𝑥)𝑑x 𝑌 𝑌|𝑋𝑖 𝑗 𝑗 𝐹 𝑗 X 𝑖 −∞ 𝜕𝜃 𝑅𝑛 𝑓 (x |𝑥) 𝑋𝑖 𝜕𝜃 𝜕𝜃 X 𝑖 [ 𝑋𝑘 𝑋𝑘

2 𝜕𝑓𝑋 (𝑥𝑖) 1 𝐼 (x) 𝜕𝑓 (x |𝑥) +(𝑃 −𝑃 ) 𝑖 ] 𝑓 (𝑥 )𝑑𝑥. 𝐹 X 𝑖 𝑓 𝑓 𝑗 𝑋 𝑖 𝑖 =𝐸( ). 𝑌 𝑌|𝑋𝑖 𝑓 (𝑥 ) 𝑖 𝑓 (x |𝑥) 𝑗 𝜕𝜃 𝑋𝑖 𝑖 X 𝑖 𝜕𝜃 𝑋𝑖 ] 𝑋𝑘 (8) (11) 4 Mathematical Problems in Engineering

Here, the effects of the distribution parameters on the IMs model interpolates exactly. This part shows the following of input variables can be known, combining local and global statistical characteristics: sensitivity analysis technology. One can directly optimize 𝐸 [𝑧 (𝑥)] =0, (13) the inputs’ IMs by controlling or changing some inputs’ 2 distribution parameters. It can provide useful information for Var [𝑧 (𝑥)] =𝜎, (14) robust design, reliability-based design, and reliability-based optimization. 2 Cov [𝑍 𝑖(𝑥 ),𝑍(𝑥𝑗)] = 𝜎 [𝑅 𝑖(𝑥 ,𝑥𝑗)] . (15) In computing the parametric sensitivity of IM, the deriva- 2 tive of the 𝑃𝑓 and 𝑃𝑓 with respect to the distribution 𝜎 𝑌 𝑌|X𝑖 The mean of this component is zero, the variance is , parameters must be calculated. The performance function and (14) defines its covariance between two points in space. expression is characterized by complex implicit limit state In (15), 𝑥𝑖 and 𝑥𝑗 denote two arbitrary points; 𝑅(𝑥𝑖,𝑥𝑗) is the functions in most engineering problems, so the derivative of correlation function of 𝑥𝑖 and 𝑥𝑗. 𝑅(𝑥 ,𝑥 ) the 𝑃𝑓 and 𝑃𝑓 is without a definite analytic expression. Several types of correlation function 𝑖 𝑗 can be 𝑌 𝑌|X𝑖 The Monte Carlo simulation (MCS) method can be used used,and,inthisstudy,theGaussianfunctioncanbe for computing the parametric sensitivity of IM. This method formulated as 𝑛 has the advantages of high-accuracy and usability based dv 󵄨 󵄨2 𝑅 (𝑥 ,𝑥 ) = (−∑𝜃 󵄨𝑥𝑖 −𝑥𝑗 󵄨 ) , on model evaluation. However, for problems with small 𝑖 𝑗 EXP 𝑘 󵄨 𝑘 𝑘󵄨 (16) −3 −4 failure probability (10 –10 or smaller), only a very small 𝑘=1 portion of samples will drop into the failure domain. The 𝑛 𝑥𝑖 where dv is the dimension of random input variables, 𝑘 and computational cost is unacceptable for failure probability and 𝑗 𝑥𝑘 denote the 𝑘th component of training samples 𝑥𝑖 and 𝑥𝑗, the derivative of the 𝑃𝑓 and 𝑃𝑓 .Forthelatterproblem,an 𝑌 𝑌|X𝑖 respectively, and 𝜃𝑘 is the correlation parameter to ensure active learning Kriging method combining MCS is proposed that the metamodel is flexible enough to approximate the in the next section. true function, which is usually obtained by an optimization process. Asetof𝑛𝑠 experimental samples is denoted by x = 3. The Active Learning Kriging [x , x ,...,x ]𝑇 x 𝑖 1 2 𝑛𝑠 ,where 𝑖 is the th training point, and Method for the Parametric Sensitivity of 𝑇 responses Y =[𝑦1,𝑦2,...,𝑦𝑛 ] is the corresponding IM on Failure Probability 𝑠 response to x.Thepredictedvalue𝜇𝐺(x) and predicted 𝜎2 (x) 𝐺(x) Section 2 tellsusthatthekeypointinestimatingtheglobal variance 𝐺 of the objective function at an unknown x reliability sensitivity indices is to provide the failure proba- point are bility 𝑃𝑓 , the unconditional failure probability 𝑃𝑓 ,andthe 𝑇 ̂ 𝑇 −1 ̂ 𝑌 𝑌|X𝑖 𝜇𝐺 (x) =𝑓 𝛽+𝑟 (x) 𝑅 (𝑌 − 𝐹𝛽) . (17) derivative of 𝑃𝑓 and 𝑃𝑓 to the distribution parameters. In 𝑌 𝑌|X𝑖 𝑅 𝑚×𝑚 𝑅 = practice, there are many engineering problems characterized In (17), denotes a correlation matrix by 𝑖,𝑗 𝑅(𝑥 ,𝑥 ) 𝑟𝑇(x)=[𝑅(x, x ), 𝑅(x, x ),...,𝑅(x, x )]𝑇 by complex implicit performance functions, so the compu- 𝑖 𝑗 ; 1 2 𝑛𝑠 is the tational cost of double-loop MCS method becomes larger correlative relations between unknown point x and sample x =[x , x ,...,x ]𝑇 𝛽̂ especially for finite element (FE) model. To solve this prob- points 1 2 𝑛𝑠 ; the unknown parameters can lem, the solution of ALK method is employed to approximate be expressed as the implicit performance function and improve the efficiency −1 𝛽̂ = (𝐹𝑇𝑅−1𝐹) 𝐹𝑇𝑅−1𝑌. greatly using few training points. Generally speaking, Kriging (18) metamodel is combined by a linear regression component 2 The Kriging variance 𝜎 (x) is defined as the minimum and a stochastic process. It is an interpolation technique 𝐺 square error between true response 𝐺(x) and estimated value basedonthestatisticaltheoryandhasbeenusedtoconstruct 𝐺(̂ x) the input and output (I/O) systems. The Kriging models are and is expressed as follows: 2 2 𝑇 𝑇 −1 −1 𝑇 −1 performed with the toolbox DACE, which is a MATLAB 𝜎 (x) =𝜎 (1 + 𝑢 (𝐹 𝑅 𝐹) 𝑢−𝑟 (x) 𝑅 𝑟 (x)), (19) toolbox developed by Lophaven et al. [27]. 𝐺 𝑧 𝑇 −1 2 where 𝑢=𝐹𝑅 𝑟(x)−𝑓; the unknown parameters 𝜎𝑧 can 3.1. Kriging Metamodel. The present Kriging model expresses be expressed as 𝐺(x) 𝑇 the unknown function as [22] (𝑌 −𝛽) 1 ̂ 𝑅−1 (𝑌 −𝛽) 1 ̂ 2 𝜎𝑧 = . (20) 𝑛𝑠 𝐺 (x) =𝐹(x, 𝛽) +𝑧(x) , (12) However, at a given unknown point X,thepredicted value 𝜇𝐺(X) is not the true value of 𝐺(X) and there exist 2 where 𝐹(x,𝛽) is the deterministic part which gives an some uncertainties. 𝜎𝐺(x) demonstrates uncertainty of the approximation of the response in mean. The second part predictor value; it also provides an important index to 𝑧(𝑥) is the realization of stochastic process, which provides adjudge the fitting accuracy and enables quantifying the the approximation of the local fluctuation so that the whole uncertainty of predictions with an easy approach. Mathematical Problems in Engineering 5

3.2. The ALK Method and the Solution for the Parametric P 𝛿𝑁𝑃 Sensitivity of IM 𝑋𝑖 . The ALK method has been applied to different fields in engineering, like efficient global optimization (EGO) [22], probabilistic analysis (PRA) [24–26], and reliability-based design optimization [28]. In the ALK R method, the learning function plays an important role in constructing the active learning Kriging model [26]. A new Kriging model is updated by adding a new training point to the design of experiment (DOE) in subsequent iterations by active learning until the Kriging model satisfies necessary accuracy. According to the indicator function of this failure 𝜇 0 G domain in (5), we only need to focus on the sign of the G performancefunctioninthereliabilityanalysis. (a) For the uncertainty of predicted value 𝜇𝐺(X),theremay P exist some risk that the value of 𝐺(X) is positive (𝐺(X)>0) even if predicted value is negative (𝜇𝐺(X)<0). So, the points owing a high potential risk to cross the predicted separator 𝐺(X)=0have to be added to the DOE and evaluated by the real performance function. These potentially “dangerous” R points determining the precision of the failure probability are in the region: close to the limit state, have high Kriging variance or both. To identify them, a new learning function named the expected risk function (ERF) is proposed in our recent work [29]. 0 G First, if 𝜇𝐺(X)<0, we define an indicator to measure such 𝜇G potential risk as (b) 𝑅 (X) = [(𝐺 (X) −0) ,0] . max (21) Figure 2: Risk of the sign of 𝐺(X) to be wrongly predicted in a Kriging model: (a) the sign of 𝐺(X) is negative; (b) the sign of 𝐺(X) As shown in Figure 2(a), 𝑅(X) measures the shift of value is positive. of 𝐺(X) tobepositiveandthelargerthevalueof𝑅(X) is, the more risky the sign of 𝐺(X) is to be changed from negative to positive. Covering the range (𝐺(X)>0)tocalculatethemeanof The ERF is employed to measure the potential possibility X 𝑅(X),theERFforthecase𝜇𝐺(X)<0is obtained as follows: that the sign of the limit state function in a point is changed from positive to negative (or negative to positive) in the 𝐸 (𝑅 (X)) =𝐸(max [(𝐺 (X) −0) ,0]) Kriging model. The point maximizing ERF should be added to the initial set of training points in DOE. +∞ 𝐺−𝜇 = ∫ 𝐺𝜙 ( 𝐺 )𝑑𝐺 TheALK-basedsolutionfortheparametricsensitivitycan 0 𝜎𝐺 (22) be simply divided into five steps and provided as follows. 𝜇 𝜇 𝐺 𝐺 𝑦=𝑔(x) 𝑁 x𝑡 = =𝜎𝐺𝜙( )+𝜇𝐺Φ( ), Step 1. For a model , generate samples 𝜎𝐺 𝜎𝐺 (𝑥1𝑡,𝑥2𝑡 ⋅⋅⋅𝑥𝑛𝑡)(𝑡= 1,2⋅⋅⋅𝑁)of the input variables by PDF 𝑓 (x) 𝜇 𝜎 X using Sobol’s low-discrepancy samples. One can refer where 𝐺 and 𝐺 arethepredictedvalueandvariancein(17) 𝑛×𝑁 X Φ(⋅) 𝜙(⋅) to Sobol [15] for more details. A dimension matrix is and (19) and and are the cumulative distribution formed and shown as function (CDF) and PDF of the standard normal distribution. If 𝜇𝐺(X)>0,asshowninFigure 2(b), we define an 𝑥1,1 ⋅⋅⋅ 𝑥𝑛,1 indicator to measure such potential risk as 𝑇 [ . . ] X =(x1, x2,...,x𝑁) = [ . d . ] , (25) 𝑅 (X) = max [(0−𝐺(X)) ,0] . (23) [𝑥1,𝑁 ⋅⋅⋅ 𝑥𝑛,𝑁]

And the ERF for the case 𝜇𝐺(X)>0is obtained as follows: where the 𝑖th column of matrix X represents the generated random realizations for input variable 𝑥𝑖. 𝐸 (𝑅 (X)) =𝐸(max [(0−𝐺(X)) ,0]) +∞ Step 2. Construct an active learning Kriging model as fol- 𝐺+𝜇𝐺 = ∫ 𝐺𝜙 ( )𝑑𝐺 lows. 0 𝜎𝐺 (24) x 𝜇 𝜇 (a) The samples 𝑡 in Step 1 are set as candidate points. =𝜎 𝜙( 𝐺 )−𝜇 Φ( 𝐺 ). Randomly choose some training points in the sam- 𝐺 𝜎 𝐺 𝜎 𝐺 𝐺 ples x𝑡 and evaluate the corresponding performance 6 Mathematical Problems in Engineering

𝛿푃 function. In the initial step, only a dozen of points in Table 1: Computational results of the importance measure 푥푖 in the DOE are enough according to our experience. Example 1.

푃 Importance measure 𝛿푥 The numbers of (b)ConstructtheKrigingmodel,computetheERFof Method 푖 𝑥 𝑥 function evaluations candidate points by (22) and (24), and judge whether 1 2 4 4 it is smaller than the given tolerance. If so, the active MC 0.00492 0.00537 2×10 ×10 learning process can stop and turn to Step 3 directly. ALK 0.00476 0.00548 12 + 36 If the stopping condition is not satisfied, it should go Error 0.0033 0.0052 to Step 2(c) to add new training point.

(c) Add the point with max value of ERF to the DOE and 4. Examples loop back to Step 2(b). 4.1. Numerical Examples Step 3. BasedontheactivelearningKrigingmodel,the corresponding 𝑁 values of Kriging predictions y푡 = Example 1. The mathematical problem in Example 1 is mod- 𝜇퐺(x푡) (𝑡 = 1,2⋅⋅⋅𝑁) are obtained. Compute the values ified from25 [ ], which behaves nonlinearly around the limit 푡 I퐹 (𝑡 = 1,2⋅⋅⋅𝑁) of the indicator function of this failure state function. The nonlinear performance function is given 𝑃 =∑푁 𝐼푡 /𝑁 as domain and 푓푌 푡=1 퐹 .Usingthesameactivelearning 𝜕𝑃 /𝜕𝜃푗 𝜕𝑃 /𝜕𝜃푗 2 Kriging model, 푓푌 푋 and 푓푌 푋 can be computed 5𝑥 (𝑥 +4)(𝑥 −1) 푖 푘 𝑔(𝑥 ,𝑥 )= ( 1 )− 1 2 +2, (27) combining the reliability sensitivity analysis method by (10). 1 2 sin 2 20 (𝑖, 𝑡) 𝑥 X Step 4. By fixing the th component ( 푖,푡)of , generate where the basic variables 𝑥1 and 𝑥2 are independent and 𝑀 k푗 =(𝑥,...,𝑥 ,𝑥 ,𝑥 ⋅⋅⋅𝑥 )(𝑗 = 𝑥 ∼ 𝑁(1.5, 1) 𝑥 ∼ samples 푥푖푡 1푗 푖−1,푗 푖푡 푖+1,푗 푛푗 follow normal distributions 1 and 2 1,2⋅⋅⋅𝑀)according to the conditional PDF 𝑓X(x |𝑥푖푡),and 𝑁(2.5, 1),respectively. 𝑀 y푗 = the corresponding values of Kriging predictions 푥푖,푡 푗 The estimates of the importance measure on the failure 𝜇퐺(k푥 )(𝑗 = 1,2⋅⋅⋅𝑀) can be obtained based on the same 푃 푖,푡 probability 𝛿푋 computedbytheMCSandALKprocedures 𝑀 y푗 (𝑗 = 푖 active learning Kriging model. With these samples 푥푖푡 and the error in each estimate are reported in Table 1.The 1,2⋅⋅⋅𝑀) 𝑃 𝑥 ,the 푓푌|푥푖푡 of conditional on 푖푡 can be obtained as numbers in the last column are the evaluation numbers of 𝛿푃 the performance function when calculating 푋푖 .Theresults of their parametric sensitivities are computed using the same ∑푀 𝐼푗 푗=1 퐹|푥푖푡 active learning Kriging model and listed in Table 2. 𝑃푓 |푥 = . (26) 푌 푖푡 𝑀 As revealed by Tables 1 and 2,theresultscomputed bytheALKprocedurearepreciseenoughcomparedwith

푗 those computed by the MCS procedure. The results of MCS At the same time, 𝑃푓 /𝜕𝜃푋 canbecomputedcombiningthe procedurecanbeseenastheaccurateresults.Comparedwith 푌|푋푖 푘 4 4 reliability sensitivity analysis method by (11). the tremendously large computational cost (2×10 ×10 samples) of Monte Carlo method, the ALK method only 𝛿푃 begins with 12 training points, while 36 points whose sign of Step 5. The IM 푋푖 canbecalculatedaccordingtothedefini- 푗 푗 푗 response has the largest potential risk to be wrongly predicted tion of (2); 𝑃푓 , 𝑃푓 |푥 , 𝜕𝑃푓 /𝜕𝜃푋 , 𝜕𝑃푓 /𝜕𝜃푋 ,and𝑃푓 /𝜕𝜃푋 푌 푌 푖푡 푌 푖 푌 푘 푌|푋푖 푘 are added into the initial DOE. Additionally, Figure 3 gives have been computed in Steps 3 and 4.Substitutingthem iteration history of the true values of the 36 training points 𝛿푁푃 into (8) or (9), the parametric sensitivity of IM 푋푖 can be for Example 1 bytheALKmethod.Thesignofthecandidate computed. points (the samples in MCS procedure) is estimated using the ALK model. Compared with the true sign of the candidate Itcanbeseenthatalargenumberofsamplesmustbe 푃 points using originality performance function, there are only taken for providing precise estimates when calculating IM 𝛿 8 candidate points whose signs of responses are wrongly and its parametric sensitivity with Monte Carlo method. In predicted. This can be linked to Figure 4 in which it shows the the second step of ALK procedure, one only needs to compute sign of the response at each Monte Carlo sampling point. It is the limit state function values using the metamodel instead of seen that most of the added points are located in the vicinity original model; thus, this procedure can further improve the ofthelimitstatefunction.Consequently,thismethodcertifies efficiency without loss of precision, especially for the finite a correct approximation of the response using a minimum element model (FEM). number of calls to the performance function. In the next section, we introduce two numerical exam- From the results in Table 1, it is noted that there is no ples and two engineering examples for demonstrating the difference between the rankings of the basic variables on efficiency and precision of the calculation procedure and failureprobabilitybytheMCSandALKprocedures,namely 푃 illustrating the engineering significance of the IM 𝛿 and its 𝑥2 >𝑥1, which illustrates that the ALK method is meaningful parametric sensitivity. and reasonable. Therefore, 𝑥2 should be paid more attention Mathematical Problems in Engineering 7

𝛿푃 Table 2: The parametric sensitivities of the importance measure 푥푖 in Example 1. 𝜇 𝜎 𝜇 𝜎 Parametric sensitivity of IM Method 푥1 푥1 푥2 푥2 𝜕𝛿푃 푥1 MC 0.00821 0.0179 0.00940 0.00610 𝜕𝜃 ALK 0.00858 0.0183 0.00947 0.00624 푥푖 𝜕𝛿푃 − −4 − −4 푥2 MC 0.00741 0.00301 4.594e 3.576e 𝜕𝜃 − −4 − −4 푥푖 ALK 0.00752 0.00293 4.648e 3.470e

3 𝛿푃 Table 3: Computational results of the importance measure 푥푖 in Ishigami function.

2 푃 Importance measure 𝛿 Method 푥푖 𝑥 𝑥 𝑥 1 1 2 3 4 3 MC (6×10 ×10 ) 0.0474 0.0263 0.0107 4 3 at training point training at 0 ALK (6×10 ×10 ) 0.0475 0.0261 0.0110 G Error 0.0053 0.0076 0.028 −1 10 0 −2 0

The true value of The true value 8

−3 0 5 10 15 20 25 30 35 40 45 50 6 0 The training point iteration 4

G(X)=0 2 Initial training points 0 X Added training points 2

Figure 3: Iteration history of the true value of added training points 0 for Example 1 by ALK. −2 to in the reliability analysis on failure probability. Results −4 −4 −3 −2 −1 0123456 shown in Table 2 point out the influences of varying some variables’ distribution parameters to the IMs on the failure X1 𝛿푃 probability. The uncertainty of IMs 푥푖 on failure probability G(X)=0 Initial training points canbeimprovedthroughmodifyingorcontrollingthe G(X)<0 Added training points distribution parameters of the variables indirectly. G(X)>0 The points wrongly predicted

Example 2 (Ishigami Function). Ishigami function [30]isa Figure 4: Sign of the response at each sample predicted for commonly used example in importance measure analysis [6, Example 1 by ALK. 20].Itcanbewrittenasfollows: 2 4 𝑔=sin 𝑥1 +𝑎sin 𝑥2 +𝑏𝑥3 sin 𝑥1, (28) the training points for Ishigami function computed by ALK method in Figure 5. This validates the high efficiency of where the basic variables 𝑥1, 𝑥2,and𝑥3 are independent and the ALK method. The importance ranking of the basic [𝑥퐿,𝑥푈] (𝑖 = 1, 2, 3) 𝑥퐿 =−𝜋 푃 uniformly distributed in 푖 푖 , 푖 , variables on failure probability obtained by 𝛿푋 is 𝑥1, 𝑥2,and 푈 푖 and 𝑥푖 =𝜋. The values of constants are set as 𝑎=5and 𝑥3. Therefore, 𝑥1 should be paid more attention to in the 𝑏 = 0.1. The estimates of the importance measure on failure reliability analysis on failure probability. Results shown in 𝛿푃 probability 푋푖 and the results of their parametric sensitivities Table 4 point out the influences of varying some variables’ arelistedinTables3 and 4,respectively. distribution parameters to the IMs on the failure probability. From Examples 1 and 2,itcanbefoundthattheimpor- As revealed by Tables 3 and 4,thecomputationalcost tance of a variable to failure probability is not just affected 8 of Monte Carlo method (3×10 samples) is tremendously by the distribution parameters of this variable but affected large, which illustrates that our model is computationally by those of the rest of input variables, and sometimes the challenging. The ALK method begins with 12 training points, latter may even play a more important role than the former. while 44 points are added into the initial DOE to satisfy Surely, the engineering examples in Sections 4.2 and 4.3 the accuracy. We plot the tendency of the true values of are also following the fact. Accordingly, it is necessary to 8 Mathematical Problems in Engineering

𝛿푃 Table 4: The parametric sensitivities of the importance measure 푥푖 in Ishigami function.

퐿 푈 퐿 푈 퐿 푈 Parametric sensitivity of IM Method 𝑥1 𝑥1 𝑥2 𝑥2 𝑥3 𝑥3 𝜕𝛿푃 − −3 −3 −2 − −2 −2 − −2 푥1 MC 7. 8 2 6 e 7. 8 2 6 e 1.513e 1.513e 1.513e 1.513e −3 −3 −2 −2 −2 −2 𝜕𝜃 ALK −7. 7 52 e 7. 7 52 e 1.537e −1.537e 1.537e −1.537e 푥푖 𝜕𝛿푃 −3 − −3 − −3 −3 −3 − −3 푥2 MC 8.422e 8.422e 4.402e 4.402e 8.422e 8.422e −3 −3 −3 −3 −3 −3 𝜕𝜃 ALK 8.418e −8.418e −4.398e 4.398e 8.418e −8.418e 푥푖 𝜕𝛿푃 −3 − −3 −3 − −3 − −3 −3 푥3 MC 3.425e 3.425e 3.425e 3.425e 1.896e 1.896e −3 −3 −3 −3 −3 −3 𝜕𝜃 ALK 3.479e −3.479e 3.480e −3.479e −1.914e −1.914e 푥푖

Table 5: Statistical properties of random variables for roof truss.

2 2 Random variable 𝑞 (N/m) 𝑙 (m) 𝐴푆 (m ) 𝐴퐶 (m ) 𝐸푆 (Pa) 𝐸퐶 (Pa) −4 11 10 Mean 𝜇푥 20000 12 9.82 × 10 0.04 1 × 10 1.2 × 10

Coefficient of variation Cov푥 0.07 0.01 0.06 0.12 0.06 0.06

7 tension bars are all made of steel. Assume the truss bears 𝑞 6 uniformly distributed load , which can be transformed into nodal load 𝑃=𝑞𝑙/4. The perpendicular deflection of truss 5 peak node C Δ 퐶 canbecalculatedusingtheknowledgeof 4 structural mechanics and 2 3 𝑞𝑙 3.81 1.13 at training point training at Δ = ( + ) ,

G 퐶 (29) 2 2 𝐴퐶𝐸퐶 𝐴푆𝐸푆

1 where 𝐴퐶 and 𝐴푆 are the cross-sectional areas of the 𝐸 𝐸 0 reinforced concrete and steel bars, respectively, 퐶 and 푆 are the corresponding elastic moduli of reinforced concrete and The true value of The true value −1 steel, and 𝑙 is the length of the truss as Figure 6 shows. The −2 distribution parameters of the independent normal random 010203040 50 60 basic variables are given in Table 5. The training point iteration Considering the safety of the truss, the perpendicular deflection Δ 퐶 should satisfy the constraint Δ 퐶 ≤0.03m. Initial training point G(X)=0 Added training point Hence, the structural performance function can be given as follows: Figure 5: Iteration history of the true value of added training points 𝑞𝑙2 3.81 1.13 for Ishigami function by ALK. 𝑔 (𝑥) = 0.03 − Δ 퐶 =0.03− ( + ). (30) 2 𝐴퐶𝐸퐶 𝐴푆𝐸푆

For this highly nonlinear example, the computational find out all the factors that may affect the importance results of the importance measure on failure probability measures according to the failure probability, and parametric measures by ALK and MCS are listed in Tables 6 and 7. sensitivities on IMs provide a way to solve this problem. In addition, the IMs on the failure probability by the state Except for that, Table 4 also shows an important property dependent parameter (SDP) method in literature [20]arealso as all the variables of the model are uniform variables; namely, listed for comparison in Table 6.TheMCSproceduretakes 7 the numerical values of two parametric sensitivities of the IM 6×10 samples. The SDP method needs only 1024 model are equal, and their signs are opposite. This property result of runs for calculating the importance measures. However, the the parametric sensitivity of the IM is decided by (8) and (9). Kriging method needs to call the performance function only For the uniform variable, the signs of (1/𝑓X(x))(𝜕𝑓X(x)/𝜕𝜃푋 ) 푖 72 times. The ALK method begins with 12 training points, are opposite. and 60 points are added into the initial DOE using ALK. The tendencyofthetruevaluesofthetrainingpointsforroof 4.2. Roof Truss. In order to test the applicability of the truss by ALK method is plotted in Figure 7.Comparedwith proposed method for problems with more random variables the SDP method, the proposed method has obtained a more and expressed in a more engineering way, the roof truss accurate result and lower computational cost. is selected as example. The truss is simply illustrated as in Additionally, it can be seen from Table 6 that the effects Figure 6. The top chord and the compression bars of the truss of the basic variable of 𝑞 on the failure probability are notable are reinforced by concrete and the bottom chord and the and the influences of basic variables sectional area 𝐴퐶 and 𝐴푆 Mathematical Problems in Engineering 9

𝛿푃 Table 6: Computational results of the importance measure 푥푖 for roof truss.

푃 Global reliability sensitivity indices 𝛿 Method 푥푖 𝑞𝑙𝐴퐶 𝐴푆 𝐸퐶 𝐸푆 4 3 MC (6×10 ×10 ) 0.05714 0.00483 0.02431 0.01148 0.00664 0.01329 ALK (69) 0.05684 0.00471 0.02425 0.01151 0.00665 0.01328 SDP (1024) 0.05044 0.00436 0.02749 0.01334 0.00691 0.01231

D F

A B E G

q N/m

(a) P

P C P Ac Ac l D 12 A c 0.75A F Ac 0.75 A s As c Ac l

A 12 B 3A 2A s E s G 3As

0.278l 0.222l 0.25l 0.25l

(b)

Figure 6: Schematic diagram of a roof truss.

on the failure probability are relatively smaller, whereas elastic 4.3. A Planar 10-Bar Structure. A planar 10-bar structure moduli 𝐸푆 and 𝐸퐶 and the length 𝑙 are the least influential shown in Figure 8 is investigated. The horizontal bars have ones which can attract less attention. In engineering, param- the same length 𝐿=1m. The diagonal bars and the vertical eters of the load 𝑞 and the length 𝑙 are more easily controlled members have the same length √2𝐿.Thecross-sectional 2 or modified than other inputs, so the paper especially pays area of 10 bars is denoted by 𝐴 = 0.001 m .Theelastic 𝛿푃 attention to their parametric sensitivities of IM 푋푖 . modulus of all bars is 𝐸 = 100 GPa. 𝑃1 =80KN and Due to space limitation, Table 7 lists the parametric 𝑃2 =10KN are the external loads subjected to joints 4 and sensitivities of basic variables IM on the failure probabil- 2, respectively. Joint 2 is also subjected to a horizontal load 𝛿푃 ity. Taking IM 푞 for example, the distribution param- 𝑃3 =10KN. The input variables are all normally distributed, eters of the load 𝑞 are relatively higher than those of and their coefficient of variance is 0.05. We assume the the length 𝑙. Thus, it can be seen that changing those displacement of node 3 in vertical direction not exceeding parameters with high parametric sensitivity has more influ- 3 mm as the constrain condition. The limit state function can ences on the IM of the corresponding variables than other be constructed, 𝑔=3mm −|𝑌2|,where𝑌2 is an implicit 𝛿푃 parameters. The parametric sensitivity of 푋푖 is valuable function of the basic random variables. As shown in Figure 9, in the reliability engineering because it can provide indi- the finite element model can be obtained in Ansys 11.0. The 푃 rect information for reliability design and reliability-based results of the importance measures 𝛿 of inputs are listed in optimization. Table 8. To identify the influential distribution parameters, 10 Mathematical Problems in Engineering

𝛿푃 Table 7: The parametric sensitivities of the importance measure 푥푖 for roof truss. 𝑙𝑞 Parametric sensitivity of IM Method 𝜇푙 𝜎푙 𝜇푞 𝜎푞 푃 −6 − −6 −5 −5 𝜕𝛿푞 MC 2.361e 1.369e 2.241e 5.221e −6 −6 −5 −5 𝜕𝜃 ALK 2.384e −1.378e 2.256e 5.240e 푥푖 𝜕𝛿푃 − − − − 퐴퐶 MC 0.432 0.285 4.947 4.053 𝜕𝜃 ALK −0.428 −0.271 −4.956 −4.057 푥푖 𝜕𝛿푃 − − − 2 − 2 퐴푆 MC 28.546 7. 8 02 3.235e 1.324e 𝜕𝜃 ALK −27.611 −8.096 −3.312e2 −1.387e2 푥푖

𝛿푃 Table 8: Computational results of the importance measure 푥푖 in 10-bar structure.

푃 Global reliability sensitivity indices 𝛿 Method 푥푖 𝐴𝐸1 𝑙𝑃 𝑃2 𝑃3 4 3 −4 MC (6×10 ×10 ) 0.0449 0.0382 0.0401 0.0242 0.00282 2.249e −4 ALK (69) 0.0443 0.0371 0.0396 0.0245 0.00265 2.411e

5 impressively and ensure acceptable accuracy for the implicit 4 finite element model. As revealed in Table 8, the ranking of IM not to exceed 푃 3 as 3 mm the constrain condition 𝛿 is as follows: the basic variables of 𝐴, 𝑙,and𝐸 on the failure probability are notable; 2 the influences of basic variable 𝑃1 on the failure probability

at training point training at 𝑃 𝑃 1 are less; and the influences of basic variables 2 and 3 on G the failure probability are very small, which are even near 0 zeros. From Table 9, the distribution parameters 𝜇퐴, 𝜎퐴, 𝜇푙,and𝜎푙 are the most influential ones on the IMs. Thus, −1 itcanbeseenthatchangingthoseparameterswithhigh parametric sensitivities has more influences on the IM of the

The true value of The true value −2 corresponding variable than other parameters. In the sight −3 of this structural design, we need to pay more attention to 0 1020304050607080 the distribution parameters with high parametric sensitivity. The training point of iteration In order to obtain the IM on failure probability results, Initial training point G(X )=0 especially for the high ranking IM, the important distribution Added training point parameters to them must be given precisely. To do this, it is necessary to collect the information and improve the Figure 7: Iteration history of the true value of added training points understanding of the distribution parameters. for a roof truss by ALK.

5. Conclusions the computational results of parametric sensitivity are shown This paper investigates the influence of the distribution in Table 9. parameters on the IM on failure probability. It is noted From Tables 8 and 9, it can be seen that the computational that the IM of basic variable not only is influenced by results of Monte Carlo method and ALK method are in its distribution parameters but also is influenced by other 7 good agreement. The MCS procedure takes 6×10 samples basic variables’ distribution parameters. By further devel- anditsresultscanbeseenastheaccurateresults,butthe oping the presented moment-independent IM on failure computational cost of Monte Carlo method is tremendously probability, the parametric sensitivity of IM is first presented large especially for FEM. The ALK method only needs to according to the derivative theory; thus, how the influential call the finite element model 93 times to satisfy the accuracy. distribution parameters influence the influential IM can be The efficient Kriging method begins to get convergence at12 made clear. Meanwhile, we can decrease the variability of training points, and 81 points are added into the initial DOE the IM on failure probability by collecting the information using ALK. The true values of the added training points for and improving the understanding of those most influential 𝛿푃 roof truss by the ALK method are shown in Figure 10.The parameters. The parametric sensitivity of 푋푖 is valuable ALK method can also improve the computational efficiency in reliability engineering because it can provide direct Mathematical Problems in Engineering 11

𝛿푃 Table 9: The parametric sensitivities of the importance measure 푥푖 in 10-bar structure. 푃 푃 푃 𝜕𝛿퐴 𝜕𝛿퐸 𝜕𝛿푙 𝜃푥 𝜕𝜃 𝜕𝜃 𝜕𝜃 Basic variable 푖 푥푖 푥푖 푥푖 MC ALK MC ALK MC ALK 𝜇 − − 𝐴 푙 236.763 240.579 19.812 19.820 17.774 17.756 𝜎푙 1235.778 1232.543 −1.637 −1.625 −1.471 −1.457 𝜇 − −2 − −2 𝑙 푙 0.230 0.241 0.223 0.220 1.759e 1.788e −2 −2 𝜎푙 −0.408 −0.412 −0.384 −0.397 −1.956e −2.012e 𝜇 − −10 − −10 −11 −11 − −12 − −12 𝐸 푙 1.843e 2.316e 1.477e 1.546e 2.304e 2.277e −10 −10 −13 −13 −12 −12 𝜎푙 −4.231e −6.186e −3.714e −4.827e −4.034e −5.259e 𝜇 −5 −5 −6 −6 −6 −6 𝐹 푙 2.186e 2.492e 2.127e 2.111e 2.620e 2.581e 1 −5 −5 −5 −5 −6 −6 𝜎푙 −3.262e −3.641e −3.208e −3.496e −3.207e −3.834e 𝜇 −6 −6 −6 −6 −6 −6 𝐹 푙 6.210e 7. 4 3 4 e 5.812e 5.792e 7. 4 2 0 e 6.834e 2 −5 −5 −5 −5 −5 −5 𝜎푙 −3.174e −3.412e −3.986e −3.796e −3.528e −3.419e 𝜇 − −6 − −6 − −6 − −6 − −6 − −6 𝐹 푙 1.641e 1.624e 1.574e 1.657e 1.871e 1.862e 3 −7 −7 −6 −6 −7 −7 𝜎푙 −2.631e −2.942e −1.876e −1.748e 2.576e 2.124e

5 (1) 3 (2) 1

(10) (7)

(5) (6)

(9) (8)

P 6 (3) 4 (4) 2 3

P1 P2

Figure 8: Planar 10-bar structure.

1 The computation of the IM on failure probability and its Nodes September 28, 2014 19:42:27 parametric sensitivity is often feasible by the MCS, but the 4 56computational cost of MCS method is tremendously large −3 −4 with small failure probability (10 –10 or smaller). For dealing with this problem, the ALK method is employed to calculate the IM and its parametric sensitivity. It can be seen by the numerical and engineering examples that the ALK method is more efficient than MC method. To ensure the computational accuracy, the large number of training points used in the traditional Kriging method is essential. Thanks to Y the existence of active learning process, the points which may Z X 2 3 greatly affect the metamodel’s fitting accuracy can be precisely selected, which can make the Kriging metamodel more Figure 9: The finite element model of the planar 10-bar structure. accurate and the additional computational cost is acceptable. It is noticed that a small quantity of points in the inter- estingregionareaddedtoconstructtheKrigingpredictor and useful information for reliability design and reliability- model until the Kriging model satisfies necessary accuracy. based optimization. The computational results of several examples demonstrate 12 Mathematical Problems in Engineering

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Research Article Gear Crack Level Classification Based on EMD and EDT

Haiping Li,1 Jianmin Zhao,1 Xinghui Zhang,1 and Hongzhi Teng1,2

1 The Sixth Department, Mechanical Engineering College, No. 97 Heping West Road, Xinhua District, Shijiazhuang, Hebei 050003, China 2 Lanzhou Maintenance Centre, No. 27 Fanjiaping Road, Xigu District, Lanzhou, Gansu 730060, China

Correspondence should be addressed to Jianmin Zhao; jm [email protected]

Received 3 July 2014; Accepted 27 October 2014

Academic Editor: Wenbin Wang

Copyright © 2015 Haiping Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Gears are the most essential parts in rotating machinery. Crack fault is one of damage modes most frequently occurring in gears. So, this paper deals with the problem of different crack levels classification. The proposed method is mainly based on empirical mode decomposition (EMD) and Euclidean distance technique (EDT). First, vibration signal acquired by accelerometer is processed by EMD and intrinsic mode functions (IMFs) are obtained. Then, a correlation coefficient based method is proposed to select the sensitive IMFs which contain main gear fault information. And energy of these IMFs is chosen as the fault feature by comparing with kurtosis and skewness. Finally, Euclidean distances between test sample and four classes trained samples are calculated, and on this basis, fault level classification of the test sample can be made. The proposed approach is tested and validated through a gearbox experiment, in which four crack levels and three kinds of loads are utilized. The results show that the proposed method has high accuracy rates in classifying different crack levels and may be adaptive to different conditions.

1. Introduction separated the vibration signal of planet and sun gears using time domain averaging. Halim et al. [7]combinedtime Gearboxes are one of the fundamental and important compo- synchronous average and wavelet transformation together nents of rotating machinery. Its function is to transfer torque to extract periodic waveforms at different scales from noisy and power from one shaft to another. Representative appli- vibration signals to clean up noise and detect both local and cations involve motorcars, helicopters, and steel mills. Their distributed faults simultaneously. Feng et al. [8]proposed failures will lead to great power loss and high maintenance a regularization dimension technique to make vibration fee. Therefore, condition monitoring and fault diagnosis of signals increase monotonically with respect to gear fault gearboxes are important topics in maintenance field. levels. Zhang et al. [9, 10] used narrow band interference Jardine et al. [1] summarized and reviewed the research cancellation to enhance the gearbox fault diagnosis and and developments in diagnostics and prognostics of mechan- extract effective degradation indicator which is not sensitive ical systems. They mainly focused on models, algorithms, and to the nonstationary condition. In addition, many other technologies of data processing and maintenance decision- techniques have been used in fault diagnosis of gearboxes, making. Samuel and Pines [2] reviewed vibration-based such as support vector machine (SVM) [11, 12], wavelet packet diagnosis techniques for helicopter transmission system. transformation (WPT) [13], artificial neural network (ANN) The importance of condition monitoring for gearbox was [14], and hidden Markov model (HMM) [15–17]. emphasized from cost and safety point of view. In addition, For gearbox fault diagnosis, fault level classification is features used for fault diagnosis and remaining useful life- more difficult than fault detection. However, limited papers time prediction were introduced. Meanwhile, various fault reported research topic about different fault levels identifi- detection methods of gearbox were discussed. Lebold et al. [3] cation. Typical faults of gears include pitting, chipping, and reviewed feature extraction methods for gearbox diagnosis crack [18, 19]. In particular for gear crack fault, it is difficult and prognosis. Samuel and Pines [4] and McFadden [5, 6] to diagnose. Loutridis [19, 20] utilized instantaneous energy 2 Mathematical Problems in Engineering

density and local scaling exponent algorithm to detect gear envelope 𝑙min(𝑡). The upper and lower envelopes should cover crack and identify crack levels effectively. Lei and Zuo [21] the entire signal between them. Third, compute their mean as proposed a gear crack level identification method based on 𝑚1(𝑡) and the difference between the signal 𝑥(𝑡) and 𝑚1(𝑡) is weighted KNN classification algorithm. However, the above ℎ1(𝑡). Consider methods require the expertise of an engineer to apply them 𝑙max (𝑡) +𝑙min (𝑡) successfully. A dilemma of crack level classification is early 𝑚1 (𝑡) = , fault detection. This is a challenge to traditional method. 2 (1) Takingfrequencyspectrumanalysisasanexample,itisbased ℎ1 (𝑡) =𝑥(𝑡) −𝑚1 (𝑡) . on the amplitude changing of fault characteristic frequency. Due to the fact that the amplitude changing is not very large Ideally, after the sifting operation of (1), ℎ1(𝑡) should between early fault and normal condition, the phenomenon be the first IMF. The construction of ℎ1(𝑡) described above is not obvious in its frequency spectrum and it is very difficult seems to satisfy all the requirements of IMF. However, during to detect the early fault. However, the minor changing can be the practical process, the theoretical upper envelope 𝑙max(𝑡) reflected in an IMF obtained using EMD. Then the changing and lower envelope 𝑙min(𝑡) are very difficult to calculate. In will be very obvious after being amplified by IMF. Thus, addition, any little inflection points of the monotonous signal EMD is very adaptive to early fault detection of gearbox. In can be transformed to new extrema. And these new extremas addition, EDT is a useful method to help in automotive fault should be contained by the next sifting operation. To solve diagnosis and fault level classification. Therefore, this paper this dilemma, Huang et al. [22] repeated the sifting process of proposed a fault level classification method based on EMD (1) as many times as required to reduce the extracted signal and EDT, which has a good performance in early gear crack to an IMF. Therefore, the fourth step is to repeat the sifting fault. A correlation coefficient based method is also proposed process by treating ℎ1(𝑡) as the original signal as follows: to select the sensitive IMFs which contain main gear fault information. By comparing with kurtosis and skewness, it is 𝑙1 max (𝑡) +𝑙1 min (𝑡) 𝑚11 (𝑡) = , found that energy of these IMFs is the most suitable feature 2 (2) to be used in fault level classification. The effectiveness of ℎ (𝑡) =ℎ (𝑡) −𝑚 (𝑡) . the proposed method has been validated through analyzing 11 1 11 gearbox experimental data. The sifting process will be repeated 𝑘 times until ℎ1𝑘(𝑡) The remaining sections of this paper are organized as becomes a true IMF; that is, follows. In Section 2, framework of the proposed gear crack level classification method is given. Section 3 describes the ℎ1𝑘 (𝑡) =ℎ1(𝑘−1) (𝑡) −𝑚1(𝑘−1) (𝑡) . (3) experiment and applies the proposed method to fault level diagnosis. Finally, conclusions are given in Section 4. Then, make 𝑐1(𝑡) =1𝑘 ℎ (𝑡),anditcanbeseemedasthefirst IMF. Remove 𝑐1(𝑡) from the signal 𝑥(𝑡);namely,

2. Framework of the Proposed Method 𝑟1 (𝑡) =𝑥(𝑡) −𝑐1 (𝑡) . (4) 𝑟 (𝑡) 𝑟 (𝑡) Hilbert Huang transform (HHT) is a new signal processing And generate the residue signal 1 .Treating 1 as a method developed by Huang et al. [22]. It contains two new original signal and repeating the same sifting process parts:EMDandHilbertspectrumanalysismethod.Asthe above,thesecondIMFcanbegetted.Similarly,aseriesof (𝑐 (𝑡)(𝑖 = 1,2,...,𝑛)) kernel of HHT, EMD has been developed and widely used in IMFs 𝑖 can be obtained until the final 𝑟 (𝑡) 𝑥(𝑡) fault diagnosis of rotating machinery recently [23–26]. Using residue 𝑛 is monotonous. Then the original signal can EMD,thecomplexsignalcanbedecomposedintoasetof be reconstructed as complete, simple, and almost orthogonal components named 𝑛 intrinsic mode functions (IMFs). The IMFs represent the 𝑥 (𝑡) = ∑𝑐𝑖 (𝑡) +𝑟𝑛 (𝑡) . (5) natural oscillatory mode embedded in the signal and work as 𝑖=1 the basis functions, which are determined by the signal itself. The IMFs 𝑐1,𝑐2,...,𝑐𝑛 represent different frequency bands And the IMFs should satisfy the following two conditions: (1) ranging from high to low. The frequency components con- in the whole data set, the number of extrema and the number tained in each frequency band are different and they change of zero-crossings must either be equal or differ at most by one with the variation of the original signal 𝑥(𝑡),and𝑟𝑛(𝑡) and (2) at any point, the mean value of the envelope defined by represents the central tendency of signal 𝑥(𝑡). local maxima and minima must be zero. Namely, local signal After getting all the IMFs of a signal, sensitive IMFs which issymmetricalaboutthetimeaxis. containmainfaultinformationshouldbeselectedtopromote EMD is developed based on the assumption that any the velocity of calculation. This paper proposed a correlation signal consists of many different IMFs. The procedures of coefficient based method to select sensitive IMFs, as follows. decomposing a given signal 𝑥(𝑡) to different IMFs can be 𝑥 (𝑡) categorized into the following steps. First, identify all the (1) Assume one test sample of fault state fault pro- local extrema from the given signal and then connect them duced 𝐾 IMFs 𝑐𝑘(𝑡)(𝑘= 1,2,...,𝐾)after being pro- with a cubic spline line as the upper envelope 𝑙max(𝑡).Second, cessed by EMD; compute the correlation coefficients 𝛼 𝑐 (𝑡) 𝑥 (𝑡) repeat the first step for the local minima to produce the lower 𝑘 of 𝑘 and fault . Mathematical Problems in Engineering 3

(2) Similarly, compute the correlation coefficients 𝛽𝑘 of Select sensitive 𝑐 (𝑡) 𝑥 (𝑡) IMFs 𝑘 and normal . (3) Calculate the fault factors 𝜆𝑘 based on 𝛼𝑘 and 𝛽𝑘; namely, IMF1 IMF1 Vibration data EMD 𝜆𝑘 =𝛼𝑘 −𝛽𝑘, (𝑘=1,2,...,𝐾) . (6) IMF2 IMF2 Feature vectors (4) Analyze the fault factors and select the 𝑃 bigger value . . corresponding 𝑐𝑝(𝑡) as the IMFs which contain the . . main fault information. IMFk IMFp Then,theselectedsensitiveIMFscanbeinputtedinto EDT. The algorithm is implemented by computing the Euclidean distances between the test sample and the trained sample as Euclidean distances 𝑇 󵄩 󵄩2 𝐷2 =(𝑋−𝑋) (𝑋 −𝑋)=󵄩𝑋 −𝑋󵄩 𝑖𝑗 𝑖 𝑗 𝑖 𝑗 󵄩 𝑖 𝑗󵄩 Optimum state

𝑃 2 (7) Figure 1: Flowchart of the classification process using EMD and = ∑ (𝑥𝑖𝑝 −𝑥𝑗𝑝) , EDT. 𝑝=1 where 𝑋𝑖 is the test sample belonging to the unknown class Magnetic powder Speed and and 𝑋𝑗 is the trained sample belonging to known class, class brake Gearbox torque sensor Electromotor 𝜔.And𝑝=1,2,...,𝑃is the number of the selected IMFs. Therefore, a feature parameter set {𝐸𝑚𝑝𝑞, 𝑚 = 1,2,..., 𝑀; 𝑝 = 1,2,...,𝑃; 𝑞 = 1,2,...,𝑄}canbeacquiredbefore computing the Euclidean distances between the test samples and the trained samples, which is an 𝑀-by-𝑃-by-𝑄 matrix, where 𝑚 is the 𝑚th crack level of gears, 𝑝 is the 𝑝th IMF, and 𝑞 is the 𝑞th test sample. Then the feature vector matrix canbe built as

𝑇𝑚𝑞 =[𝐸𝑚𝑞1,𝐸𝑚𝑞2,...,𝐸𝑚𝑝𝑞], (8) 𝑚=1,2,...,𝑀 𝑞=1,2,...,𝑄. Figure 2: The gearbox test rig.

Euclidean distances between the test sample and trained samples can be calculated. If the distances between this test 3. A Case Study sample and each trained sample satisfy

2 2 3.1. Experimental Setup and Data Acquisition. Amechanical 𝐷 (𝑋, 𝜔𝑖)<𝐷 (𝑋,𝑗 𝜔 ), 𝑖,𝑗=1,2,...,𝑀, 𝑖=𝑗,̸ (9) test bed in the RCM laboratory of Mechanical Engineering College is used in this research to validate the effectiveness of then the test sample belongs to class 𝜔. the proposed method in this paper. The gearbox is driven by a Following the procedure described above, the crack level 4 KW three-phase asynchronous drive motor. In addition, the classification of gears can be performed. The classification speed and torque sensors are used to acquire the speed and process can be summarized as follows. torque information; a magnetic powder brake is utilized to provide load. These components are connected by couplings, (1) Acquire vibration signal. as shown in Figure 2. (2) Obtain IMFs by signal processing and EMD. The crack fault is implemented on one teeth of gear #2. Three crack levels are introduced and the length of each level (3)SelectthesensitiveIMFswhichcontainmainfault is 1 mm, 2 mm, and 5 mm, respectively. Figure 3(a) shows the information. structure of the gearbox used in this experiment. Gear #2 is (4) Extract feature parameters of sensitive IMFs and build the test gear and its tooth number is 64. The tooth numbers of the feature vector matrix. other three gears are 35 (#1), 18 (#3), and 81 (#4), respectively. (5) Obtain the diagnosis result using EDT. Four accelerometers are mounted on the gearbox casing and the specific location of every accelerometer is also shown in The flowchart of the new proposed method is described Figure 3(a). Figure 3(b) is the photo of the fault gear used in in Figure 1. this study. 4 Mathematical Problems in Engineering

Gearbox #1 Motor Accelerometer 1 #3

Accelerometer 2 #2 Brake

Accelerometer 4

#4 Accelerometer 3 (a) (b)

Figure 3: (a) The structure of the gearbox. (b) The fault gear used in this study.

The sampling frequency of this experimental system is Table 1: The correlation coefficients and fault factor of IMFs. 20 kHz and sampling time is 6 s. Each fault mode has 60 Number 𝛼𝑘 𝛽𝑘 𝜆𝑘 samples. The input rotary speed of motor is 800 rpm and the loads generated by brake are 10 N⋅m, 15 N⋅m, and 20 N⋅m. IMF1 0.471 0.001 0.470 IMF2 0.494 −0.007 0.501 IMF3 0.556 0.012 0.543 3.2. Results Analysis and Discussion. Following the procedure − described in Section 2,theprocessofgearcracklevelclassifi- IMF4 0.384 0.012 0.396 cation can be introduced as follows. First, raw vibration data IMF5 0.285 0.001 0.284 are collected from the data acquisition system of the gearbox IMF6 0.154 0.000 0.154 test rig. This paper chooses the vibration data acquired by IMF7 0.028 −0.004 0.032 accelerometer 1. Then, the vibration data is processed by EMD IMF8 0.005 0.000 0.005 and a number of IMFs are obtained which range from 15 to IMF9 0.006 0.002 0.004 19. Taking one vibration signal which obtained 15 IMFs after IMF10 0.017 0.005 0.013 processed by EMD as a sample, and the IMFs are shown in IMF11 0.011 −0.005 0.016 Figure 4. IMF12 0.000 0.000 0.000 In order to select the sensitive IMFs, the correlation IMF13 0.001 0.000 0.000 coefficients and fault factors are calculated, which are shown IMF14 0.000 0.000 −0.001 in Table 1. IMF15 −0.001 0.000 −0.001 ItcanbeseenfromTable 1 that the first to sixth IMFs have great correlation of the fault signal and little correlation of the normal signal. Namely, these IMFs contain the main fault information, and they are selected as the sensitive IMFs. from the figure that gear meshing frequencies are prominent. The vibration signal of a gearbox is a mixture of many However, shafts and bearings rotating frequencies are filtered components, such as shafts and bearings, not limited to out. Because the filtered order of EMD is from high frequency gear meshing vibration only. To validate the selected IMFs to low, so these results can ensure that the selected IMFs containing gear fault information, this paper analyzed the contain gear fault information. frequency spectrum of original signal and each IMF, respec- Then, the feature parameter vectors can be calculated. tively.Thesignalisacquiredfrom1mmcrackstatewith This paper selects energy of IMF as feature parameter. In 800rpmspeedand20Nmloadcondition. addition,thegearhas4cracklevelsand30testsamplesare Usually, shaft and bearing rotating frequencies are all in chosen for each crack level. So, the energy set, 𝐸𝑚𝑝𝑞,isa4- low frequency area. And gear meshing frequency will be a by-6-by-30 matrix. The mean values of feature vectors of all little high relatively. Figure 5 istheenvelopeanalysisoforig- the samples for the same class are used as the trained sample. inal signal. The figure shows that gear meshing frequencies Therefore, the euclidean distances between test samples and are very obvious. Shafts and bearings rotating frequencies can four classess trained samples can be obtained. Table 2 shows also be seen in low frequency area. In addition, the noise the distance values between normal test samples and the pollution is very serious. trained samples of each level when the load is 10 N⋅m. It can be Figure 6 is the envelope analysis of IMF1.Similarly,gear seen from Table 2 that distance values between test samples meshing and shafts and bearings rotating frequencies are and the trained samples of normal state are the minimum, obvious. But the noise pollution is restrained effectively. and the accuracy rate of the classification result is about Figure 7 istheenvelopeanalysisofIMF2.Itcanbeseen 96.67%. Mathematical Problems in Engineering 5

5 Table 2: The distance values when normal test samples are inputted 0 ⋅ −5 and the load is 10 N m. signal Original 2 Number Normal 1mmcrack 2mmcrack 5mmcrack

(t) 0 2.38E + 03 1.30𝐸 + 04 4.94𝐸 + 03 1.12𝐸 +04 1 1 c −2 2 2.14E + 03 1.32𝐸 + 04 5.41𝐸 + 03 1.15𝐸 +04 2 3 1.81E + 03 1.39𝐸 + 04 7.28𝐸 + 03 1.25𝐸 +04

(t) 0 2 c −2 4 1.98E + 03 1.20𝐸 + 04 6.09𝐸 + 03 1.05𝐸 +04 2 5 1.79E + 03 1.39𝐸 + 04 7.98𝐸 + 03 1.26𝐸 +04

(t) 0 3 6 2.06E + 03 1.38𝐸 + 04 6.28𝐸 + 03 1.23𝐸 +04 c −2 7 2.64E + 03 1.38𝐸 + 04 8.10𝐸 + 03 1.25𝐸 +04 2 1.94E + 03 1.38𝐸 + 04 6.21𝐸 + 03 1.22𝐸 +04

(t) 0 8 4 c −2 9 2.43E + 03 1.33𝐸 + 04 6.92𝐸 + 03 1.18𝐸 +04 2 10 2.65E + 03 1.42𝐸 + 04 6.73𝐸 + 03 1.25𝐸 +04

(t) 0

5 1.89E + 03 1.34𝐸 + 04 5.37𝐸 + 03 1.16𝐸 +04 c −2 11 1.66E + 03 1.45𝐸 + 04 7.72𝐸 + 03 1.30𝐸 +04 1 12 1.90E + 03 1.34𝐸 + 04 6.19𝐸 + 03 1.17𝐸 +04 (t) 0 13 6 c −1 14 3.40E + 03 1.49𝐸 + 04 8.77𝐸 + 03 1.35𝐸 +04 0.5 15 1.33E + 03 1.40𝐸 + 04 7.71𝐸 + 03 1.25𝐸 +04

(t) 0

7 1.64E + 03 1.38𝐸 + 04 7.10𝐸 + 03 1.22𝐸 +04 c −0.5 16 17 2.29E + 03 1.39𝐸 + 04 5.83𝐸 + 03 1.21𝐸 +04 0.2 2.43E + 03 1.40𝐸 + 04 6.34𝐸 + 03 1.23𝐸 +04

(t) 0 18 8 c −0.2 19 1.98E + 03 1.33𝐸 + 04 6.45𝐸 + 03 1.18𝐸 +04 0.1 20 1.88E + 03 1.45𝐸 + 04 6.89𝐸 + 03 1.28𝐸 +04

(t) 0 9 c −0.1 21 2.54E + 03 1.45𝐸 + 04 6.92𝐸 + 03 1.29𝐸 +04 3.99E + 03 1.53𝐸 + 04 8.91𝐸 + 03 1.40𝐸 +04 0.05 22

(t) 0 23 4.40𝐸 + 03 1.37𝐸 +04 4.16E + 03 1.16𝐸 + 04 10

c −0.05 24 2.97E + 03 1.43𝐸 + 04 5.99𝐸 + 03 1.25𝐸 +04 0.05 25 3.23E + 03 1.42𝐸 + 04 6.73𝐸 + 03 1.25𝐸 +04 (t) 0

11 4.78E + 03 1.42𝐸 + 04 5.33𝐸 + 03 1.22𝐸 +04 c −0.05 26 0.05 27 3.91E + 03 1.37𝐸 + 04 5.24𝐸 + 03 1.18𝐸 +04

(t) 0 28 3.16E + 03 1.41𝐸 + 04 6.63𝐸 + 03 1.23𝐸 +04 12

c −0.05 29 4.66E + 03 1.48𝐸 + 04 6.45𝐸 + 03 1.31𝐸 +04 0.02 3.39E + 03 1.37𝐸 + 04 5.40𝐸 + 03 1.17𝐸 +04 (t) 0 30 13

c −0.02 0.01

(t) 0 14 c −0.01 original signal is extracted without EMD the accuracy rate ×10−3 is about 80% and the classification results are unsatisfactory. 8 Therefore, the process of EMD is effective by this comparing

RES. 4 study. 0123456 Inordertovalidatetheproposedmethodforwhich t (s) sensitive IMFs are selected, energy of first to third IMFs Figure 4: The decomposition result by EMD. is extracted and the final classification results are shown as in Figures 16, 17, 18,and19.Theclassificationresults show the effectiveness of the proposed method for selecting the sensitive IMFs. If all IMFs are selected, the computing To show the distance values more directly and save velocity will be slow. And if few IMFs are selected, for the case space, the results are all shown by figures. When the load of 3 IMFs, the classification results will be not very accurate. is 10 N⋅m, the results can be depicted as in Figures 8, 9, 10, Allthesamplesaboveareundertheloadof10N⋅m; for the and 11. It can be seen from the figures that the accuracy purpose of checking the adaptability to different conditions of rate of the classification results using the proposed method themethod,theloadof15N⋅mand20N⋅m is also considered is approximately 100%. and the classification results are shown as in Figures 20, 21, To validate the effectiveness of EMD, the energy of 22,and23 and Figures 24, 25, 26,and27,respectively.Itcan original signal that is not processed by EMD is extracted and be seen that the method proposed in this paper also has good the classification results are shown as in Figures 12, 13, 14, performance. The accuracy rates are nearly 100% for the two and 15. It can be seen that for the case that the energy of cases. 6 Mathematical Problems in Engineering

0.25 ×104 2 0.2

Shafts and 3× 1×gear meshing 2×gear meshing gear meshing 1.5 0.15 bearings rotating frequency frequencies frequency frequency 0.1 1 Amplitude (V) Amplitude

0.05 values Distance 0.5

0 0 100 200 300 400 500 0 Frequency (Hz) 5 1015202530 Test samples number Figure 5: Envelope analysis of original signal. Normal 2 mm crack 1 mm crack 5 mm crack 0.12 Figure 8: The classification result when normal test samples are 0.1 inputted. Shafts and 0.08 bearings rotating frequencies 1×gear meshing frequency 2×gear meshing 0.06 frequency 3×gear meshing 15000 0.04 frequency Amplitude (V) Amplitude

0.02 10000 0 0 100 200 300 400 500 Frequency (Hz) 5000 Figure 6: Envelope analysis of IMF1. values Distance

0.12 0 5 1015202530 0.1 Test samples number 1×gear meshing 0.08 frequency 2 mm crack 2×gear meshing Normal 1 mm crack 5 mm crack 0.06 frequency 3×gear meshing 0.04 frequency Figure 9: The classification result when 1 mm crack test samples are Amplitude (V) Amplitude inputted. 0.02

0 0 100 200 300 400 500 Frequency (Hz) 15000

Figure 7: Envelope analysis of IMF2.

10000

It can be obtained from above analysis that the gear crack level classification method is effective to identify different 5000 crack levels no matter whether the fault is in early stage values Distance (1 mm) or sever stage (5 mm). In addition, EMD and the method of selecting sensitive IMFs are crucial during process 0 of the original signal. 5 1015202530 Test samples number 4. Conclusion Normal 2 mm crack 1 mm crack 5 mm crack In this paper, a new gear crack level classification method based on empirical mode decomposition (EMD) and Euclid- Figure 10: The classification result when 2 mm crack test samples eandistancetechnique(EDT)isproposed.Theapproachwas are inputted. Mathematical Problems in Engineering 7

14000 ×104 6 12000 10000 5 8000 4

6000 3

Distance values Distance 4000 2 2000 values Distance 1 0 5 1015202530 0 Test samples number 5 1015202530 Test samples number Normal 2 mm crack 1 mm crack 5 mm crack Normal 2 mm crack 1 mm crack 5 mm crack Figure 11: The classification result when 5 mm crack test samples areinputted. Figure 14: The classification result of original signal when 2mm crack test samples are inputted.

×104 4 6 ×10 5 5 4 4

3 3

2 2 Distance values Distance

1 values Distance 1 0 5 1015202530 0 Test samples number 5 1015202530 Test samples number Normal 2 mm crack 1 mm crack 5 mm crack Normal 2 mm crack 1 mm crack 5 mm crack Figure 12: The classification result of original signal when normal test samples are inputted. Figure 15: The classification result of original signal when 5mm crack test samples are inputted.

×104 6 15000

4 10000

2 5000 Distance values Distance Distance values Distance 1

0 0 5 1015202530 5 1015202530 Test samples number Test samples number

Normal 2 mm crack Normal 2 mm crack 1 mm crack 5 mm crack 1 mm crack 5 mm crack

Figure 13: The classification result of original signal when 1mm Figure 16: The classification result using first to third IMFs when crack test samples are inputted. normal test samples are inputted. 8 Mathematical Problems in Engineering

14000 ×104 3 12000 10000 2.5 8000 2

6000 1.5

Distance values Distance 4000 1 2000 values Distance 0.5 0 5 1015202530 0 Test samples number 5 1015202530 Test samples number Normal 2 mm crack 1 mm crack 5 mm crack Normal 2 mm crack 1 mm crack 5 mm crack Figure 17: The classification result using first to third IMFs when 1 mm crack test samples are inputted. Figure 20: The classification result of⋅ 15N mloadwhennormaltest samples are inputted.

14000 ×104 2.5 12000 10000 2 8000 1.5 6000 1

Distance values Distance 4000 Distance values Distance 2000 0.5 0 5 1015202530 0 5 1015202530 Test samples number Test samples number Normal 2 mm crack 2 1 mm crack 5 mm crack Normal mm crack 1 mm crack 5 mm crack Figure 18: The classification result using first to third IMFs when ⋅ 2 mm crack test samples are inputted. Figure 21: The classification result of 15N mloadwhen1mmcrack test samples are inputted.

12000 ×104 2 10000

8000 1.5

6000 1 4000 Distance values Distance

Distance values Distance 0.5 2000

0 0 5 1015202530 5 1015202530 Test samples number Test samples number

Normal 2 mm crack Normal 2 mm crack 1 mm crack 5 mm crack 1 mm crack 5 mm crack

Figure 19: The classification result using first to third IMFs when Figure 22: The classification result of⋅ 15N m load when 2 mm crack 5 mm crack test samples are inputted. test samples are inputted. Mathematical Problems in Engineering 9

×104 ×104 2.5 5

2 4

1.5 3

1 2 Distance values Distance Distance values Distance 0.5 1

0 0 5 1015202530 5 1015202530 Test samples number Test samples number

Normal 2 mm crack Normal 2 mm crack 1 mm crack 5 mm crack 1 mm crack 5 mm crack

Figure 23: The classification result of⋅ 15N m load when 2 mm crack Figure 26: The classification result of⋅ 20N mloadwhen2mmcrack test samples are inputted. test samples are inputted.

×104 ×104 6 6 5 5 4 4 3 3 2 2 values Distance Distance values Distance 1 1 0 0 5 1015202530 5 1015202530 Test samples number Test samples number Normal 2 mm crack Normal 2 mm crack 1 mm crack 5 mm crack 1 mm crack 5 mm crack Figure 27: The classification result of⋅ 20N m load when 5 mm crack Figure 24: The classification result of⋅ 20N mloadwhennormaltest test samples are inputted. samples are inputted.

tested and validated successfully using a test rig implanted ×104 crack fault experiment case. The results show the proposed 6 method obtains high accuracy rate in classifying different 5 crack levels and adapts to different conditions. Additionly, it is found through comparison that EMD and the method 4 of selecting sensitive IMFs are crucial during process of the original signal. 3

Distance values Distance Conflict of Interests 1 The authors declare that there is no conflict of interests 0 regarding the publication of this paper. 5 1015202530 Test samples number References Normal 2 mm crack 1 mm crack 5 mm crack [1] A. K. S. Jardine, D. Lin, and D. Banjevic, “Areview on machinery diagnostics and prognostics implementing condition-based Figure 25: The classification result of⋅ 20N m load when 1 mm crack maintenance,” Mechanical Systems and Signal Processing,vol.20, test samples are inputted. no.7,pp.1483–1510,2006. 10 Mathematical Problems in Engineering

[2] P. D. Samuel and D. J. Pines, “A review of vibration-based [19] S. J. Loutridis, “Instantaneous energy density as a feature for techniques for helicopter transmission diagnostics,” Journal of gear fault detection,” Mechanical Systems and Signal Processing, Sound and Vibration,vol.282,no.1-2,pp.475–508,2005. vol.20,no.5,pp.1239–1253,2006. [3] M. Lebold, K. McClintic, R. Campbell et al., “Review of vibra- [20] S. J. Loutridis, “Self-similarity in vibration time series: application analysis methods for gearbox diagnostics and prognostics,” tion to gear fault diagnostics,” JournalofVibrationandAcoustics, in Proceedings of the 54th Meeting of the Society for Machinery Transactions of the ASME,vol.130,no.3,ArticleID031004, Failure Prevention Technology, pp. 623–634, 2000. 2008. [4] P.D. Samuel and D. J. Pines, “Vibration separation methodology [21] Y. Lei and M. J. 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Zheng, “Wear detection in gear system gearsandsungearinanepicyclicgearbox,”Journal of Vibration using Hilbert-Huang transform,” JournalofMechanicalScience and Acoustics, Transactions of the ASME,vol.116,no.2,pp.179– and Technology,vol.20,no.11,pp.1781–1789,2006. 187, 1994. [24]Z.P.FengandF.L.Chu,“Transienthydraulicpressurefluctu- [7]E.B.Halim,M.A.A.ShoukatChoudhury,S.L.Shah,andM.J. ation signal analysis of hydroturbine based on Hilbert-huang Zuo, “Time domain averaging across all scales: a novel method transform,” Proceedings of the CSEE, vol. 25, no. 10, pp. 111–115, for detection of gearbox faults,” Mechanical Systems and Signal 2005. Processing,vol.22,no.2,pp.261–278,2008. [25]Z.P.Feng,X.J.Li,andF.L.Chu,“Applicationofstation- [8] Z. P. Feng, M. J. Zuo, and F. L. Chu, “Application of regulariza- ary wavelet packets decomposition based hilbert spectrum tion dimension to gear damage assessment,” Mechanical Systems to nonstationary hydraulic turbine vibration signal analysis,” and Signal Processing, vol. 24, no. 4, pp. 1081–1098, 2010. Proceedings of the Chinese Society of Electrical Engineering,vol. [9]X.Zhang,J.Kang,E.Bechhoefer,andJ.Zhao,“Anewfeature 26, no. 12, pp. 79–84, 2006. extraction method for gear fault diagnosis and prognosis,” [26] Z. Feng and F. Chu, “Frequency demodulation analysis method Maintenance and Reliability,vol.16,no.2,pp.295–300,2014. forfaultdiagnosisofplanetarygearboxes,”Proceedings of the [10] X. H. Zhang, J. S. Kang, E. Bechhoefer, L. Xiao, and J. M. Zhao, Chinese Society of Electrical Engineering,vol.33,no.11,pp.112– “Gearbox degradation analysis using narrowband interference 117, 2013. cancellation under non-stationary conditions,” Journal of Vibro- engineering,vol.16,no.4,pp.2089–2102,2014. [11] L. M. R. Baccarini, V. V. Rocha E Silva, B. R. De Menezes, and W.M. Caminhas, “SVM practical industrial application for mechanical faults diagnostic,” Expert Systems with Applications, vol. 38, no. 6, pp. 6980–6984, 2011. [12]X.L.Tang,L.Zhuang,J.Cai,andC.B.Li,“Multi-fault classification based on support vector machine trained by chaos particle swarm optimization,” Knowledge-Based Systems,vol.23, no. 5, pp. 486–490, 2010. [13] G.G.YenandW.F.Leong,“Faultclassificationonvibrationdata with wavelet based feature selection scheme,” ISA Transactions, vol.45,no.2,pp.141–151,2006. [14] X. H. Zhang, L. Xiao, and J. S. Kang, “Application of an improved Levenberg-Marquardt back propagation neural network to gear fault level identification,” Journal of Vibroengineering,vol.16,no. 2, pp. 855–868, 2014. [15]T.BoutrosandM.Liang,“Detectionanddiagnosisofbearing and cutting tool faults using hidden Markov models,” Mechan- ical Systems and Signal Processing,vol.25,no.6,pp.2102–2124, 2011. [16] J. S. Kang and X. H. Zhang, “Application of hidden markov models in machine fault diagnosis,” Information—An Interna- tional Interdisciplinary Journal,vol.15,no.12,pp.5829–5838, 2012. [17] S. Hassiotis, “Identification of damage using natural frequencies and Markov parameters,” Computers and Structures,vol.74,no. 3, pp. 365–373, 2000. [18] X. Fan and M. J. Zuo, “Gearbox fault detection using Hilbert and wavelet packet transform,” Mechanical Systems and Signal Processing,vol.20,no.4,pp.966–982,2006. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 681232, 8 pages http://dx.doi.org/10.1155/2015/681232

Research Article Weibull Failure Probability Estimation Based on Zero-Failure Data

Ping Jiang,1 Yunyan Xing,2 Xiang Jia,1 andBoGuo1

1 College of Information System and Management, National University of Defense Technology, Changsha 410073, China 2College of Continuing Education, National University of Defense Technology, Changsha 410073, China

Correspondence should be addressed to Ping Jiang; [email protected]

Received 27 May 2014; Revised 4 September 2014; Accepted 15 October 2014

Academic Editor: Wenbin Wang

Copyright © 2015 Ping Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reliability testing is often carried out with small sample sizes and short duration because of increasing costs and the restriction of development time. Therefore, for highly reliable products, zero-failure data are often collected in such tests, which could notbe used to evaluate reliability by traditional methods. To cope with this problem, the match distribution curve method was proposed by some researchers. The key step needed to exercise this method is to estimate the failure probability, which has yet to be solved in the case of the Weibull distribution. This paper presents a method to estimate the intervals of failure probability for the Weibull distribution by using the concavity or convexity and property of the distribution function. Furthermore, to use the method in practice, this paper proposes using the approximate value of the shape parameter determined by either engineering experience or by hypothesis testing through a p value. The estimation of the failure probability is thus calculated using a Bayesian approach. A numerical example is presented to validate the effectiveness and robustness of the method.

1. Introduction probability of the product at time 𝑡𝑖.Then𝑝𝑖 satisfies the following statements: Reliability testing is usually required in product development to evaluate product reliability. Product lifetimes are becoming (1) 𝑝0 =0,when𝑡=0; longer than in previous decades because of the improvement (2) 𝑝1 <⋅⋅⋅<𝑝𝑘. of reliability. Therefore, with the restrictions of increasing costs and short development times, reliability testing is often 𝑘 Let 𝑠𝑖 = ∑𝑗=𝑖 𝑛𝑗 denote the number of samples at time 𝑡𝑖; carriedoutwithsmallsamplesizesandshortduration,which that is, there are 𝑠𝑖 samples in the reliability test at time 𝑡𝑖. will often lead to zero-failure data1 [ ]. Accordingly, it is Estimating product reliability based on zero-failure data desirable to estimate product reliability using the zero-failure is challenging. Welker and Lipow [1]firstraisedthisproblem data. and, since then, some researchers have made progress on The zero-failure scenario is described as follows. the topic [2, 3]. For the binomial distribution, Bailey [4] Let 𝐹 (𝑡, 𝜃) denote the lifetime distribution of a product, proposes a model to predict failure probability from zero- where 𝜃∈Θis the parameter of the distribution and failure data; however, his model requires a large sample size. Θ is the parameter space. The reliability test is composed Basedonzero-failuredata,WangandLanganke[5]presentan of 𝑘 truncated tests with corresponding censored time approach that compares the reliability index—mean time to 𝑡𝑖 (𝑖=1,2,...,𝑘), which satisfies 𝑡1 <𝑡2 < ⋅⋅⋅ < 𝑡𝑘.The failure (MTTF)—between a newly designed product and the sample size for the 𝑖th test is 𝑛𝑖.Asnofailureisobserved old product, but their approach is based on the assumption in the tests, the zero-failure data is expressed by (𝑡𝑖,𝑛𝑖), 𝑖= that the shape parameters for the two products are the same, 1,2,...,𝑘.Let𝑝𝑖 =𝑃(𝑇≤𝑡𝑖)=𝐹(𝑡𝑖) denote the failure which limits the application in practice. Miller et al. [6] 2 Mathematical Problems in Engineering

𝑚−1 study the estimation of failure probability for software when The failure rate function 𝜆 (𝑡) =𝑚/𝜂(𝑡/𝜂) is an no errors are observed in testing. Jiang et al. [7]construct increasing function, when 𝑚>1, which describes the the shrinkage preliminary test estimator (SPTE) to estimate character of various products. The Weibull distribution will the reliability of a product following a Weibull distribution, become another type of distribution when the value of shape when a prior estimate is available. Chen et al. [8]introduce parameter 𝑚 varies. For example, it becomes an exponential the optimal confidence limit method to obtain the optimal distribution when 𝑚=1; it becomes a Rayleigh distribution lower confidence limit of some parameters of reliability when 𝑚=2; and it approximates a normal distribution distributions, in the case of zero-failure data. However, their when 𝑚∈[3,4].Ning[10]summarizesthataWeibulldis- method only considers one sample in each test and could not tribution can be used to describe traumatic failures, when be applied to the case of multiple samples in each test. As 𝑚≤1, and it can be applied to describe degradation failures, no failure is observed in the tests, the maximum likelihood when 𝑚 ≥ 3.25. Therefore, it can be adopted to describe estimation (MLE) approach could not be applied in such the combined effects of traumatic failures and degradation ascenario.Tosolvethisproblem,L.WangandB.Wang failures, when 𝑚 ∈ (1, 3.25),wheretheratioofthetwofailures [9] propose the modified maximum likelihood estimation is determined by the value of 𝑚. Due to the flexibility of (MMLE), introducing a new parameter, 𝐿,tomodifythe Weibull distribution, it is widely used in reliability evaluation resultsobtainedbyapplyingMLEonthezero-failuredata. in practice, even in the cases of zero-failure data. The key to MMLE is the value of parameter 𝐿.However, As the key step in the MDC method is failure probability some researchers have found that the parameter, 𝐿,willoften estimation,whichisyettobesolvedinthecaseofaWeibull cause the overestimation of parameters. The overestimation distribution, we present this estimation procedure in the usually results from the improper use of the zero-failure data, following section. which has raised debates among researchers for a long time. One solution is to introduce failure information when dealing 3. Failure Probability Estimation with zero-failure data [10]. The key to this method is to acquire the failure information (i.e., failure time), whereas Let 𝑦=ln ln (1/1 − 𝐹), 𝑥=ln 𝑡,and𝑏=𝑚ln 𝜂;then(2) is it is often estimated by the zero-failure data. This will result transformed into in an inaccurate estimate when the test duration is far less than the real lifetime of a product [11]. Mao and Luo [12] 𝑦=𝑚𝑥−𝑏. (3) present the match distribution curve (MDC) method to solve the evaluation on zero-failure data: first, estimating the The linear regression method could be applied to cal- failure probability 𝑝𝑖 at censoring time 𝑡𝑖 is carried out; then, culate the parameters of the Weibull distribution. Here, the the data pairs (𝑡𝑖,𝑝𝑖) are used to construct a distribution weighted least square estimation method is used to obtain the curvetoestimatetheparametersofthedistribution;finally, estimates of parameters 𝑚̂ and 𝜂̂, which minimizes thereliabilitycanbeevaluatedbasedonthedistribution. 𝑘 TheMDCmethodiswidelyusedinvariousdistributions 2 𝑄=∑𝑤𝑖 (𝑦𝑖 −𝑚𝑥𝑖 +𝑚ln 𝜂) . (4) with zero-failure data; however, it is not discussed in the 𝑖=1 case of a Weibull distribution because of the computational complexity of the distribution. Motivated by this problem, As it is already proven that the weighted least square we focus our research on the failure probability estimation estimates, 𝑚̂ and 𝜂̂,areunbiased,weproposetheuseofthe method in a Weibull distribution. method and briefly introduce it here. 𝑘 Denote the weight 𝑤𝑖 =𝑛𝑖𝑡𝑖/∑𝑖=1 𝑛𝑖𝑡𝑖 (𝑖=1,2,...,𝑘), 𝑘 𝑘 2 2. Weibull Distribution where 𝑡𝑖 is failure time. Let 𝐴=∑𝑖=1 𝑤𝑖𝑥𝑖, 𝐵=∑𝑖=1 𝑤𝑖𝑥𝑖 , 𝑘 𝑘 𝐶=∑𝑖=1 𝑤𝑖𝑦𝑖,and𝐷=∑𝑖=1 𝑤𝑖𝑥𝑖𝑦𝑖.Theweightedleast When evaluating reliability using test data, we often assume 𝑚̂ 𝜂̂ thattheproductlifefollowssomekindofdistribution.Then, square estimates, and , are obtained by referring to the parameters of the distribution are estimated based on 𝐵−𝐴2 thedata,andreliabilityisevaluatedbyusingthedetermined 𝑚=̂ , 𝐷−𝐴𝐶 distribution. The Weibull distribution is one of the most (5) commonly used distributions in reliability evaluation because 𝐵𝐶 − 𝐴𝐷 𝜂=̂ ( ). of its ability to take on various forms by adjusting its exp 𝐵−𝐴2 parameters [13]. The two-parameter Weibull distribution is defined as After 𝑚̂ and 𝜂̂ areacquired,giventhetesttime𝑡𝑖,itiseasy 𝑡 𝑚 𝑡 𝑚−1 𝑡 𝑚 to calculate the failure probability at time 𝑖 for the Weibull 𝑓 (𝑡) = ( ) [− ( ) ] (𝑡>0) , distribution by 𝜂 𝜂 exp 𝜂 (1) 𝑡 𝑚̂ where 𝑚 is the shape parameter and 𝜂 is the scale parameter. 𝑖 𝑝𝑖 =𝐹(𝑡𝑖) =1−exp [− ( ) ] . (6) The CDF of the Weibull distribution is defined as 𝜂̂ 𝑡 𝑚 𝐹 (𝑡) =𝑃(𝑇≤𝑡) =1− [− ( ) ] . The above procedure is typical to obtain the failure exp 𝜂 (2) probability at time 𝑡𝑖 for the Weibull distribution given Mathematical Problems in Engineering 3

that 𝑡𝑖 (𝑖=1,2,...,𝑘) is the corresponding failure time. To Inspired by this theory, and given the distribution type, calculate the failure probability at time 𝑡𝑖 for the Weibull researchers tried to use the properties of the distribution distribution with zero-failure data, we have to consider an to determine the interval of 𝑝𝑖 if there is no other prior alternativesolution.Thisisdiscussedinthenextsubsection information: in the case of an exponential distribution, Ning for Weibull distribution. [10]obtainstheintervalof𝑝𝑖 as [𝑝𝑖−1,(𝑡𝑖/𝑡𝑖−1)𝑝𝑖−1],and in the case of an extreme distribution, Li [11]obtainsthe 𝑝 [𝑝 ,(𝑡/𝑡 )𝑝 ] 3.1. Failure Probability Estimation with Zero-Failure Data. In interval of 𝑖 as 𝑖−1 𝑖 𝑘 𝑢 .However,inthecaseof cases of zero-failure data, Ning [10]proposesthefollowing a Weibull distribution, there is no reference to determine 𝑝 equation to estimate 𝑝𝑖 at censoring time 𝑠𝑖: the interval of 𝑖,whichiswhywechoosetostudythis 0.5 problem according to the convex and concave properties of ̂ 𝑝𝑖 = (𝑖=1,2,...,𝑘) . (7) the Weibull distribution. 𝑠𝑖 +1 Equation (7) isdesignedtocalculatethemeanvalueofthe 3.2. Weibull Failure Probability Estimation. Based on the upper limit 1/ (𝑠𝑖 +1)and lower limit 0, which is too simple accumulated test data from Weibull-distributed products in and arbitrary, and often results in inaccurate estimation. worldwide practice, Han [16] states that the shape parameter, Bayesian theory is the most popular method to estimate 𝑚, for Weibull-distributed products is usually within the [1, 10] 𝑚 [1, 10] 𝑝𝑖 as it combines prior information with test data, making interval of .So,theintervalfor is set to be . the estimation more accurate than if only prior information Toobtaintheconvexandconcavepropertiesofthe or test data was used alone. In Bayesian theory, distribution Weibull distribution, by taking the second derivative of (2), parameter 𝜃 is regarded as a random variable, and the prior we have distribution is determined by using historical data, experts’ 𝑚−2 𝑚 𝑑2𝐹 𝑚𝑡 exp (− (𝑡/𝜂) ) 𝑚𝑡𝑚 judgments, or data from similar products to obtain the = (𝑚−1− ) . (10) posterior distribution of 𝜃 by 𝑑𝑡2 𝜂𝑚 𝜂𝑚 𝜋 (𝜃) 𝑓 (𝑥|𝜃) 𝑑2𝐹/𝑑𝑡2 <0 𝜋 (𝜃|𝑥) = , (8) From (10),itiseasytofindthat ,when ∫ 𝜋 (𝜃) 𝑓 (𝑥|𝜃) 𝑑𝜃 0<𝑚≤1 𝑚>1 𝑡 = 𝜃 ,andwhen , let the inflexion point turn 𝜂 ((𝑚−1) /𝑚)1/𝑚 𝑑2𝐹/𝑑𝑡2 >0 𝑡∈[0,𝑡 ) where 𝑓 (𝑥|𝜃) is the PDF of the population 𝑋,oritcan ;then ,when turn ; 𝑑2𝐹/𝑑𝑡2 <0 𝑡∈(𝑡 ,+∞) be replaced by likelihood function 𝐿(𝜃),and𝜋 (𝜃) is the ,when turn . So, according to the ̂ 𝐹 (𝑡) prior distribution of 𝜃. The point estimate of 𝜃 is 𝜃= convex and concave criteria, we have the following: is ∫ 𝜃𝜋 (𝜃|𝑥) 𝑑𝜃 convex, when 0<𝑚≤1,and𝐹 (𝑡) is concave, when 𝑚>1 𝜃 , which is the expectation of the posterior 𝑡∈[0,𝑡 ) and turn . This conclusion provides us with the criteria distribution under the square loss. 𝑝 𝑝 for determining the interval of 𝑖. In the case of zero-failure data, to estimate 𝑖,the (1) 0<𝑚≤1 𝐹 (𝑡) 𝑠𝑖 When , is convex, so we have likelihood function is usually defined as 𝐿=(1−𝑝𝑖) . Therefore, in terms of Bayes’ theorem, the key to estimating 𝑝1 𝑝2 𝑝𝑖−1 𝑝𝑖 𝑝𝑘 𝑝 𝑝 > >⋅⋅⋅> > >⋅⋅⋅> . (11) 𝑖 is to choose the prior distribution of 𝑖.Toachievethis,two 𝑡1 𝑡2 𝑡𝑖−1 𝑡𝑖 𝑡𝑘 important answers should be sought here: the right choice of the prior distribution and the determination of the interval of As we have assumed that 𝑝𝑖−1 <𝑝𝑖,thefollowing 𝑝𝑖 in the prior distribution. inequality stands: 𝑡 (1) The Right Choice of the Prior Distribution of 𝑝𝑖.Forthe 𝑖 𝑝𝑖−1 <𝑝𝑖 < 𝑝𝑖−1 (𝑖≥2) . (12) choice of the prior distribution of 𝑝𝑖, one may simply assume 𝑡𝑖−1 that 𝑝𝑖 follows the uniform distribution. However, since the 𝑝 (𝑝 ,(𝑡/𝑡 )𝑝 ) data exhibits zero failure, the value of failure probability 𝑝𝑖 is Therefore, the interval of 𝑖 is 𝑖−1 𝑖 𝑖−1 𝑖−1 ,which more likely to be small. Therefore, Han and Li [14]presentthe couldbecalculatedbyaniterativemethodbeginningfrom𝑝1 idea of utilizing the decreasing function to construct the prior (𝑖=1). distribution. We also adopt this idea and use it to construct The calculation begins with 𝑝1,whichcanbeestimated 𝑝 (0, 𝑝 ) the prior distribution in the next section. by the Bayesian method. Let the interval of 1 be upper , 𝑝 where upper is the upper limit set by experts. In practice, 𝑝 𝑝 𝑝 = 0.5 𝑝 (2) The Interval of 𝑖. To determine the interval of 𝑖 for a upper is often used. However, this value of upper is normal distribution, Zhang [15]provesthattheCDFof𝐹 (𝑡) conservative here, as no failure occurs; we can assume that 𝑡 𝑡<𝜇 𝑝 <𝑝 ≤ 0.5 (𝑖≥2) 𝑝 is a concave function in ,when .Then,basedonthe 1 𝑖 ,so upper could be set at less than 0.5. properties of the concave function, Zhang concludes that the (2) When 𝑚>1, 𝐹 (𝑡) is not strictly convex or concave, following inequalities stand: which makes it difficult to analyze. Therefore, we present our 𝑝 𝑝 𝑝 𝑝 solution as follows. 0< 1 < 2 <⋅⋅⋅< 𝑘 < 𝑢 . 𝑡 =𝜂 ( 2/𝑚) 𝑡 𝑡 𝑡 𝑡 (9) We can acquire the median time mid exp ln ln 1 2 𝑘 𝑘 𝐹(𝑡 )=0.5 by referring to (1),when mid . The inflexion point Then the interval of 𝑝𝑖 is [0, (𝑡𝑖/𝑡𝑘)𝑝𝑢],where𝑝𝑢 is the upper (time) is obtained in a similar manner, which is denoted by 𝑡 =𝜂((𝑚 − 1)/𝑚)1/𝑚 limit preset by expert or engineering experience. turn . 4 Mathematical Problems in Engineering

1 As the data reveals no failure, we assume that 𝑡1 < ⋅⋅⋅ < 𝑡 <𝑡 𝐹 (𝑡) 𝑘 mid. Therefore, to find the interval, within which 0.9 exhibits convexity or concavity, we will determine the interval 𝑡 ≤𝑡 0.8 that satisfies mid turn. 1/𝑥 Let 𝑔(𝑥) = exp (ln ln 2/𝑥) and ℎ (𝑥) = ((𝑥 − 1)/𝑥) , 0.7 where 𝑥∈(1, 10].Thenitiseasytoprovethatboth𝑔 (𝑥) 0.6 and ℎ (𝑥) are strictly increasing in (1, 10] and 𝑔 (𝑥) =ℎ(𝑥), 0.5 when 𝑥 ≈ 3.3,asshowninFigure 1. Therefore, when 𝑚 ≥ 3.3, 𝑡 ≤𝑡 𝐹 (𝑡) 0.4 we have mid turn,and is concave. 3.3≤𝑚≤10𝐹 (𝑡) When , is concave. According to 0.3 thepropertyoftheconcavefunction(asshownin(9)), we

The value of the two functions of two the The value 0.2 have (𝑡𝑖/𝑡𝑖−1)𝑝𝑖−1 <𝑝𝑖.Thus,theupperlimitof𝑝𝑖 needs to be determined. Li [11]andZhang[15]suggestsettinga 0.1 𝑝 𝑝 universal value of the upper limit, upper,forall 𝑖,which,in 0 fact, extends the interval of 𝑝𝑖 and makes analysis inaccurate. 12345678910 𝑝 In this paper, we introduce the parameter con, which satisfies Parameter m 𝑝 <𝑝 +𝑝 𝑝 𝑖 𝑖−1 con,where con is the possible difference of 𝑝 𝑝 𝑝 ≜Δ𝑝=𝑝−𝑝 t(F = 0.5) 𝑖 and 𝑖−1,thatis, con 𝑖 𝑖−1.Asthereare 𝑘 censored tests with zero-failure data, we can assume that Inﬂection point 𝑝 0.5/𝑘 the value of con is around .So,itsvalueislimited Figure 1: Comparison between abscissas of inflection point and to a much smaller range and can be preset by expert or time when failure probability is 0.5. engineering experience. So we have when 𝑚∈[3.3, 10], 𝑝 ∈ ((𝑡 /𝑡 )𝑝 ,𝑝 +𝑝 ) 𝑖 𝑖 𝑖−1 𝑖−1 𝑖−1 con . (3) When 1≤𝑚≤3.3, there is no similar method to find the concave or convex properties of 𝐹 (𝑡).Wecan Underthesquarelossassumption,wecanacquirethe transform (2) into the following equations, given 𝑝𝑖−1 and 𝑝𝑖, expectation of 𝑝𝑖 astheestimateasfollows: respectively: 𝑠 +3 𝑝̂ = 𝑖 [𝑞 (𝑝 )−𝑞(𝑝)] , 1 𝑖 𝑠 +3 𝑠 +3 𝑢 𝑙 =𝑚 𝑡 −𝑚 𝜂, 𝑖 𝑖 (16) ln ln ln 𝑖−1 ln (1 − 𝑝𝑙) −(1−𝑝𝑢) 1−𝑝𝑖−1 (13) 1 ln ln =𝑚ln 𝑡𝑖 −𝑚ln 𝜂. where 1−𝑝𝑖 𝑠 +4 𝑠 +3 (1−𝑥) 𝑖 (1−𝑥) 𝑖 We can combine these two equations to obtain 𝑞 (𝑥) = − . (17) 𝑠𝑖 +4 𝑠𝑖 +3 (1 − 𝑝 ) 𝑡 𝑡 ln 𝑖−1 =𝑚 𝑖−1 < 𝑖−1 , ln ln ln (14) According to this, the interval of 𝑝𝑖 depends on the value ln (1 − 𝑝𝑖) 𝑡𝑖 𝑡𝑖 of 𝑚. But it is not necessary to find the exact value of 𝑚;only 𝑚 which is equivalent to the interval of is to be determined to calculate the estimate of failure probability.

ln (1 − 𝑝𝑖−1) ln (1 − 𝑝𝑖) Inourearlypaper[17], we presented the above estimate > . 𝑚 𝑡 𝑡 (15) without discussing the estimate of .Thereisalargeamount 𝑖−1 𝑖 of engineering experience in practice to help us estimate the interval of 𝑚 [18]. If there is little knowledge that we can refer Then we can obtain the lower limit of 𝑝𝑖 as 1− 𝑡 /𝑡 to, we propose using the following method [19] to determine (1 − 𝑝 ) 𝑖 𝑖−1 𝑝 𝑖−1 .Fortheupperlimitof 𝑖,inasimilarmanner, 𝑚,whichisbasedonhypothesistesting. 𝑝 +𝑝 1<𝑚<3.3 let it be 𝑖−1 con. So here we have, when , 𝑡 /𝑡 1−(1−𝑝 ) 𝑖 𝑖−1 <𝑝 <𝑝 +𝑝 𝑖−1 𝑖 𝑖−1 con. (1) Propose the hypothesis: 𝐻0 :𝑚=𝑚0 𝐻1 :𝑚=𝑚̸ 0. Now, we can determine the intervals of 𝑝𝑖,thefailure probability in prior distribution, with respect to the values (2) Construct the statistic 2 of 𝑚.Denotetheintervalby(𝑝𝑙,𝑝𝑢).Let(1 − 𝑝𝑖) denote the core of the prior distribution. To meet the requirement ∑𝑘−1 𝑖𝑊 2 𝑖=1 𝑖+1 ∫ 𝐴(1−𝑝) 𝑑𝑝 =1 𝐺𝑘 = , (18) of the distribution (i.e., 𝑝 𝑖 )thePDF 𝑘 (𝑘−1) ∑𝑖=1 𝑊𝑖 of the prior distribution of 𝑝𝑖 is defined by 𝜋(𝑝𝑖)= 2 3 3 3(1−𝑝𝑖) /((1−𝑝𝑙) −(1−𝑝𝑢) ). Then, by referring to (8), 𝑝 𝑚 𝑚 we can obtain the PDF of the posterior distribution for 𝑖 as where 𝑊𝑖 = (𝑘−𝑖+1) (𝑡𝑖 −𝑡𝑖−1) (𝑖=1,...,𝑘) and 𝑠𝑖+2 𝑠𝑖+3 𝑠𝑖+3 𝜋(𝑝𝑖 |𝑠)=(𝑠𝑖 +3)(1−𝑝𝑖) /((1−𝑝𝑙) −(1−𝑝𝑢) ). 𝑡0 =0. Mathematical Problems in Engineering 5

Table 1: Sample data.

Subgroup number 1 2 3 4 5 6 Parameters 𝑚 = 0.8, 𝜂 =2800 Censoring time (h) 33.329 34.388 141.569 417.819 657.716 778.398 𝑠 12 11 10 8 7 6 Parameters 𝑚 = 1.8, 𝜂 =2800 Censoring time (h) 531.821 634.651 826.586 1029.37 1360.66 1768.31 𝑠 12 11 9 7 5 3 Parameters 𝑚 = 3.8, 𝜂 =2800 Censoring time (h) 2.38799 785.303 1428.62 1706.08 1724.18 2237.69 𝑠 12 10 9 8 7 5

(3) When 3≤𝑘≤20,therejectionintervalfor𝐻0 is (2) Based on the estimate of 𝑚, determine the interval of {𝐺𝑘 >𝜁1−𝛼/2,𝐺𝑘 <𝜁𝛼/2},where𝜁𝛼/2 is the 100 (𝛼/2)% failure probability by quantile, and the CDF of 𝐺𝑘 is defined by 𝑝𝑖

{ 𝑡𝑖 {(𝑝𝑖−1, 𝑝𝑖−1), 0<𝑚≤1 −1 { 𝑡 𝑘−1 { 𝑖−1 { 𝑡 /𝑡 𝑃(𝐺 ≤𝑥)=𝑥𝑘−1 {∏𝑐 } {(1 − (1 − 𝑝 ) 𝑖 𝑖−1 , 𝑘 𝑖 ∈ 𝑖−1 𝑖=1 { { 𝑝𝑖−1 +𝑝 ) , 1 < 𝑚 < 3.3,1 𝑝 ∈(0,𝑝 ) { con upper −1 { 𝑡𝑖 𝑘−1 𝑘−1 {( 𝑝 ,𝑝 +𝑝 ) , 3.3 ≤ 𝑚 ≤ 10 𝑘−1 { } 𝑖−1 𝑖−1 con { 𝑡𝑖−1 − ∑ (𝑥 −𝑗 𝑐 ) {𝑐𝑗∏ (𝑐𝑟 −𝑐𝑗)} , 𝑗=𝑙+1 { 𝑟=𝑗̸ } 𝑖≥2. (19) (20)

(3) Estimate 𝑝𝑖 by (16).

where 𝑐𝑗 =(𝑘−𝑗)/(𝑘−1) and 𝑙=max (arg 𝑡≤𝑐𝑙). 4. A Numerical Example

(4) Let 𝑝 value = 𝑃{𝐺𝑘 >𝑔𝑟 |𝑚=𝑚0},whichismaxi- To illustrate the validity of our proposed method, it is applied mized, if 𝑚=𝑚0, 𝑔𝑟 isthesampleresultof𝐺𝑘.When to estimate the failure probability and compared with the true 𝑝 value is maximized, the estimate of 𝑚 is the best. value from a given Weibull distribution. Simulation data is used in this numerical example. There are twelve samples in the example and they will experience six censored tests. We use MATLAB software to generate several groups of From the above introduction to our proposed method, random values for twelve variables following a given Weibull one may find that to estimate the failure probability of a distribution and, for each group, randomly divide them Weibull distribution with zero-failure data, it requires only into six subgroups. The largest variable in each subgroup is roughly determining the interval of parameter 𝑚 (instead of regarded as the censoring time for the subgroup. To ensure obtaining its exact value). Besides, there is no need to match that the random variables generated correspond to no failure, the exact value of 𝑚 by the above hypothesis testing method the failure probability of each variable should be less than 0.5. as it greatly increases the complexity of the calculation. This means that only those groups whose failure probability Therefore, in practice, if the 𝑝 value in the hypothesis testing of the largest variable in each subgroup is less than 0.5 are is comparatively large, we can conclude that parameter 𝑚 is chosen as our samples. The samples are listed in Table 1. determined. After the data is obtained, the failure probability at censoring time 𝑡𝑖 is estimated and is then compared with thetruevalue,calculatedbythegivenWeibulldistribution 𝑡 3.3. Summary of the Method. The proposed method is sum- at censoring time 𝑖.Theresultsarealsocomparedwiththose marized as follows. calculated by the classical method in (7). In this example, we assume that the interval of parameter 𝑚 is predetermined by engineering experience or by our proposed hypothesis (1) 𝑚 𝑝 Roughly determine the interval of parameter in the testing, and we also try to figure out the effects that upper and 𝑝 Weibull distribution. If engineering experience is not con have on the final estimation. available, roughly estimate 𝑚 by referring to the 𝑝 Figure 2 shows the samples generated by the Weibull value in the hypothesis testing. distribution with the parameters 𝑚=0.8, 𝜂 = 2800.The 6 Mathematical Problems in Engineering

m = 0.8 the absolute error m = 0.8 real value and estimation between two estimations 0.35 0.25

0.3 0.2

0.25

0.15 0.2

0.15 0.1 The absolute error absolute The The failure probability The failure 0.1

0.05 0.05

0 0 0 246 0246 Serial number Serial number

p = 0.3 p = 0.25 Real value upper upper Classical estimation p = 0.25 p = 0.3 upper Classical estimation upper

Figure 2: Comparison of failure probability among real value, estimator, and classical estimation value when 𝑚 = 0.8.

m = 1.8 m = 1.8 real value and estimation the absolute error 0.4 0.25 between two estimations

0.35 0.2 0.3

0.25 0.15

0.2

0.1 0.15 The absolute error absolute The The failure probability The failure 0.1 0.05 0.05

0 0 0246 0 246 Serial number Serial number

p = 0.11 p = 0.1 Real value con con Classical estimation p = 0.1 p = 0.11 con Classical estimation con

Figure 3: Comparison of failure probability among true value, estimator, and classical estimation value when 𝑚 = 1.8. Mathematical Problems in Engineering 7

m=3.8real value and estimation m=3.8the absolute error between two estimations 0.35 0.35

0.3 0.3

0.25 0.25

0.2 0.2

0.15 0.15 The absolute error absolute The The failure probability The failure 0.1 0.1

0.05 0.05

0 0 0246 0246 Serial number Serial number

p = 0.11 p = 0.1 Real value con con Classical estimation p = 0.1 p = 0.11 con Classical estimation con

Figure 4: Comparison of failure probability among real value, estimator, and classical estimation value when 𝑚 = 3.8.

𝑝 values of upper areassignedto0.3and0.25tocomparethe theory to estimate the failure probability. The proposed 𝑝 effects of upper.Figures3 and 4 are generated by the Weibull method applies the concave and convex properties of the distributions with the parameters 𝑚 = 1.8, 𝜂 = 2800 and Weibull distribution with respect to the shape parameter, 𝑚, 𝑚 = 3.8, 𝜂 = 2800, respectively. In the calculation of Figures and provides the corresponding interval of failure probability, 𝑝 = 0.25 𝑝 𝑝 𝑝 3 and 4,welet upper ,wherethevaluesof con are . Then, the prior distribution of is constructed by using the 𝑝 assigned to 0.1 and 0.11, to compare the effects of con.Inall decreasing function method, based on which the estimate of these figures, the left graph compares the estimates by our 𝑝𝑖 is calculated by applying the Bayesian method. A numerical proposed method with those by the classical method with example is presented to compare the estimations made by thetruevalues;therightgraphshowstheabsolutedifferences the proposed method and the classical method and the true between the estimates by our proposed method and the true values, which illustrates the validity and robustness of the values, as well as the differences between the estimates by the proposed method. classical method with the true values. From the above comparisons, we find that Notations 𝑝 𝑝 (1) upper and con have limited effects on the failure probability estimation, which validates the robustness CDF: Cumulative distribution function of our proposed method; PDF: Probability density function MSE: Mean square error. (2) the estimations of our proposed method have less MSEs than those of the classical method, especially when the number of tests increases. The comparisons Conflict of Interests indicate that our method is more accurate than the classical method. The authors declare that there is no conflict of interests regarding the publication of this paper. 5. Conclusion Acknowledgments Reliability analysis based on zero-failure data attracts more and more attention as products become more reliable and This research is supported by the projects of the National very few failures are observed during testing. To solve the Natural Science Foundation of China (with Grant nos. failure probability estimation problem in the Weibull distri- 71371182 and 71401170) and the Research Project of National bution with zero-failure data, this paper presents a method University of Defense Technology (with Grant no. JC13-02- of combining the decreasing function method with Bayesian 05). 8 Mathematical Problems in Engineering

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Research Article Fast Prediction with Sparse Multikernel LS-SVR Using Multiple Relevant Time Series and Its Application in Avionics System

Yang M. Guo,1 Pei He,1 Xiang T. Wang,1 Ya F. Zheng,1 Chong Liu,2 and Xiao B. Cai3

1 School of Computer Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China 2School of Software and Microelectronics, Northwestern Polytechnical University, Xi’an 710072, China 3Science and Technology Commission, Aviation Industry Corporation of China, Beijing 100068, China

Correspondence should be addressed to Yang M. Guo; yangming [email protected]

Received 3 July 2014; Revised 8 October 2014; Accepted 24 November 2014

Academic Editor: Enrico Zio

Copyright © 2015 Yang M. Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Health trend prediction is critical to ensure the safe operation of highly reliable systems. However, complex systems often present complex dynamic behaviors and uncertainty, which makes it difficult to develop a precise physical prediction model. Therefore, time series is often used for prediction in this case. In this paper, in order to obtain better prediction accuracy in shorter computation time, we propose a new scheme which utilizes multiple relevant time series to enhance the completeness of the information and adopts a prediction model based on least squares support vector regression (LS-SVR) to perform prediction. In the scheme, we apply two innovative ways to overcome the drawbacks of the reported approaches. One is to remove certain support vectors by measuring the linear correlation to increase sparseness of LS-SVR; the other one is to determine the linear combination weights of multiple kernels by calculating the root mean squared error of each basis kernel. The results of prediction experiments indicate preliminarily that the proposed method is an effective approach for its good prediction accuracy and low computation time, and it is a valuable method in applications.

1. Introduction property. However, it may fall into local minimum traps and there is difficulty in determining the hidden layer size and Over the past decade, many diagnosis methodologies have thelearningrate[15–17]. On the other hand, SVR aims at beenappliedsuccessfullyinthedomainofthereliability the global optimum and exhibits better prediction accuracy and safety of systems [1–7]. However, with the development due to its implementation of the structural risk minimization of prognosis and health management (PHM), health con- principle [18, 19]. dition diagnosis and prognosis technology has become the SVR is an effective methodology for handling function key technique for enhancement of reliability and safety of estimation problems [13, 14, 20]. SVR is performed endoge- complex systems, especially for high reliability systems. How- nouslyasapartoftheoptimizationproblemwhichhas ever, these complex systems often exhibit multiple dynamic a unique solution of a (convex) quadratic programming evolution behaviors, multilevel structures, and incomplete or (QP) problem. The solution of the QP problem provides the uncertain information. Therefore, it is difficult to establish an necessary information for choosing the most important data accurate physical model. The time series analysis methods are points, known as support vectors, among all the sample data. often utilized to analyze and predict their health conditions in Based on the SVR formulation, support vectors uniquely practice [7–10]. define the estimated regression function. Numerous methods have been reported to solve these The application of SVR is initially developed for per- problems, such as artificial neural networks (ANN) and forming linear regression. The technique called the kernel support vector regression (SVR) [5, 6, 8, 10–14]. ANN has trick has been extended to handle nonlinear regression appli- been used in several fields due to its universal approximation cations [18–20]. Firstly, a kernel function 𝜑(𝑥) is used to 2 Mathematical Problems in Engineering map the input data 𝑥 into an arbitrarily high dimensional kernels by calculating the root mean squared error feature space. Secondly, the linear SVR can be applied to (RMSE) of prediction with each basis kernel. create an approximate linear function in this arbitrarily high dimensional feature space. This way, nonlinear regression (2) We propose to prune the support vectors by judging in the low dimensional input space corresponds to linear the linear correlation among the samples in the high regression in the high dimensional feature space [21]. dimensional feature space. This scheme makes the The complexity of SVR depends not only on the input information included in the sample data to be amply space dimension, but also on the number of sample data represented, reduces the number of support vectors, points. Therefore, when the sample data size is large, the enhances the generalization performance, and short- quadratic programming (QP) problem becomes more com- ens the prediction time with better or acceptable plex, which will cost a lot of computation time. For this prediction accuracy. reason, least squares support vector regression (LS-SVR) was proposed by Suykens et al. [22]. In LS-SVR, the inequality The remainder of the paper is organized as follows. constrains are replaced by equality constrains. This way, Section 2 gives a brief introduction of LS-SVR; Section 3 pro- solvingaQPisconvertedintosolvinglinearequations.This poses the scheme in detail which includes pruning support will directly reduce training computational complexity. Thus, vectors of LS-SVR and computing combination coefficients of the training time of LS-SVR is reduced greatly. multikernel, and then the fast prediction steps are presented; LS-SVR has been demonstrated by simulations and appli- Section 4 shows three experiments and results analysis; and cations that it has better prediction results in most cases [23– conclusions are given in Section 5. 26]. Moreover, in order to obtain abundant information on the physical system, researchers proposed multiple relevant 2. A Brief Introduction of LS-SVR time series prediction methods which achieve better prediction results than single time series prediction [27–29]. In the Given the labeled data set 𝑆={(𝑥𝑖,𝑦𝑖),𝑖=1,2,...,𝑛},where 𝑑 practice of health trend fast prediction for high reliability 𝑥𝑖 ∈𝑅 is the input sample and 𝑦𝑖 ∈𝑅is the corresponding systems via LS-SVR with multiple relevant time series, we output label, the main idea of regression based on support also need to overcome some serious drawbacks in reported vector machines theory is to map the input data to a higher methods as described below. dimensional feature space by means of a nonlinear mapping 𝜑:𝑅𝑑 →𝐹 (1) Because all the training sample data are selected as and solve the linear regression model shown as support vectors, this leads to redundant information follows: in the fitting of LS-SVR and lack of sparseness [30]. 𝑦̂ =𝑤𝑇𝜑(𝑥)+𝑏, This drawback may influence the generalization capa- 𝑖 𝑖 (1) bility of LS-SVR as well as the training complexity, ̂ which will increase prediction time, especially for where 𝑦𝑖 is the estimation of 𝑦𝑖, 𝑓 istheregressionfunction, multiple relevant time series. 𝑤 is the coefficient value, and 𝑏 is the bias term. The rationale behind the support vector regression (SVR) is to find 𝑤 and (2) The information contained in the sample data is very 𝑏 that optimize the generalization ability of the regressor by often incomplete and complex. Thus, LS-SVR with minimizing the regularized loss function. The case of SVR is a single kernel may not be sufficient to represent also a convex quadratic programming (QP) problem, and it the information contained in data, and it cannot has a unique optimal solution. But the computation process solve complex problems satisfactorily, especially for is complex. So, many researchers proposed modifications of prediction with multiple relevant time series sample SVR. data. Moreover, how to select appropriate kernel The LS-SVR method is a variant of the SVR [11]. Com- functions remains unsolved in theory. pared with SVR, LS-SVR preserves the following charac- Aiming at multiple relevant time series fast prediction teristics: the error variable 𝑒𝑖 is used to control deviations with LS-SVR, we develop a new scheme to overcome the from the regression function and a squared loss function is drawbacks and obtain better prediction performance, that used instead of the insensitive loss function. According to is, accurate prediction results and fast prediction simulta- theapproachofLS-SVR,themodelestimateisgivenbythe neously. In this paper, combining the merits of LS-SVR following optimization problem: with a single kernel for multiple relevant time series [27] and LS-SVR with multikernels for single time series [31], 1 𝑐 𝑛 𝐽 (𝑤,) 𝑏 = ‖𝑤‖2 + ∑𝑒2 we propose a new fast prediction method to improve the min 2 2 𝑖 prediction performance with multiple relevant time series 𝑖=1 (2) and multiple basis kernels. This method provides a regres- 𝑇 s.t.𝑦𝑖 =𝑤 𝜑(𝑥𝑖)+𝑏+𝑒𝑖 𝑖=1,2,...,𝑛. sion scheme which integrates multiple kernel learning and enforces sparseness from the training sample data. It includes two aspects. The above formulation is nothing but a regression cost function formulated in the feature space defined by the (1) We apply a simple computational method to deter- mapping function 𝜑(𝑥). Parameter 𝑐 determines the trade- mine the combination weights of multiple basis off between the model complexity and the goodness of fit to Mathematical Problems in Engineering 3 thesampledata.InLS-SVR[22], the Lagrangian function is Any unlabeled input 𝑥 canbesubsequentlyestimatedby presented as follows: the following function: 𝑛 𝑛 𝑛 𝑇 1 𝑇 1 2 𝑦̂ (𝑥) = ∑𝛼𝑖𝜑(𝑥𝑖) 𝜑 (𝑥) +𝑏=∑𝛼𝑖𝑘(𝑥𝑖,𝑥)+𝑏. 𝐿 (𝑤, 𝑏,) 𝑒,𝛼 = 𝑤 𝑤+ 𝑐∑𝑒𝑖 (11) 2 2 𝑖=1 𝑖=1 𝑖=1 (3) 𝑛 For multiple relevant time series, we can consider all 𝑇 − ∑𝛼𝑖 (𝑤 𝜑(𝑥𝑖)+𝑏+𝑒𝑖 −𝑦𝑖), sample data at the same time point as one multidimensional 𝑖=1 vector, which means the input will become multidimensional vectors. where 𝛼𝑖 is the 𝑖th Lagrange multiplier. Because (2) is also a convex function, it is obvious that the Slater constraint qualification holds [32]. Therefore, the optimal solution of (2) 3. Proposed Fast Prediction Method for satisfies the KKT conditions. The optimality conditions are Multiple Relevant Time Series shown as follows: In order to obtain better prediction performance, we use mul- 𝜕𝐿 𝑛 𝑛 tiple relevant time series and multiple kernel learning (MKL) =𝑤−∑𝛼 𝜑(𝑥)=0󳨐⇒𝑤=∑𝛼 𝜑(𝑥), 𝑖 𝑖 𝑖 𝑖 to fully utilize the information contained in data. In this 𝜕𝑤 𝑖=1 𝑖=1 section,weproposeanimprovedschemewithmultikernel 𝜕𝐿 𝑛 𝑛 LS-SVR to accomplish fast prediction. This scheme utilizes =−∑𝛼 =0󳨐⇒∑𝛼 =0, 𝜕𝑏 𝑖 𝑖 new approaches to compute the combination coefficients of 𝑖=1 𝑖=1 multikernel and to decrease the number of support vectors. 𝜕𝐿 𝑇 𝑇 =𝑤 𝜑(𝑥𝑖)+𝑏+𝑒𝑖 −𝑦𝑖 =0󳨐⇒𝑦𝑖 =𝑤 𝜑(𝑥𝑖)+𝑏+𝑒𝑖, 𝜕𝛼𝑖 3.1. Combination Coefficients of Multiple Kernels. The kernel function and the corresponding kernel parameters are the key 𝜕𝐿 1 issues affecting the model prediction accuracy. An effective =𝑐𝑒𝑖 −𝛼𝑖 =0󳨐⇒𝑒𝑖 = 𝛼𝑖. 𝜕𝑒𝑖 𝑐 kernel function should represent sample data adaptively. (4) General kernel methods use a single kernel function and choose the same corresponding parameter for the whole After eliminating 𝑒𝑖 and 𝑤 from (4), the KKT conditions sample data set. However, the distribution of the sample canbeexpressedas data in a different mapping space is different. So, MKL was proposed by Lanckriet et al. [34]. 𝑇 −1 𝑏 0 1𝑛 0 MKL is an active research topic in the field of machine [ ]=[ I ] [ ], (5) 𝛼 1 K + y learning. It provides a more flexible framework than a single [ 𝑛 𝑐] kernel and mines information in data more adaptively and

𝑇 𝑇 effectively, especially in improving performance of the regres- where 𝛼 =[𝛼1,𝛼2,...,𝛼𝑛] , K𝑖,𝑗 =𝑘(𝑥𝑖,𝑥𝑗)=𝜑(𝑥𝑖) 𝜑(𝑥𝑗), sion function. 1𝑛 is an 𝑛-dimensional vector of all ones, I is a unity matrix, In MKL framework, a combined kernel function is 𝑇 and y =[𝑦1,𝑦2,...,𝑦𝑛] .Equation(5) can be factorized into defined as the weighted sum of the individual basis kernels. a positive definite system33 [ ]. MKL aims to optimize kernel weights while training the SVR- Let H = K + I/𝑐, and we get the following equations from based methods, such as LS-SVR with multikernel [28, 31, 35]. (5): Though researchers have proposed a variety of methods of integrating multiple kernels from three aspects, which are 1𝑇𝛼 =0, 𝑛 (6) the composite kernels, the multiscale kernels, and the infinite 1 𝑏+H𝛼 = y. kernels [35], linear convex combination of basis kernels is still 𝑛 (7) one of the most frequently used methods. For this method, 1𝑇H−1 each basis kernel can exploit the full set of features or just use a Multiplying 𝑛 to both sides of (7),weget subset of features. In this paper, using the equations provided 1𝑇H−11𝑏+1𝑇𝛼 = 1𝑇H−1y. by Sonnenburg et al. [36], we consider the combined kernel 𝑛 𝑛 𝑛 (8) expressed as follows: According to (6), (7),and(8),wehave 𝑚 𝑘(𝑥𝑝,𝑥𝑞)=∑𝜇𝑗𝑘𝑗 (𝑥𝑝,𝑥𝑞) 𝑇 −1 𝑇 −1 𝑗=1 1𝑛 H 1𝑏=1𝑛 H y. (9) 𝛼 𝑏 𝑚 (12) The Lagrange dual variables and the bias term are then {∑𝜇 =1 obtained solely by . . 𝑗 s t {𝑗=1 −1 {𝜇𝑗 ≥0 𝑗=1,2,...,𝑚, 𝛼 = H (y − 1𝑛𝑏) , (10) where 𝑚 is the number of basis kernels and 𝜇𝑗 is the 𝑇 −1 𝑇 −1 −1 𝑏=1𝑛 H y (1𝑛 H 1𝑛) . combining weight for the 𝑗th basis kernel. According to 4 Mathematical Problems in Engineering the properties of kernel functions [37], matrix 𝑘 is symmetric Thus, the RMSE of prediction is used as the evaluation positive semidefinite; that is, 𝑘⪰0. We need to normalize all criterion. The RMSE of the multiple variables prediction is kernel matrices 𝑘𝑗 by replacing 𝑘𝑗(𝑥𝑝,𝑥𝑞) with the following defined as follows: equation to get unit diagonal matrices: 𝑟 𝑛 𝑘𝑗 (𝑥𝑝,𝑥𝑞) 1 2 𝑘 (𝑥 ,𝑥 ) 󳨀→ . 𝛿 = √ ∑ ∑ (𝑦 (𝑘) − 𝑦̂ (𝑘)) , (16) 𝑗 𝑝 𝑞 (13) RMSE 𝑛𝑟 𝑖 𝑖 𝑖=1 𝑘=1 √𝑘𝑗 (𝑥𝑝,𝑥𝑝)𝑘𝑗 (𝑥𝑞,𝑥𝑞)

Based on MKL, the LS-SVR optimization problem with where 𝑟 isthenumberofrelevantvariables,𝑛 is the number of the new 𝑘 matrix derived from (2)–(4) is represented as fol- the original training sample data points, and 𝑦𝑖(𝑘) and 𝑦̂𝑖(𝑘) lows: are the prediction and the actual values, respectively. The min max 𝐿 (𝑤, 𝑏,) 𝑒,𝛼 = max min 𝐿 (𝑤, 𝑏,) 𝑒,𝛼 . 𝜇 𝑤,𝑏,𝑒 𝛼 𝛼 𝑤,𝑏,𝑒 (14) combination weights 𝑗 ofthemultiplekernelsarecomputed as follows: We then transform the new kernel into the matrix form 𝑚 𝑚 K = ∑ 𝜇𝑗K𝑗. When the weights 𝜇𝑗 are constrained to be 𝑗=1 ∑𝑖=1 𝛿𝑖 −𝛿𝑗 nonnegative and K𝑗 are positive semidefinite, the constraint 𝜇𝑗 = 𝑚 , (17) 𝑚 (𝑚−1) ∑𝑖=1 𝛿𝑖 ∑𝑗=1 K𝑗 ⪰0issatisfiedautomatically.Inthiscase,with respect to the corresponding parameters motivated by Lanck- 𝑚 riet et al. [34]andYeetal.[38], the solution of (14) can be where 𝛿𝑗 is the prediction RMSE of the 𝑗th kernel, ∑𝑖=1 𝛿𝑖 is 𝑚 presented as follows: the sum RMSE of all basis kernels, and ∑𝑖=1 𝛿𝑖 −𝛿𝑗 presents 𝑗 −1 the contribution of the th kernel. 1 𝑚 I (y +𝜅1 )𝑇 (∑𝜇 K + ) (y +𝜅1 ) min𝜇,𝜅 2 𝑛 𝑗 𝑗 𝑐 𝑛 𝑗=1 3.2. Linear Correlation-Based Method of Pruning Support (15) 𝑚 𝑚 Vectors. LS-SVR has a major drawback that is lack of sparseness [42] because almost each sample data point will be s.t. ∑𝜇𝑗K𝑗 ⪰0, ∑𝜇𝑗 =1. 𝑗=1 𝑗=1 a support vector. In order to get a sparse solution, several methods have been proposed. For example, Suykens et al. Using the Schur complement lemma [32, 39], (15) can be [43] proposed a pruning scheme based on support vector cast to the form of semidefinite program (SDP), and then spectrum that prunes those support vectors with smaller we reduce the SDP under strict constraints to a quadrati- Lagrange multiplier values. de Kruif and de Vries [42] cally constrained quadratic program (QCQP). The objective introduced a procedure of pruning according to the smallest function of QCQP is convex in 𝜇 and 𝛼.Inotherwords, approximation error when the support vector is omitted. the minimization problem is strictly feasible in 𝜇,andthe Hoegaerts et al. [44]improvedthemethodproposedbyKruif maximization problem is feasible in 𝛼 [34, 40]. Such a et al. They suggested a variant that improved the performance QCQP problem can be solved efficiently by the interior point significantly and assessed its relative performance compared methods. The obtained dual variables can be used to fix the with two other subset selection schemes. Keerthi and Shevade optimal kernel coefficients. [45] extended the well-known SMO algorithm to LS-SVM For MKL, with linear convex combination of basis ker- to solve problems with large training samples. Based on nels, the key is to obtain the optimal combing weights 𝜇𝑗. this study, Zeng and Chen [46]proposedanewpruning Some researchers proposed to simultaneously optimize both algorithm for sparse LS-SVR via SMO algorithm. Jiao et al. the combing weights 𝜇𝑗 and the parameters of LS-SVR, for [47] presented two fast sparse approximation schemes for LS- example, regularization of the parameter 𝑐 and the basis SVR. kernels parameters and so on. Now, the more common The basic idea of the many approaches reviewed above way is to adopt optimization software packages, such as is to find the support vector with the smaller indicator MOSEK [41], which can solve the primal and dual problems value and remove the corresponding training sample. It is simultaneously using the interior point methods. But all these obvious that the idea is simple, but all the reported schemes solutionmethodsalwaysrequiremuchcomputationtimeand require computation duplication in solving linear equations are complex for applications in practice. [48, 49]. For this reason, these schemes are not suitable for Because the combination of the multiple kernels is a linear fast prediction. Thus, Yaakov et al.50 [ ]andCawleyandTalbot combination in this paper, the realistic application will be [51] presented a more efficient scheme by evaluating the takenintofullaccount;here,weproposeasimpleapproach linear correlation among the samples in the mapping space. to determine the combination weights of the new kernel by The merits of [48–51] are that the linear correlation-based calculating the root mean squared error (RMSE) of prediction sparseness strategy ensures the information in the data to using each single basis kernel; that is, a smaller RMSE value be maximally retained and the sparseness prediction model will result in a bigger weight value. improves the generalization capability to improve the model’s That the prediction value has less relative error means that adaptability to noisy sample data. So, the proposed method the LS-SVR model with the basis kernel is a better model. bases on [48–51] and uses the following idea. Mathematical Problems in Engineering 5

Suppose that 𝜑(𝑥𝑘) is one of the 𝑛 support vectors in the 𝑙 𝑐 {𝑥 ,𝑥 ,...,𝑥,...,𝑥 } high dimensional feature space. If and only if 𝜆𝑖 =0(𝑖 =𝑘̸ ), parameters: , , support vectors set 1 2 𝑡 𝑛 the following equation holds: Initialize: 𝑙=2, 𝑐 equals to the LS-SVR parameter 𝑐 (1) choose the initial BV: 𝑥1 and 𝑥2; 𝑛 (2) establish the initial BVS: {𝑥1,𝑥2}; 𝑥 𝜑(𝑥𝑘) ≠ ∑ 𝜆𝑖𝜑(𝑥𝑖). (18) (3) get new support vector 𝑡; 𝑖=1,𝑖=𝑘̸ (4) approximate the linear dependence compute via (23) 󸀠 if 𝛿𝑡 >0 𝑙=𝑙+1 Here, 𝑥𝑘 is called base vector (BV) and the set consisting , 𝑥 =𝑥 of 𝑥𝑘 is called base vector set (BVS). In order to obtain 𝑙 𝑡, 𝑥 sparseness in LS-SVR, any support vector linearly related add 𝑙 into the BVS, 𝑛 else to the BV can be deleted. Assume that there are support 𝑥 𝑙 drop 𝑡, vectors in the sample data and the BVS consists of support 𝑙=𝑙 𝑙<𝑛 𝜑(𝑥 ) vectors ( ). If the high dimensional mapping 𝑡 of (5)gobackto(3). the (𝑙+1)th sample 𝑥𝑡 cannot be represented by a linear combination of the 𝑙 support vectors, it will be added into the BV; otherwise, it does not join the BVS. Pseudocode 1: Pseudocode of the proposed pruning method. In many real cases, the new sample data may not be absolutely linearly dependent on the existing support vectors. However, it is very similar to a linear combination of the exist- where 𝛾 is a positive constant coefficient. It is obvious that the ing support vectors; that is, for sample 𝑥𝑡,wewillbecontent sparseness level and the model accuracy are controlled by the with finding coefficients 𝜆𝑡,𝑖 with at least one nonzero element parameters 𝜐 and 𝛾.Butnoruleshavebeenreportedinthe satisfying the approximate linear dependence condition literature for the selection of these two parameters. 2 2 In this paper, we select 𝜐=‖𝜆𝑡‖ /𝑙𝑐 ,andthenwepropose 󵄩 󵄩2 󵄩 𝑙 󵄩 a new sparseness method, which is described as follows: 󵄩 󵄩 𝛿𝑡 = 󵄩∑𝜆𝑡,𝑖𝜑(𝑥𝑖)−𝜑(𝑥𝑡)󵄩 ≤𝜐, (19) 󵄩 󵄩 2 󵄩𝑖=1 󵄩 󵄩 𝑙 󵄩 󵄩 󵄩2 󵄩 󵄩 󵄩𝜆𝑡󵄩 𝛿 = 󵄩∑𝜆 𝜑(𝑥 )−𝜑(𝑥)󵄩 ≤ . (22) 𝑡 󵄩 𝑡,𝑖 𝑖 𝑡 󵄩 𝑙𝑐2 where 𝜐 is a small positive constant. Then, [48, 50]rewrote 󵄩𝑖=1 󵄩 (19) into the following optimization problem: Equation (22) can be solved as the following optimization 𝑙 𝑙 problem: 𝑇 min 𝛿𝑡 = ∑ ∑𝜆𝑡,𝑖𝜑(𝑥𝑖)𝜆𝑡,𝑗𝜑(𝑥𝑗) 𝜆 𝑙 𝑙 𝑡 𝑖=1 𝑗=1 󸀠 𝑇 min 𝛿𝑡 = ∑ ∑𝜆𝑡,𝑖𝜑(𝑥𝑖)𝜆𝑡,𝑗𝜑(𝑥𝑗) 𝜆 𝑙 𝑡 𝑖=1 𝑗=1 𝑇 𝑇 −2𝜑(𝑥𝑡) ∑𝜆𝑡,𝑖𝜑(𝑥𝑖) +𝜑(𝑥𝑡)𝜑(𝑥𝑡) 󵄩 󵄩2 𝑙 󵄩𝜆 󵄩 𝑖=1 −2𝜑(𝑥) ∑𝜆 𝜑(𝑥)𝑇 +𝜑(𝑥)𝜑(𝑥)𝑇 − 󵄩 𝑡󵄩 (20) 𝑡 𝑡,𝑖 𝑖 𝑡 𝑡 𝑙𝑐2 𝑙 𝑙 𝑖=1 = ∑ ∑𝜆 𝜆 𝑘(𝑥,𝑥 ) 𝑡,𝑖 𝑡,𝑗 𝑖 𝑗 𝑙 𝑙 𝑖=1 𝑗=1 = ∑ ∑𝜆𝑡,𝑖𝜆𝑡,𝑗𝑘(𝑥𝑖,𝑥𝑗) 𝑙 𝑖=1 𝑗=1 −2∑𝜆𝑡,𝑖𝑘(𝑥𝑡,𝑥𝑖)+𝑘(𝑥𝑡,𝑥𝑡). 󵄩 󵄩2 𝑙 󵄩𝜆 󵄩 𝑖=1 −2∑𝜆 𝑘(𝑥,𝑥)+𝑘(𝑥,𝑥)−󵄩 𝑡󵄩 𝑡,𝑖 𝑡 𝑖 𝑡 𝑡 2 𝑖=1 𝑙𝑐 By minimizing the left-hand side of (20),wecanobtain 𝜆 =[𝜆 ,𝜆 ,...,𝜆 ]𝑇 󵄨 󵄨 the linear correlation coefficients 𝑡 𝑡,1 𝑡,2 𝑡,𝑙 s.t. 󵄨𝜆𝑡,𝑖󵄨 ≤𝑐, simultaneously. (23) To sacrifice the optimality of 𝜆𝑡 in return for a reduction in the size of the sample data, a 𝑙2 norm regularization term where 𝑐 isthesameastheLS-SVRparameter𝑐.Solving 2 2 2 of the form 𝛾‖𝜆𝑡‖ isaddedtotheminimizationproblem (23) and if 𝛿𝑡 is bigger than ‖𝜆𝑡‖ /𝑙𝑐 ,add𝑥𝑡 into the BVS; defined in (20), ending up with the following optimization otherwise, 𝑥𝑡 is dropped. problem: Following the ideas outlined above, we propose the following approach to prune support vectors. A detailed 𝑙 𝑙 pseudocode account of the proposed method is given in min 𝛿𝑡 = ∑ ∑𝜆𝑡,𝑖𝜆𝑡,𝑗𝑘(𝑥𝑖,𝑥𝑗) Pseudocode 1. 𝜆𝑡 𝑖=1 𝑗=1 Based on the pseudocode, we choose two sample data (21) 𝑙 points corresponding to the two biggest Lagrange multipliers 󵄩 󵄩2 as the initial BV firstly, and then if the new nonlinear mapping −2∑𝜆𝑡,𝑖𝑘(𝑥𝑡,𝑥𝑖)+𝑘(𝑥𝑡,𝑥𝑡)+𝛾󵄩𝜆𝑡󵄩 , 𝑖=1 𝜑(𝑥𝑟) depends linearly on the selected support vectors, it can 6 Mathematical Problems in Engineering

be dropped; that is, if the new sample data 𝑥𝑟 satisfies the (4) assess the linear correlation and prune support vec- relation formula tors and compute new BV-based Lagrange multiplier 𝑚 𝛽𝑖; 𝜑(𝑥𝑟)=∑𝜆𝑗𝜑(𝑥𝑗), (24) (5) set up the regression function and perform predic- 𝑗=1 tion. where 𝜆𝑖 is a constant coefficient and 𝑚 is the number of independent support vectors, then the new data point is 4. Experiments and Result Analysis dropped. Combining (24) with (11),wehave In order to examine the prediction efficiency of the proposed 𝑛 method, we provide simulation and application experiments. 𝑦̂ (𝑥) = ∑ 𝛼𝑖𝑘(𝑥𝑖,𝑥)+𝛼𝑟𝑘(𝑥𝑟,𝑥)+𝑏 All the experiments use MatlabR2011b with LS-SVMlab1.8 𝑖=1,𝑖=𝑟̸ Toolbox(thesoftwareandguidebookcanbedownloaded 𝑛 from http://www.esat.kuleuven.be/sista/lssvmlab)underWin- = ∑ 𝛼𝑖𝜑(𝑥𝑖)𝜑(𝑥) +𝛼𝑟𝜑(𝑥𝑟)𝜑(𝑥) +𝑏 (25) dows XP operating system. 𝑖=1,𝑖=𝑟̸

𝑛 𝑚 4.1. Simulation Experiments and Results Analysis. The sim- = ∑ 𝛼𝑖𝜑(𝑥𝑖)𝜑(𝑥) +𝛼𝑟∑𝜆𝑗𝜑(𝑥𝑗)𝜑(𝑥) +𝑏, ulation experiments include two parts. One is to test the 𝑖=1,𝑖=𝑟̸ 𝑗=1 proposed method based on linear correlation (Experiment 1) and the other is to test the proposed computing method where 𝑥𝑗 is the sample corresponding to BV.Then, we will get of multikernel combination coefficients (Experiment 2). All the new regression function with BV, and it can be described tests are repeated 50 times with the results averaged. The two by experiments to test the proposed scheme are presented below 𝑚 in detail. 𝑦̂ (𝑥) = ∑𝛽𝑖𝑘(𝑥𝑖,𝑥)+𝑏, (26) 𝑖=1 Experiment 1. We first used LS-SVR to learn the 1- dimensional function 𝑥=sin(𝑡)/𝑡 defined in the interval where 𝛽𝑖 is combined with Lagrange dual variable coefficient 𝑡 ∈ [0.1, 10]. We collect 100 values of 𝑥 as our time series. and linear combination coefficient. The data points from 1 to 60 in the time series are taken as The proposed method combines advantages of removing the 50 initial training sample data points. The first sample smaller support vectors and the advantages of [48–51]. It data set consists of points 1 through 11, with the first 10 as can retain much of the useful information in the sample the input sample vector and the 11th point as the output. The data. Thus, the prediction accuracy will not be sacrificed too second sample data set consists of points 2 through 12, with much while the model’s generalization ability is improved. the points 2 through 11 as the input sample vector and the Although this processing step would increase the calculation 12th point as the output. This way we have 50 training data time for the LS-SVR model, the training time will not be points out of the first 60 data points. increased much because almost each middle parameter’s valueisalreadycomputedandstoredintheprocessofsetting In order to test the efficiency of the proposed pruning up the prediction model, and the prediction time will be method, we compare it with the method which Yaakov et al. presented in [50]. Here, the kernel is Gaussian RBF kernel reduced because of fewer support vectors. Moreover, the 2 2 proposed method can avoid a difficult problem in Yaakov’s 𝑘(𝑥, 𝑦) = exp(−‖𝑥 − 𝑦‖ /2𝜎 ) with standard deviation 𝜎= scheme, the parameters selection, because no rules have been 1. The other parameters are 𝜐 = 0.01 and 𝜐 = 0.001. reported to fix the problem. Thus, it may be more valuable in In this experiment, the training time, prediction time, and practice. prediction mean RMSE are compared. The results are shown in Table 1. 3.3. Prediction Steps of the Proposed Method. By applying Table 1 shows that the proposed method attains good theschemeproposedabove,wecansetuptheLS-SVR accuracy with less computation time. In particular, it can modelwithmultikernelformultiplerelevanttimeseries, reduce the prediction time greatly without losing much called the improved MKLS-SVR (IMKLS-SVR), and we will prediction accuracy. In addition, this method avoids the use this new model to carry out multiple relevant time problemofparameterselection. series prediction. Here, we look at the multiple relevant Experiment 2. Henon function is a typical time series, and its time series as an integral whole; that is, they are taken as variables influence each other. Its corresponding differential amultidimensionalinput[52, 53], and the prediction is equations are given as follows: executed simultaneously. 2 The prediction steps can be described as follows: 𝑥 (𝑡+1) =1−𝑎𝑥 (𝑡) +𝑦(𝑡) , (27) (1) choose appropriate basis kernels; 𝑦 (𝑡+1) =𝑏𝑥(𝑡) . 𝜇 (2) determine 𝑗 andthenobtainthecombinationkernel; Let 𝑎=1.4and 𝑏 = 0.3, and points of the initialization of (3) compute K, H and 𝛼, 𝑏; 𝑥 and 𝑦 are [0.1, 0.1]. 100 data points of variables 𝑥 and 𝑦 are Mathematical Problems in Engineering 7

Table 1: Prediction results of simulation Experiment 1.

Method Training time (s) Prediction time (s) Mean RMSE LS-SVR without sparseness 0.001053 0.018954 1.5032 Method of [50] 𝜐 = 0.01 0.002072 0.009461 1.8673 𝜐 = 0.001 0.003179 0.009241 1.7780 The proposed method 0.002242 0.009407 1.6160

Table 2: Prediction results of simulation Experiment 2.

Mean RMSE Method Training time (s) Prediction time (s) 𝑥𝑦 LS-SVR with RBF kernel 1.273911 1.264925 3.1418 4.6412 Method of [31] 2.783566 0.962933 1.4001 0.8322 The Proposed method 1.659715 0.877201 1.5273 1.0125

Table 3: Prediction results of application experiment.

Mean RMSE Method Mean total prediction time reduced 𝑥𝑦𝑧 (compare with LS-SVR) LS-SVR of [25] 3.0662 1.5252 2.3811 MKLS-SVR of [31] 2.4722 0.7416 1.8869 6.18% The proposed IMKL-SVR 2.5668 0.8651 1.8412 13.83%

selected, respectively, as sample time series. We set the first 80 to compare the prediction time and prediction accuracy with data points as training samples. We also take any continuous the traditional LS-SVR model described in [25]andLS-SVR 11 data points of 𝑥 and 𝑦 as a sample, where the first 10 data with multikernel, called MKLS-SVR, shown in [31]. pointscomposeaninputsamplevectorandthelastoneisthe In order to collect the sample data, we measure voltage output vector; that is, in this simulation experiment, we have values at three measuring points named point A, point B, and 70 training data points for each variable. And then we predict point C (with pentagram representations, see Figure 1). The number 81 to number 100 time series data using the trained acquisition step is 0.1 s with time unit of 4.5 s. model. We take the first 30 measured data points group as the In order to test the practicability of the proposed com- training samples. Again each sample set consists of any puting method of multikernel combination coefficients, we continuous 11 data points; that is, we have 20 groups of 𝑇 training data. And then we predict number 31 to number 45 choose one linear kernel function 𝑘(𝑥, 𝑦) =𝑥 𝑦 representing 2 time series data. Here, we look at the time series obtained global information and two Gaussian RBF functions (𝜎 from point A, point B, and point C (see Figure 1)asanintegral equal to 5 and 9, resp.) representing local information whole; that is, they are examined as a multidimensional [54] and establish multikernel LS-SVR, respectively, by two input and do prediction simultaneously. We run 100 times approaches which are the proposed method and the one of prediction and take the average of them. Moreover, all the described in [31]. After determining the new combination parameters of kernel functions are jointly optimized with the weights of the basis kernels, we compare the training time, traditional grid search method, where the search range for prediction time, and prediction mean RMSE with LS-SVR 2 2 𝑐 and 𝜎 is [0.01, 2000], and the prediction performance is (with single Gaussian RBF kernel, 𝜎 =5) and multikernel measured by RMSE. LS-SVR.TheresultsareshowninTable 2. Gaussian RBF kernel is adopted as the kernel function The simulation results in Table 2 show that the proposed of traditional LS-SVR. For MKL-SVR and IMKLS-SVR, we method can reduce computation time more than the other also choose one linear kernel function and two Gaussian RBF two methods, although it does not achieve the best prediction functions. The prediction results are shown in Figures 2, 3, results. The proposed method has good prediction accuracy and 4 (dimensionsomitted)andthemeanpredictionRMSE and requires less prediction time. for three data sets are reported in Table 3. Figures 2–4 and Table 3 show that (1) compared with 4.2. Avionics System Experiment and Results Analysis. In this traditional LS-SVR, the proposed method, IMKLS-SVR, has section, we use a certain circuit of avionics system (shown good prediction accuracy. This may be due to the fact that in Figure 1) to demonstrate the efficiency of the proposed theproposedmethodusesrelevantinformationfullyand method with multiple relevant time series. The circuit is used maps them in multikernel high dimensional space; (2) in spite 8 Mathematical Problems in Engineering

R11 R18 C2 100 K 200 K CAP C1 R19 C4 −15 A B V 100 GND1 +15V K +15V C11 C13 C5 +15 CAP U2 V U3 − R17 − C 200 K + OP07 + OP07 −15V W4 R12 C10 R20 530 K 510 K C12

−15V W3 −15V

Figure 1: Local circuit of avionics system.

Prediction result of point A Prediction result of point B 11 4

10 3.5

3 9 2.5 8 2 7 1.5 6 1

5 0.5

4 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Raw data MKLS-SVR prediction result Raw data MKLS-SVR prediction result LS-SVR prediction result IMKLS-SVR prediction result LS-SVR prediction result IMKLS-SVR prediction result

Figure 2: Prediction results of data collected from point A (see Figure 3: Prediction results of data collected from point B (see Figure 1). Figure 1). of almost the same accuracy with MKLS-SVR, IMKLS-SVR point may have different distribution characteristics, we apply requires less prediction time than MKLS-SVR and LS-SVR, MKL,thatis,alinearcombinationofbasiskernels,toenhance especially than LS-SVR. This may be due to the fact that the generalization capability of the learning machine and to we have reduced the support vectors of sample data by exploit all discriminative information in the sample data. We measuring the linear correlation of the sample data. Thus, determine the linear combination weights by calculating the IMKLS-SVR is more valuable in application. RMSE of each basis kernel to yield the new kernel. Secondly, we remove the samples by judging the linear correlation 5. Conclusions toreducethenumberofsupportvectors.Thisapproach can control the loss of useful information of the sample In this paper, we study the issue of fast prediction with data, reduce computation time, and improve the model’s multikernel LS-SVR for multiple relevant time series. We generalization ability simultaneously. utilize two approaches to overcome the drawbacks of LS-SVR We conducted several experiments to evaluate the pro- to meet the requirements of good prediction accuracy and posedmethods;weespeciallyutilizedacertainprediction less prediction time cost. Firstly, because each time series data application to demonstrate the effectiveness of the proposed Mathematical Problems in Engineering 9

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Research Article Fuzzy Dynamic Reliability Models of Parallel Mechanical Systems Considering Strength Degradation Path Dependence and Failure Dependence

Peng Gao1 and Liyang Xie2

1 Department of Chemical Mechanical Engineering, Liaoning Shihua University, Liaoning 113001, China 2Department of Mechanical Engineering and Automation, Northeastern University, Liaoning 110004, China

Correspondence should be addressed to Peng Gao; [email protected]

Received 27 June 2014; Revised 18 September 2014; Accepted 5 October 2014

Academic Editor: Enrico Zio

Copyright © 2015 P. Gao and L. Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Fuzzy dynamic reliability models of mechanical parallel systems with respect to stress parameters and strength parameters are developed in this paper. Strength degradation path dependence (SDPD) and failure dependence of components in the system are two main problems to be addressed in developing fuzzy dynamic reliability of mechanical systems, which are taken into account in the proposed reliability models. In addition, the SDPD sensitivity function and the failure dependence sensitivity function are defined to analyze the influences of the fuzzy characteristics of stress on the effects of SDPD and failure dependence of mechanical components on dynamic reliability. The bolted joint systems for connection between launch vehicle and satellite are chosen as illustrative examples to demonstrate the proposed model. Moreover, Monte Carlo simulations are carried out to validate the effectiveness of the proposed models. The results show that SDPD and failure dependence of components have significant impacts on fuzzy dynamic reliability of mechanical systems. Besides, the influences of the fuzzy characteristics of stress on SDPD sensitivity and that on failure dependence sensitivity are different.

1. Introduction undertaking to collect enough data to calculate the probability of failure. Besides, due to the complexity of the operational Over the last few decades, reliability models of mechanical environment, it is difficult to obtain a large number of samples systems have been well reported in the probability context to precisely describe the statistical characteristics of stress [1–5]. The basic idea for reliability analysis of mechanical caused by environmental load on the mechanical systems systems is to calculate the system reliability according to the in aerospace products, especially in the preliminary design component reliability in the system and structure function stage where many factors have to be determined by the (SF). For mechanical components, the well-known stress- judgment of engineers. In this case, it can only increase the strength interference (SSI) model is the most important the- uncertainty in reliability estimation to adopt the imprecise ory for reliability assessment, where both stress and strength pdf of stress. Alternatively, fuzzy theory can be used to deal aremodeledasrandomvariables.IntheSSImodel,thefailure with this problem by modeling stress in an analytical form via of mechanical components is defined as the probability that integrating the limited available data and expert judgments. the stress is higher than strength as shown in Figure 1.The As a matter of fact, fuzzy variables are always used to deal interference area between the probability density function with the uncertainty brought by insufficient information. In (pdf) curve of stress and the pdf curve of strength is an this paper, we model the stress at each load application as a indication of the possibility that the failure of the component fuzzy variable. We assume that the uncertainty of stress comes might occur. from the insufficient information about the stress. When the However, for mechanical systems in aerospace products, amount of available information increases, which is reflected failures are relatively rare events. It is a costly and difficult via the enhancement of membership degree, the uncertainty 2 Mathematical Problems in Engineering

Probability density function of stress

Interference section Probability density function of strength Probability density function density Probability

Stress/strength

Figure 1: Schematic SSI model.

Probability density function of strength Probability density function of strength Strength process Membership function of stress Strength/stress

Static fuzzy reliability analysis Stress process Membership function of stress t1 t2 Time Dynamic fuzzy reliability analysis

Figure 2: Schematic stress process and strength process. of stress decreases and the stress tends to be a deterministic process. Most of current fuzzy reliability models number. of mechanical components and mechanical systems Since Zadeh introduced the fuzzy theory in 1965 [6], great are static, which are only suitable for the reliability efforts have been made to develop fuzzy reliability models. calculation at a specific moment as shown in Figure 2. Cai et al. classified the fuzzy reliability models into three basic As mentioned in the authors’ former research in [14], classes:therobustreliabilitymodels,theposbistreliability when considering the strength degradation process models, and the posfust reliability models [7, 8]. Besides, of mechanical components, the strength degradation Caietal.proposedfuzzyreliabilitymodelsofvarioustypical path dependence (SDPD) is a main problem to systems based on the possibility assumption and the fuzzy be addressed in developing dynamic reliability of state assumption [9]. Wu discussed the relationship between mechanical components, which also exists in estab- the fuzzy reliability of a system and the fuzzy reliability of its lishingfuzzyreliabilitymodels.Asamatteroffact, components via the proposed fuzzy reliability models [10]. fuzzy dynamic reliability models of parallel mechan- Utkin proposed the method for fuzzy reliability analysis and ical systems considering SDPD is seldom reported. fuzzy reliability optimization [11]. Huang et al. presented Besides, the fuzzy characteristics of stress have great an approach for reliability estimation by using the Bayesian influences on SDPD, which is also seldom reported method based on fuzzy lifetime data [12]. Bing et al. extended in current literatures. In this paper, impacts of fuzzy the conventional SSI model to the fuzzy stress—random characteristics of stress on the effects of SDPD on strength interference model to analyze the fuzzy reliability of reliability will be analyzed via the proposed fuzzy mechanical components [13]. These fuzzy reliability models establish the basis and framework for reliability assessment dynamic reliability models. in the possibility context. However, for mechanical systems, (2) Due to the unique operating mode of mechani- itisdifficulttocalculatereliabilitybydirectlyemployingthese cal product, failure dependence exists commonly in models because of the problems as follows. mechanical systems and the components, under the (1) Strength degradation exists commonly for mechani- stress with a common source of load, in the system are cal components in practical engineering. The interac- statistically dependent on each other. Fuzzy dynamic tion between the stress and the strength is a dynamic reliability models of parallel mechanical systems with Mathematical Problems in Engineering 3

respect to stress and material parameters, which con- The TMF can also be expressed as a triplet (𝑎, 𝑏, 𝑐) for sider failure dependence between different compo- descriptive convenience. In addition, appoint the MF to nents in a system, are seldom reported. Correspond- be equal to a specific value of 𝛼. Then, the elements with ingly, the influences of fuzzy characteristics of stress degree of membership larger than 𝛼 constitute an 𝛼-cut. on failure dependence need further investigation. In Through the 𝛼-cut, the computation under fuzzy uncertainty this paper, the sensitivity analysis of dynamic system can be transformed into the computation under the interval reliability with respect to fuzzy parameters of stress uncertainty. Correspondingly, the calculation with respect to will be performed in order to present how the fuzzy fuzzy numbers can be reduced to the calculation with respect characteristics of stress affect the failure dependence to interval numbers. To consider the weight of elements on behavior of components in mechanical systems. a specified interval and for the purpose of computational The remainder of this paper is organized as follows. convenience in case of both random variables and inter- In Section 2, fuzzy dynamic reliability models of parallel val variables existing, a certain distribution is sometimes mechanical systems are developed, which take SDPD and assumed for the elements on the interval, such as the uniform failure dependence into consideration. In Section 3,Monte distribution [15, 16]orthelineardistribution[17]. As pointed Carlo simulations are carried out to verify the reliability out by Nguyen et al. when the distribution is unknown on a models. Furthermore, the influences of fuzzy characteristics given interval, the maximum entropy approach is always used to determine an appropriate distribution for the elements on of stress on both the effects of SDPD and the effects of failure the interval. In the case where the uniform distribution is dependence on reliability are analyzed via the numerical adopted, the entropy reaches its maximum value [18]. In this examples. paper, we assign uniform distribution to the elements in an 𝛼-cut and denote the fuzzy stress at the 𝑖th load application 2. Fuzzy Dynamic Reliability Models of by (𝑎𝑖,𝑏𝑖,𝑐𝑖). Then, the distribution function of the 𝛼-cut can Parallel Mechanical Systems be given by Denote the pdf of a random variable 𝑥 by 𝑓(𝑥) and the 𝐹 (𝑠 ) membership function (MF) of 𝑥 belonging to a fuzzy set 𝑋̃ 𝑖𝛼 𝑖 ̃ by 𝜇𝑋̃(𝑥). Then, the fuzzy probability that 𝑥 belongs to 𝑋 can {0, 𝑠𝑖 <𝑎𝑖 +(𝑏𝑖 −𝑎𝑖)𝛼, be calculated as [6] { ∞ { {𝑠𝑖 −𝑎𝑖 −(𝑏𝑖 −𝑎𝑖)𝛼 𝑃 (𝑥) = ∫ 𝜇 (𝑥) 𝑓 (𝑥) 𝑑𝑥. { ,𝑎+(𝑏 −𝑎)𝛼≤𝑠 𝑋̃ (1) { (1−𝛼) (𝑐 −𝑎) 𝑖 𝑖 𝑖 𝑖 (4) −∞ = 𝑖 𝑖 { Meanwhile, the fuzzy probability in (1)canalsobecalculated { <𝑐−(𝑐 −𝑏)𝛼, { 𝑖 𝑖 𝑖 by employing the 𝛼-cut of fuzzy set as follows: { { ̃𝑈 {1, 𝑠 ≥𝑐−(𝑐 −𝑏)𝛼. 1 𝑋𝛼 𝑖 𝑖 𝑖 𝑖 𝑃 (𝑥) = ∫ ∫ 𝑓 (𝑥) 𝑑𝑥 𝑑𝛼. (2) { ̃𝐿 0 𝑋𝛼 In fact, the expression of fuzzy probability in (2) provides According to the reliability theory of systems, a parallel the basis for reliability evaluation with multiple format of system fails to operate when all the components in the system variables, such as the random variables and fuzzy variables, failtowork.Inthispaper,weassumethatthereare𝑘 identical involved in the calculation. As mentioned above, in some componentswiththesamestructureandwiththestrength cases, it is difficult to obtain the accurate pdf of stress on pdf of 𝑓𝑟(𝑟) in a parallel system. When the components in a components in mechanical systems because of insufficient parallelsystemaresubjectedtothesameload,thestressof𝑠 load history samples, while most pdf of material parameters with the pdf of 𝑓𝑠(𝑠) on each component is mutually identical. can be acquired from handbooks for mechanical design or Then, the probability that the parallel system does not fail to experiment. Therefore, in this paper, the stress is modeled as work is equal to the probability that the maximum strength a fuzzy variable and strength is modeled as a random variable. is higher than 𝑠, which can be expressed by using the order As a matter of fact, the triangular membership function statistics theory as follows: (TMF) is always adopted to express the fuzzy characteristics 𝑠 of stress and the parameters in TMF of stress can be ∞ 𝑟 𝑘−1 evaluatedbyusingthemethodin[13]. The mathematical 𝑃(𝑟max >𝑠)=∫ 𝑘[∫ 𝑓𝑟 (𝑟) 𝑑𝑟] 𝑓𝑟 (𝑟) 𝑑𝑟. (5) expression of TMF of 𝑠 can be given by 𝑠 −∞ 𝑠−𝑎 { , 𝑎≤𝑠<𝑏, {𝑏−𝑎 Therefore, the reliability of the parallel system can be given by { {1, 𝑠 = 𝑏, 𝜇𝑠 (𝑠) = { 𝑠−𝑐 (3) 𝑘−1 { ,𝑏≤𝑠<𝑐, ∞ ∞ 𝑟 { 𝑅 = ∫ 𝑓 (𝑠) {∫ 𝑘[∫ 𝑓 (𝑟)𝑑𝑟] 𝑓 (𝑟) 𝑑𝑟} 𝑑𝑠. {𝑏−𝑐 parallel 𝑠 𝑟 𝑟 { −∞ 𝑠 −∞ 0, otherwise. { (6) 4 Mathematical Problems in Engineering

Equation (6)issimilartotheSSImodelinmathematical When considering the distribution of the equivalent strength expression. Hence, the equivalent strength of the parallel of the system characterized by the pdf shown in (7), the system can be expressed as fuzzy reliability of the parallel system at the level of 𝛼 can be obtained according to the total probability formula for 𝑟 𝑘−1 𝑓 (𝑟) =𝑘[∫ 𝑓 (𝑟) 𝑑𝑟] 𝑓 (𝑟) . continuous variables as follows: parallel 𝑟 𝑟 (7) −∞ 𝑘−1 𝑛 ∞ 𝑟0 𝑅𝛼 (𝑛) = ∫ 𝑘[∫ 𝑓𝑟 (𝑟0)𝑑𝑟0] 𝑓𝑟 (𝑟0) ∏ 𝑅𝑖 𝑑𝑟0. (13) In the failure mode of fatigue, strength could degrade −∞ −∞ 0 0 under the application of random load which causes damage 𝑖=1 tothecomponentsinthesystem.Theremainingstrengthis According to the decomposition theorem, the fuzzy dynamic the function of the load application times and the magnitude reliability of the parallel system can be given by of stress, which is generally expressed as follows [14]: 1 ∞ 𝑟 𝑘−1 𝑛 𝑟 (𝑛) =𝑟[1−𝐷(𝑛)]𝛾, 0 0 (8) 𝑅 (𝑛) = ∫ ∫ 𝑘[∫ 𝑓𝑟 (𝑟0) 𝑑𝑟0] 𝑓𝑟 (𝑟0) ∏ 𝑅𝑖 𝑑𝑟0 𝑑𝛼. 0 −∞ −∞ 0 0 𝑟 𝛾 𝑖=1 where 0 and are initial strength and material parameter, (14) respectively. 𝐷(𝑛) is the cumulative damage induced by the applied load. In addition, the 𝑆-𝑁 Curve, which is usually Hence, the failure rate of the parallel system can be expressed adopted to compute the lifetime of mechanical components as follows: under the stress with a specific magnitude, can be written in 𝜆 (𝑛) an analytical format as follows: 𝑚 1 ∞ 𝑟 𝑘−1 𝑛 𝑠 𝑁=𝐶, (9) 0 =(∫ ∫ 𝑘[∫ 𝑓𝑟 (𝑟0)𝑑𝑟0] 𝑓𝑟 (𝑟0) ∏ 𝑅𝑖 𝑑𝑟0 𝑑𝛼 0 −∞ −∞ 0 0 where 𝑚 and 𝐶 are material parameters. When considering 𝑖=1 𝑘−1 𝑛+1 the effects of SDPD, the equivalent remaining strength under 1 ∞ 𝑟0 the stress at the level of 𝛼 canbeobtainedaccordingtotherule − ∫ ∫ 𝑘[∫ 𝑓 (𝑟 )𝑑𝑟 ] 𝑓 (𝑟 ) ∏ 𝑅 𝑑𝑟 𝑑𝛼) 𝑟0 0 0 𝑟0 0 𝑖 0 of damage accumulation and the rule of damage equivalence 0 −∞ −∞ 𝑖=1 as follows [14]: 𝑘−1 𝑛 −1 1 ∞ 𝑟0 𝑛 ×(∫ ∫ 𝑘[∫ 𝑓 (𝑟 )𝑑𝑟 ] 𝑓 (𝑟 ) ∏ 𝑅 𝑑𝑟 𝑑𝛼) . 𝑚+1 𝑟0 0 0 𝑟0 0 𝑖 0 0 −∞ −∞ 𝑟𝛼 (𝑛) =𝑟0 {1 − {∑ {([𝑐𝑖 −(𝑐𝑖 −𝑏𝑖)𝛼] 𝑖=1 𝑖=1 (15) 𝑚+1 −[𝑎𝑖 +(𝑏𝑖 −𝑎𝑖)𝛼] ) It should be noted that (14) is derived with the effects 𝛾 of SDPD taken into consideration. For computational con- ×((𝑚+1)(1−𝛼) (𝑐 −𝑎)𝐶)−1}} }. venience, conventional dynamic reliability models always 𝑖 𝑖 model the strength degradation process as a stochastic pro- (10) cess with the distribution of strength at each load application determined by the specified stochastic process and calculate When the initial strength 𝑟0 is deterministic, the reliability thereliabilitybasedonthestrengthateachloadapplication. at the 𝑖th load application can be given according to the Aspointedbytheauthors’formerstudy,itcouldresultinlarge distribution of the fuzzy stress on the interval of 𝛼-cut as computational error due to neglecting the effects of SDPD follows: [14]. When the fuzzy system reliability is calculated according 𝑅 to the distribution of component strength at each load 𝑖 application, which can be obtained from the distribution of initial strength and (10), the fuzzy reliability can be calculated {0, 𝑟 (𝑛−1)<𝑎 +(𝑏 −𝑎)𝛼, { 𝛼 𝑖 𝑖 𝑖 as follows: { {𝑟 (𝑛−1) −𝑎 −(𝑏 −𝑎)𝛼 { 𝛼 𝑖 𝑖 𝑖 ,𝑎+(𝑏 −𝑎)𝛼≤𝑟 (𝑛−1) 1 𝑛 ∞ 𝑠 𝑘 { (1−𝛼) (𝑐 −𝑎) 𝑖 𝑖 𝑖 𝛼 𝑅 (𝑛) = ∫ ∏ ∫ 𝑓 (𝑠 )[1−(∫ 𝑓 (𝑟 )𝑑𝑟) ]𝑑𝑠𝑑𝛼, 𝑖 𝑖 1 𝑖𝛼 𝑖 𝑟𝑖 𝑖 𝑖 = { 0 𝑖=1 −∞ −∞ { { <𝑐𝑖 −(𝑐𝑖 −𝑏𝑖)𝛼, { (16) { { 1, 𝑟𝛼 (𝑛−1)≥𝑐𝑖 −(𝑐𝑖 −𝑏𝑖)𝛼. where 𝑓𝑖𝛼(𝑠𝑖), which can be obtained via derivation of 𝐹𝑖𝛼(𝑠𝑖) { in (4)withrespectto𝑠𝑖,isthepdfofstresscausedbythe𝑖th (11) 𝑓 (𝑟 ) 𝑖 load application and 𝑟𝑖 𝑖 is the pdf of strength at the th Therefore, the probability that the system operates normally load application. when load applies for 𝑛 times can be expressed as Asamatteroffact,thefuzzycharacteristicsofstressare 𝑎 𝑏 𝑐 𝑏 𝑛 mainly determined by the parameters of , ,and . is the center parameter with the largest membership degree. When 𝑅𝛼 (𝑛) = ∏ 𝑅𝑖. (12) 𝑖=1 more information is available, which means more investment Mathematical Problems in Engineering 5

Table 1: Stress parameters and material parameters of explosive bolts.

2 𝜇(𝑟0) [MPa] 𝜎(𝑟0) [MPa] 𝑎 [MPa] 𝑏 [MPa] 𝑐 [MPa] 𝑚𝛾𝐶[MPa ] 8 400 5 250 300 350 2 1 10 in experiments, the parameters of 𝑎, 𝑐 will be closer to 𝑏 Environmental load and the fuzzy number tends to be a deterministic number. Explosive bolts In order to analyze the influences of the parameters of 𝑎 and 𝑐 ontheeffectsofSDPDonreliability,theSDPDsensitivity functions with respect to 𝑎 and 𝑐 are defined as follows:

𝐼 (𝑛) . SDPD𝑎 . . 𝜕[𝑅(𝑛) −𝑅 (𝑛)] = 1 𝜕𝑎

1 ∞ 𝑟0 𝑘−1 𝑛 𝜕{∫ ∫ 𝑘[∫ 𝑓 (𝑟 )𝑑𝑟 ] 𝑓 (𝑟 ) ∏ 𝑅 𝑑𝑟 𝑑𝛼} Bolted joint system 0 −∞ −∞ 𝑟0 0 0 𝑟0 0 𝑖=1 𝑖 0 = 𝜕𝑎 Figure 3: The bolted joint system. 1 ∞ 𝑠 𝑘 𝜕{∫ ∏𝑛 ∫ 𝑓 (𝑠 )[1−(∫ 𝑓 (𝑟 )𝑑𝑟) ]𝑑𝑠𝑑𝛼} 0 𝑖=1 −∞ 𝑖𝛼 𝑖 −∞ 𝑟𝑖 𝑖 𝑖 − , 𝐼 (𝑛) 𝜕𝑎 FD𝑐 (17) 𝜕[𝑅(𝑛) −𝑅2 (𝑛)] 𝐼 (𝑛) = SDPD𝑐 𝜕𝑐

∞ 𝑟 𝑘−1 𝜕[𝑅(𝑛) −𝑅1 (𝑛)] 𝜕{∫ 𝑘[∫ 0 𝑓 (𝑟 )𝑑𝑟 ] 𝑓 (𝑟 ) ∏𝑛 𝑅 𝑑𝑟 } = −∞ −∞ 𝑟0 0 0 𝑟0 0 𝑖=1 𝑖 0 𝜕𝑐 = 𝜕𝑐 1 ∞ 𝑟0 𝑘−1 𝑛 𝜕{∫ ∫ 𝑘[∫ 𝑓 (𝑟 )𝑑𝑟 ] 𝑓 (𝑟 ) ∏ 𝑅 𝑑𝑟 𝑑𝛼} 𝑘−1 0 −∞ −∞ 𝑟0 0 0 𝑟0 0 𝑖=1 𝑖 0 ∞ 𝑛 = −(𝑘(1−∫ 𝑓 (𝑟 ) ∏𝑅 𝑑𝑟 ) 𝜕𝑐 𝑟0 0 𝑖 0 −∞ 𝑖=1 1 ∞ 𝑠 𝑘 𝜕{∫ ∏𝑛 ∫ 𝑓 (𝑠 )[1−(∫ 𝑓 (𝑟 )𝑑𝑟) ]𝑑𝑠𝑑𝛼} 0 𝑖=1 −∞ 𝑖𝛼 𝑖 −∞ 𝑟𝑖 𝑖 𝑖 1 ∞ 𝑛 − . −1 ×𝜕{∫ ∫ 𝑓𝑟 (𝑟0) ∏𝑅𝑖 𝑑𝑟0 𝑑𝛼}) × (𝜕𝑐) . 𝜕𝑐 0 −∞ 0 (18) 𝑖=1 (20) In addition, when the components in the system are mutually independent, the fuzzy reliability can be computed as 3. Illustrative Examples 𝑘 1 ∞ 𝑛 The explosive bolts are comprehensively used for connection 𝑅 (𝑛) =1−(1−∫ ∫ 𝑓 (𝑟 ) ∏ 𝑅 𝑑𝑟 𝑑𝛼) . (19) 2 𝑟0 0 𝑖 0 and separation in aerospace products. For example, the bolted 0 −∞ 𝑖=1 joints for connection between launch vehicle and satellite are In order to evaluate the impacts of the parameters of 𝑎, shown in Figure 3. The redundancy design of the explosive 𝑐 on failure dependence, we define the failure dependence bolt system largely improves the reliability of the success sensitivity functions with respect to 𝑎 and 𝑐 as follows: launch of satellites. In this section, the bolted joint system is chosen as an illustrative example to analyze the influence 𝐼 (𝑛) FD𝑎 ofthefuzzycharacteristicsofstressonSDPDandfailure 𝜕[𝑅(𝑛) −𝑅 (𝑛)] dependence of parallel mechanical systems. In the bolted = 2 𝜕𝑎 joint system, the explosive bolts are mutually identical with ∞ 𝑟 𝑘−1 the material parameters [14] and stress parameters listed in 𝜕{∫ 𝑘[∫ 0 𝑓 (𝑟 )𝑑𝑟 ] 𝑓 (𝑟 ) ∏𝑛 𝑅 𝑑𝑟 } −∞ −∞ 𝑟0 0 0 𝑟0 0 𝑖=1 𝑖 0 Table 1. = 𝜕𝑎 In this section, three cases will be analyzed in order to 𝑘−1 investigate the following problems: ∞ 𝑛 −(𝑘(1−∫ 𝑓 (𝑟 ) ∏𝑅 𝑑𝑟 ) 𝑟0 0 𝑖 0 (1) verify the proposed model via Monte Carlo simula- −∞ 𝑖=1 tion; 1 ∞ 𝑛 ×𝜕{∫ ∫ 𝑓 (𝑟 ) ∏𝑅 𝑑𝑟 𝑑𝛼}) × (𝜕𝑎)−1, (2) analyze the influences of the parameters of 𝑎, 𝑐 on the 𝑟0 0 𝑖 0 0 −∞ 𝑖=1 effects of SDPD on reliability; 6 Mathematical Problems in Engineering

Start

Generate the level of 𝛼

Determine k, n, and N; Set b=0and m=1

m=m+1

Set j=1and z =k; No max m>N? Generate randon strength rcj (c = 1, 2, 3, . . . , k) 8 According to equation ( ), calculate the No actual remaining strength r d=1,2,3,...,z d ( max) and set rdj =rd; Yes Generate random load sj j>n? at the level of 𝛼; Yes Set z=0,i=1and z1 =1

Set i=i+1 z =z and z1 =z1 +1 Set max and j=j+1

R=1−b/N No No No r >s z =z ? Yes ij j 1 max z=0?

Yes Yes

b=b+1 Set z=z+1 and rzj =rij End

Figure 4: Flowchart of the Monte Carlo simulation for parallel subsystem.

(3) investigate the influences of the parameters of 𝑎, 𝑐 on Case 2. Consider a system with three dependent components. the failure dependence. The system reliability considering SDPD and the system reliability without SDPD taken into account are shown in Case 1. Consider a system with three dependent components. Figure 6. In addition, the SDPD sensitivity with respect to 𝑎 The flowchart for the Monte Carlo simulation is shown in and 𝑐 is shown in Figures 7 and 8,respectively. Figure 4. In the flowchart, 𝑘 represents the number of compo- From Figure 6,itcanbeseenthatSDPDhavegreat nents in the system. 𝑁 is the total number of simulation trials. influences on fuzzy system reliability. Although it could 𝑛 isthenumberofloadapplication.𝑏 is the total number of facilitate the reliability calculation by directly using the failure of the system in the 𝑁 trials. It should be noted that in strength distribution at each load application, reliability could the Monte Carlo simulation, the strength degradation process be underestimated due to neglecting the effects of SDPD. is only dependent on the randomly generated stress and (8) Moreover, from Figures 7 and 8, it can be learnt that the left for remaining strength calculation under deterministic stress. parameter 𝑎 and right parameter 𝑐 have different impacts on Therefore, the simulation is essentially identical with physical theeffectsofSDPDonreliability.TheeffectsofSDPDon experiment and does not rely on any analytical dynamic reliability are more sensitive to variation of 𝑐. reliability model. The comparison between the result of the For a deterministic 𝑎, the local maximum of the error MonteCarlosimulationandtheresultfromtheproposed caused by SDPD arises at middle stage and the end of the models is shown in Figure 5. operational duration of the system. The reliability calculated From Figure 5, it can be learnt that the result from the considering SDPD is higher than that without SDPD taken proposed reliability method shows good agreement with into account. Besides, from (17)and(18), we can learn that, the reliability from Monte Carlo simulations. The reliability for the SDPD sensitivity, a positive value means that the decreases rapidly with the load application. error increases with the increase of the parameter considered Mathematical Problems in Engineering 7

1.0 ×10−4 ×10−3 0.9 12 1.5 0.8 10 (n) a 0.7 1 8 ) SDPD n I ( 0.6 6 R 0.5 4 0.5 2 0.4 0 0 Reliability 0.3 −0.5 −2

0.2 SDPD sensitivity 270 −4 −1 260 a 250 −6 0.1 0 50 100 240 150 200 250 Load application times 300 350 230 −8 0.0 n 0 50 100 150 200 250 300 350 Left parameter Load application times n Figure 7: SDPD sensitivity with respect to 𝑎. The proposed model Monte Carlo simulation 0.01 Figure 5: Reliability from Monte Carlo simulation and reliability from proposed models. 0.015 0.005 (n) c 0.01 1.0 SDPD 0.005 I 0.9 0 0 0.8 −0.005 0.7 −0.005

) −0.01 n ( 0.6 370 R SDPD sensitivity −0.015 360 c 0.5 0 50100 350 Load application150200 times 340 −0.01 0.4 250300 350400 330 Reliability n 0.3 Right parameter 0.2 Figure 8: SDPD sensitivity with respect to 𝑐. 0.1 0.0 0 50 100 150 200 250 300 350 which represent the most sensitive time instant. Therefore, in Load application times n dynamic system reliability analysis, attention should be paid 𝑎 Model considering SDPD to the influences of the change in on the effects of SDPD Model without considering SDPD at these two sensitive “time instants” associated with their adjacent time interval. In addition, the positive peak value of Figure 6: Reliability considering SDPD and reliability without the SDPD sensitivity comes earlier when 𝑎 increases. SDPD taken into consideration. For a deterministic 𝑐 with a low value, only one positive peak value of the SDPD sensitivity appears in the middle stage of the system operational duration. However, when 𝑐 (𝑎 or 𝑐), while a negative value means that the error decreases is large, two positive peak values arise in the middle stage with the increase of the parameter considered (𝑎 or 𝑐). of the system operational duration with one peak value Moreover, the reliability calculated considering SDPD is evidently larger than the other one. Therefore, we should always larger than the reliability calculated without SDPD pay attention to the variation of 𝑐 at the moment where the taken into account. Therefore, the SDPD sensitivity with a positive peak value of the SDPD sensitivity appears in the negative value means that the effects of SDPD on reliability reliability estimation when considering the effects of SDPD. are weakened. In mechanical design, engineers are more Similar to the case in the sensitivity analysis with respect to interested in the sensitive areas where a small variation in 𝑎,themaximumoftheSDPDsensitivitycomesearlierwhen the parameter considered could lead to a rapid increase in 𝑐 increases. the error due to neglecting SDPD. Thus, in this paper, we areconcentratedonthecharacteristicsofthepositivepeak Case 3. In this section, we are aimed at analyzing the value of SDPD sensitivity. From Figure 7,wecanseethat influences of the failure dependence of components on two evident positive peak values of the SDPD sensitivity system reliability and influences of fuzzy parameters of appear in the middle stage of the system operational duration, stress on failure dependence sensitivity function. The system 8 Mathematical Problems in Engineering

1.0 ×10−3 0.9 0

0.8 (n) c 0.005 FD −5

0.7 I ) n ( 0.6 0 R −0.005 0.5 −10 0.4 −0.01 Reliability 0.3 −0.015 −15 0.2 −0.02 0.1 370 −0.025 360 c −20

Failure dependence sensitivity Failure 0 0.0 50100 350 0 50 100 150 200 250 300 350 150200 340 Load application250 times300 n 350400 330 Load application times n Right parameter System with dependent components Figure 11: Failure dependence sensitivity with respect to 𝑐. System with independent components

Figure 9: Reliability of dependent system and reliability of independent system. we are more concentrated on the negative peak value of the failure dependence sensitivity. A single negative peak value ×10−3 of the failure dependence sensitivity appears in the middle stage of the system operational duration in both the case of 1 failure dependence sensitivity analysis with respect to 𝑎 and −3

(n) ×10 a 0.5 the case of failure dependence sensitivity analysis with respect

FD 𝑐 𝑎 𝑐 I 2 to . We should pay attention to the variation of and 0 1 at the moment where the negative peak value of the failure −0.5 dependence sensitivity appears in the reliability assessment 0 −1 when considering the effects of failure dependence. Besides, −1 the negative peak value of the failure dependence sensitivity −1.5 𝑎 𝑐 −2 comes earlier when or increases. In general, the effects of −2 the failure dependence of components in the parallel system −3 270 𝑐 −2.5 aremoresensitivetothechangeof . 260 a

Failure dependence sensitivity Failure −4 250 −3 0 50 100 150 240 4. Conclusion 200 250 Load application times 300 350 230 n Left parameter In this paper, fuzzy dynamic reliability models of mechanical parallel systems with respect to stress parameters and Figure 10: Failure dependence sensitivity with respect to 𝑎. strength parameters are established. In the proposed reliability models, the SDPD and failure dependence of components inamechanicalparallelsystemaretakenintoaccount.For reliability considering failure dependence of components can computational convenience, strength distribution of compo- be calculated according to (14), while the system reliability nents at each load application is always adopted in conven- under the assumption that the components are independent tional dynamic reliability models, which is determined by the of each other can be calculated according to (19). The failure assumed stochastic process. However, the results show that dependence sensitivity function with respect to 𝑎 and 𝑐 can SDPD has considerable influences on fuzzy dynamic reliabil- be obtained through (20). The reliability of a system with ity of parallel systems. In addition, the failure dependence of three dependent components and the reliability of a system components in a system also significantly affects the system with three independent components are shown in Figure 9. fuzzy dynamic reliability. In addition, the failure dependence sensitivity with respect to By defining the SDPD sensitivity function and the failure 𝑎 and 𝑐 is shown in Figures 10 and 11,respectively. dependence sensitivity function in this paper, it is found From Figure 9, it can be seen that the failure dependence that the fuzzy characteristics of stress have great impacts makes the parallel system less reliable. Thus, from (18)and on the effects of SDPD and failure dependence on fuzzy (19), Figures 10 and 11, it can be seen that the negative peak dynamic reliability. In general, the effects of SDPD and value of the failure dependence sensitivity means that the failure dependence of mechanical components in a parallel error caused by failure dependence increases at the highest system on reliability are more sensitive to variation of the speed,whilethepositivepeakvalueoftheSDPDsensitivity right parameter of 𝑐 than the left parameter of 𝑎.Moreover, means that the error decreases at the highest speed. Therefore, the influences of 𝑎 and 𝑐 on SDPD sensitivity and that on Mathematical Problems in Engineering 9

failure dependence sensitivity are different. The proposed [12]H.-Z.Huang,M.J.Zuo,andZ.-Q.Sun,“Bayesianreliability models provide the basis for analytically determining the analysis for fuzzy lifetime data,” Fuzzy Sets and Systems,vol.157, moment when the peak values of SDPD sensitivity and failure no.12,pp.1674–1686,2006. dependence sensitivity appear, which are important to the [13] L. Bing, Z. Meilin, and X. Kai, “Practical engineering method for dynamic reliability evaluation and reliability-based design fuzzy reliability analysis of mechanical structures,” Reliability of mechanical parallel systems. In addition, Monte Carlo Engineering and System Safety,vol.67,no.3,pp.311–315,2000. simulation is carried out to validate the effectiveness of the [14] P. Gao, S. Yan, L. Xie, and J. Wu, “Dynamic reliability analysis proposed models. of mechanical components based on equivalent strength degradation paths,” Journal of Mechanical Engineering,vol.59,no.6, pp. 387–399, 2013. Conflict of Interests [15] Y. Dong, Design of Mechanical Fuzzy Reliability, Machine Industry Press, Beijing, China, 2001. The authors declare that there is no conflict of interests [16]Y.Dong,“Fuzzyreliabilitydesignwithrandomvariableand regarding to the publication of this paper. fuzzy variable,” Chinese Journal of Mechanical Engineering,vol. 36,no.6,pp.25–29,2000. Acknowledgments [17] Q. Jiang and C.-H. Chen, “A numerical algorithm of fuzzy reliability,” Reliability Engineering and System Safety,vol.80,no. This work was supported by the Natural Science Foundation 3, pp. 299–307, 2003. of China under Contract no. 51175240, the Natural Science [18] H. T. Nguyen, V. Kreinovich, B. Wu, and G. Xiang, Computing Foundation of China under Contract no. 51175072, and the Statistics under Interval and Fuzzy Uncertainty: Applications to Natural Science Foundation of China under Contract no. Computer Science and Engineering, Springer, 2011. 51335003.

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Research Article Delayed Age Replacement Policy with Uncertain Lifetime

Xueyan Li and Chunxiao Zhang

College of Science, Civil Aviation University of China, Tianjin 300300, China

Correspondence should be addressed to Xueyan Li; [email protected]

Received 4 July 2014; Accepted 11 October 2014

Academic Editor: Wenbin Wang

Copyright © 2015 X. Li and C. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers the delayed age replacement policy, in which the lifetimes of all units are assumed to be uncertain variables, and the lifetime of the first unit has an uncertainty distribution which is different from the others. A delayed age replacement model which is concerned with finding the optimal replacement time to minimize the expected cost is developed. In the policy, the optimal replacement time is irrelevant to the uncertain distribution of lifetime of the first unit over the infinite time span.

1. Introduction belief degree may have a much larger range than the real frequency. Therefore, it is unreasonable to employ stochastic The replacement policy for the unit based on its age is method for the particularity of the problem. In order to called age replacement policy, which means that a unit is rationally deal with belief degrees, uncertainty theory was always replaced at failure or at an age 𝑇,whicheveroccurs founded by Liu in 2007 [10],andwasrefinedbyLiuin2010 first. Age replacement policy is easy to operate especially for [11] based on normality, duality, subadditivity, and product multicomponent systems, so it is one of the widely used main- axioms. Nowadays, uncertainty theory has become a branch tenance policies. Age replacement policies have been studied of axiomatic mathematics for modeling human uncertainty, theoretically by many authors. In 1965, Barlow and Proschan and some applications can be found in various fields. [1] studied the basic replacement policies. Furthermore, age Yao and R alescu [12]firstlyproposedtheuncertain replacement policy with continuous discounting was pro- age replacement policy, where the lifetimes of all units are posed by Fox in 1966 [2]. Scheaffer [3] considered optimum assumed as iid uncertain variables. However, in practice of age replacement policies with an increasing cost factor. Later maintenance engineering, the lifetime of the first unit may be in1979,Clerouxetal.[4] studied age replacement policy with quite different from the remains. If the age 𝑇 is unchanging random charges. Cleroux and Hanscom [5] studied a general in the age replacement policy, only the lifetime of the first age replacement model with minimal repair. Boland and unit has an uncertainty distribution which is different from Proschan [6]studiedthecasewhentherepaircostincreases the others. The first replacement point may be observed at with age. Jhang and Sheu [7]proposedanopportunity- a delayed time and call it delayed age replacement policy as based age replacement policy with minimal repair. For more the delayed renewal process. In this paper, we will consider development of replacement policies, readers can refer to the delayed age replacement policy with uncertain lifetimes Nakagawa [8]. of all units, and the lifetime of first unit has a different In the above literatures, the lifetime of a unit is regarded uncertainty distribution from the others. And then, a delayed as a random variable, and probability theory is employed age replacement model to find the optimal predetermined to deal with the optimization of the replacement policy. replacement time 𝑇 will be developed. Theprobabilitytheoryisapplicableonlywhenwehavethe Thispaperisorganizedasfollows:Section 2 recalls some large enough sample size. However when no samples are basic concepts and properties about uncertainty theory which available in some situation, we have to invite some domain will be used throughout the paper. In Section 3,delayedage experts to evaluate the belief degree that each event will occur. replacement policy in uncertain environment is introduced Since human tends to overweight unlikely events [9], the and the expected cost over infinite time is proposed; thus 2 Mathematical Problems in Engineering the optimal age replacement time will be derived. Numerical An uncertain process [13]isessentiallyasequenceof example is given in Section 4,followedbySection 5 where we uncertain variables indexed by time. Renewal process is one conclude the paper. of the most important uncertain processes in which events occur continuously and independently of one another in 2. Preliminaries uncertain times. 𝜉 ,𝜉 ,... Let Γ be a nonempty set. L is a 𝜎-algebra on Γ. Each element Definition 5 (Liu [13]). Let 1 2 be iid positive uncertain 𝑆 =0 𝑆 =𝜉 +𝜉 +⋅⋅⋅+𝜉 𝑛≥1 Λ in the 𝜎-algebra L is called an event. Uncertain measure is variables. Define 0 and 𝑛 1 2 𝑛 for . afunctionfromL to [0, 1]. In order to present an axiomatic Then the uncertain process definition of uncertain measure, it is necessary to assign to 𝑁𝑡 = max {𝑛 |𝑛 𝑆 ≤𝑡} each event Λ anumberM{Λ} which indicates the belief 𝑛≥0 (5) degree that the event Λ will occur. In order to ensure that the is called an uncertain renewal process. number M{Λ} has certain mathematical properties, Liu [10] proposed the following three axioms. Age replacement means that an element is always Axiom 1: (normality axiom) M{Γ} = 1 for the replacedatfailureoratanage𝑇.If𝜉1,𝜉2,... denote the universal set Γ. lifetimes of the elements which are iid uncertain variable with 𝑐 Axiom 2: (duality axiom) M{Λ} + M{Λ }=1for any a common uncertainty distribution, then the actual lifetimes event Λ. oftheelementsareiiduncertainvariables Axiom 3: (subadditivity axiom) For every countable 𝜉1 ∧𝑇,𝜉2 ∧𝑇,..., (6) sequence of events Λ 1,Λ2,...,wehave which may generate an uncertain renewal process: ∞ ∞ 𝑛 M {⋃Λ 𝑖}≤∑M {Λ 𝑖}. (1) 𝑖=1 𝑖=1 𝑁𝑡 = max {𝑛 | ∑ (𝜉𝑖 ∧𝑇)≤𝑡}. (7) 𝑛≥0 𝑖=1 Definition 1 (Liu [10]). The set function M is called an uncer- Yao and R alescu [12]investigatedtheuncertainage tain measure if it satisfies the normality, duality, subadditivity, replacement policy and obtained the long-run average and product axioms. replacement cost as follows:

Definition 2 (Liu [10]). An uncertain variable is a measurable 𝑁𝑡 𝑇 𝑓(𝜉𝑖 ∧𝑇) 𝑎 𝑏−𝑎 Φ (𝑥) function 𝜉 from the uncertainty space (Γ, L, M) to the set of lim 𝐸[∑ ]= + Φ (𝑇) + ∫ 𝑑𝑥, 𝑡→∞ 𝑡 𝑇 𝑇 𝑥2 real numbers; that is, for any Borel set 𝐵 of real numbers, the 𝑖=1 0 set (8) {𝜉∈𝐵} = {𝛾∈Γ|𝜉(𝛾) ∈𝐵} (2) where 𝑓(𝑥) is the replacement cost function. is an event. 3. Delayed Age Replacement Policy In order to describe an uncertain variable, a concept of uncertainty distribution is introduced as follows. Consider an age replacement policy in which a unit is replaced at constant time 𝑇 after its installation or at failure, Definition 3 (Liu [10]). The uncertainty distribution of an whichever occurs first. We assume that failures are instantly uncertain variable 𝜉 is defined by detected and replaced with a new one, where its replacement time is negligible. Assume that the lifetimes of the units Φ (𝑥) = M {𝜉≤𝑥} , (3) are uncertain variables 𝜉1,𝜉2,...,and𝜉1 has an uncertainty for any real number 𝑥. distribution which is different from the others. A net unit is installed at time 𝑡=0. Then, the actual lifetimes of the units 𝜉 ∧𝑇,𝜉 ∧𝑇,... Expected value is the average of an uncertain variable in are uncertain variables 1 2 ,whichgeneratean thesenseofuncertainmeasureandrepresentedthesizeof uncertain delayed renewal process: uncertain variable. 𝑛 𝑁𝑡 = max {𝑛 | ∑ (𝜉𝑖 ∧𝑇)≤𝑡}, (9) Definition 4 (Liu [10]). Let 𝜉 be an uncertain variable. Then 𝑖=1 the expected value of 𝜉 is defined by where 𝜉1 ∧𝑇has an uncertainty distribution which is different +∞ 0 𝐸[𝜉]=∫ M {𝜉≥𝑟} 𝑑𝑟 − ∫ M {𝜉≤𝑟} 𝑑𝑟 from the others. We consider the problem of minimizing the 0 −∞ expected cost per unit of time for an infinite time span. For (4) +∞ 0 simplicity, we introduce the following cost function: = ∫ (1−Φ(𝑥)) 𝑑𝑥 − ∫ Φ (𝑥) 𝑑𝑥 𝑏, 𝑥<𝑇 0 −∞ 𝑓 (𝑥) ={ if 𝑎, 𝑥=𝑇, (10) provided that at least one of the two integrals is finite. if Mathematical Problems in Engineering 3

where 𝑎>0isthecostofreplacingtheunitatage𝑇 and 𝑏 is with 0<𝑎<𝑏, the uncertainty distribution Ψ𝑡(𝑥) of the the cost of replacing the unit at failure, which is larger than uncertain variable 𝑎.Then𝑓(𝜉𝑖 ∧𝑇)denotes the cost to replace the 𝑖th unit, and 𝑁𝑡 𝑡 𝑓(𝜉𝑖 ∧𝑇) the expected total replacement cost before time is ∑ (19) 𝑖=1 𝑡 𝑁𝑡 𝐸[∑𝑓(𝜉𝑖 ∧𝑇)]. (11) satisfies 𝑖=1 𝑎 {0𝑖𝑓𝑥≤ 𝑡 { 𝑇 The average cost over the time is expressed as { 𝑎 𝑏 1−Φ(𝑇) 𝑖𝑓 <𝑥≤ lim Ψ𝑡 (𝑥) ≥ (20) 𝑁𝑡 𝑡→∞ { 𝑇 𝑇 𝑓(𝜉𝑖 ∧𝑇) { 𝑏 𝑏 𝐸[∑ ]. (12) { 𝑡→∞lim 1−Φ( )𝑖𝑓𝑥> . 𝑖=1 𝑡 { 𝑥 𝑇

Delayed age replacement policy aims at finding an opti- Proof. Since ∗ mal time 𝑇 to minimize the average replacement cost; that 𝑁𝑡 is, 𝑓(𝜉1 ∧𝑇)≥𝑎, ∑ (𝜉𝑖 ∧𝑇)≤𝑡, (21) 𝑖=2 𝑁𝑡 ∗ 𝑁𝑡 𝑓(𝜉𝑖 ∧𝑇 ) 𝑓(𝜉𝑖 ∧𝑇) lim 𝐸 [∑ ] = min lim 𝐸 [∑ ] . 𝑡→∞ 𝑇 𝑡→∞ we have 𝑖=1 𝑡 𝑖=1 𝑡 𝑁 (13) 𝑡 𝑓(𝜉 ∧𝑇) Ψ (𝑥) = M {∑ 𝑖 ≤𝑥} 𝑡 𝑡 Lemma 6 (Yaoand Ralescu [12]). Let 𝜉 be a positive uncertain 𝑖=1 variable with an uncertainty distribution Φ.Giventhat 𝑁 {(𝑎 + ∑ 𝑡 𝑓(𝜉 ∧𝑇)) } ≥ M 𝑖=2 𝑖 ≤𝑥 𝑏, 𝑖𝑓𝑥<𝑇 { 𝑡 } 𝑓 (𝑥) ={ { } 𝑎, 𝑖𝑓𝑥=𝑇, (14) 𝑁𝑡 𝑁𝑡 ∑𝑖=2 𝑓(𝜉𝑖 ∧𝑇) 𝜉𝑖 ∧𝑇 𝑎 0<𝑎<𝑏 = M {( )(∑ )≤𝑥− } with , the uncertain variable 𝑁𝑡 𝑡 𝑡 ∑𝑖=2 𝜉𝑖 ∧𝑇 𝑖=2 𝑓 𝜉∧𝑇 ( ) 𝑁 (15) ∑ 𝑡 𝑓(𝜉 ∧𝑇) 𝑎 𝜉∧𝑇 ≥ M { 𝑖=2 𝑖 ≤𝑥− } 𝑁𝑡 𝑡 ∑𝑖=2 (𝜉𝑖 ∧𝑇) has an uncertainty distribution 𝑓(𝜉 ∧𝑇) 𝑎 𝑎 = M { 2 ≤𝑥− } {0𝑖𝑓𝑥< 𝜉 ∧𝑇 𝑡 { 𝑇 2 { 𝑎 𝑏 Ψ (𝑥) = 1−Φ(𝑇) 𝑖𝑓 ≤𝑥< , 𝑎 { 𝑇 𝑇 (16) =Ψ(𝑥− ) { 𝑏 𝑏 𝑡 {1−Φ( )𝑖𝑓𝑥≥ . { 𝑥 𝑇 𝑎 𝑎 {0 if 𝑥< + { 𝑡 𝑇 Lemma 7 𝑁 { 𝑎 𝑎 𝑎 𝑏 (Yao and Ralescu [12]). Let 𝑡 be an uncertain 1−Φ 𝑇 + ≤𝑥< + = { ( ) if renewal process with iid uncertain interarrival times 𝜉1,𝜉2,..., { 𝑡 𝑇 𝑡 𝑇 { 𝑏 𝑎 𝑏 and let 𝑓 be a positive function. Then {1−Φ( ) 𝑥≥ + . { 𝑥 − 𝑎/𝑡 if 𝑡 𝑇 𝑁 ∑ 𝑡 𝑓(𝜉) 𝑓(𝜉 ) (22) 𝑖=1 𝑖 , 1 , 𝑁 (17) 𝑡 𝑔(𝜉 ) ∑𝑖=1 𝑔(𝜉𝑖) 1 The last second inequality holds because of Lemma 6. Thus have a common uncertainty distribution. 𝑎 lim Ψ𝑡 (𝑥) ≥ lim Ψ(𝑥− ) 𝑡→∞ 𝑡→∞ 𝑡 Theorem 8. Let 𝜉1,𝜉2,...be a sequence of positive uncertain 𝜉 Φ 𝜉 ,𝜉 ,... 𝑎 variables. If 1 has an uncertainty distribution 1 and 2 3 0 𝑥≤ Φ 𝑁 { if has a common uncertainty distribution , 𝑡 is an uncertain { 𝑇 (23) 𝜉 ∧𝑇,𝜉 ∧𝑇,... { 𝑎 𝑏 renewal process with uncertain interarrivals 1 2 . = 1−Φ(𝑇) if <𝑥≤ { 𝑇 𝑇 Given that { 𝑏 𝑏 {1−Φ( ) 𝑥> . 𝑏, 𝑖𝑓𝑥<𝑇 { 𝑥 if 𝑇 𝑓 (𝑥) ={ 𝑎, 𝑖𝑓𝑥=𝑇, (18) The theorem is proved. 4 Mathematical Problems in Engineering

Theorem 9. Let 𝜉1,𝜉2,...be a sequence of positive uncertain That means variables. If 𝜉1 has an uncertainty distribution Φ1 and 𝜉2,𝜉3,... ∞ 𝑁𝑡 has a common uncertainty distribution Φ, 𝑁𝑡 is an uncertain 𝑓(𝜉𝑖 ∧𝑇) 𝑏 ⋂ {𝜉𝑖 ≤𝑇−𝜀}⊂{∑ > }. (32) renewal process with uncertain interarrivals 𝜉1 ∧𝑇,𝜉2 ∧𝑇,.... 𝑖=2 𝑖=1 𝑡 𝑇 Given that 𝑏, 𝑖𝑓𝑥<𝑇 It is equivalent of 𝑓 (𝑥) ={ 𝑎, 𝑖𝑓𝑥=𝑇, (24) 𝑁 𝑡 𝑓(𝜉 ∧𝑇) 𝑏 ∞ { 𝑖 ≤ }⊂ {𝜉 >𝑇−𝜀}, 0<𝑎<𝑏 Ψ (𝑥) ∑ ⋃ 𝑖 (33) with the uncertainty distribution 𝑡 of the 𝑖=1 𝑡 𝑇 𝑖=2 uncertain variable 𝑁 𝑡 𝑇(2𝑇 − 𝜀)/𝜀 𝑡 𝑓(𝜉 ∧𝑇) provided that is larger than . ∑ 𝑖 𝑡 (25) According to the monotonicity of uncertain distribution, 𝑖=1 for any 𝑥 ∈ [𝑎/𝑇, 𝑏/𝑇),weobtainthat satisfies 𝑁𝑡 𝑏 𝑓(𝜉𝑖 ∧𝑇) 𝑏 𝑎 lim Ψ𝑡 (𝑥) ≤ lim Ψ𝑡 ( )=M {∑ ≤ } {0𝑖𝑓𝑥< 𝑡→∞ 𝑡→∞ 𝑇 𝑡 𝑇 { 𝑇 𝑖=1 { 𝑎 𝑏 1−Φ(𝑇) 𝑖𝑓 ≤𝑥< ∞ lim Ψ𝑡 (𝑥) ≤ { (26) 𝑡→∞ { 𝑇 𝑇 ≤ M {⋃ {𝜉 >𝑇−𝜀}} { 𝑏 𝑏 𝑖 1−Φ( )𝑖𝑓𝑥≥ . 𝑖=2 { 𝑥 𝑇 =1−Φ(𝑇−𝜀) . Proof. (34) Case 1. Assume that 𝑥 < 𝑎/𝑇.Let𝑡> 𝑇𝑎/(𝑎 −𝑇𝑥) ;thenwe have Letting 𝜀→0,wehave

𝑁𝑡 𝑓(𝜉𝑖 ∧𝑇) 𝑎𝑁𝑡 𝑎 (𝑡/𝑇) −1 Ψ (𝑥) ≤1−Φ(𝑇) . ∑ ≥ ≥ 𝑡→∞lim 𝑡 (35) 𝑖=1 𝑡 𝑡 𝑡 𝑎 𝑇 (27) Case 3. Assume that 𝑥 ≥ 𝑏/𝑇.Forany𝜀>0and fixing = (1 − ) 𝑇 𝑡 ∞ 𝑏 𝛾∈⋂ {𝜉 ≤ −𝜀} , >𝑥. 𝑖 (36) 𝑖=2 𝑥 It is concluded that when 𝑁 𝑡 𝑓(𝜉 ∧𝑇) 𝑖 𝑏 𝑏+𝑥𝑇−𝑥𝜀 lim Ψ𝑡 (𝑥) = M {∑ ≤𝑥}=0. (28) ( ) 𝑡→∞ 𝑡 𝑡≥ , (37) 𝑖=1 𝑥2𝜀 𝑎/𝑇 ≤ 𝑥 < 𝑏/𝑇 Case 2. Assume that .Firstly,wewillprove we have that

𝑁𝑡(𝛾) 𝑁𝑡 ∞ 𝑓(𝜉 (𝛾) ∧ 𝑇) 𝑏𝑁 (𝛾) 𝑓(𝜉𝑖 ∧𝑇) 𝑏 ∑ 𝑖 > 𝑡−𝑇 {∑ ≤ }⊂⋃ {𝜉𝑖 >𝑇−𝜀}, (29) 𝑡 𝑡 𝑖=1 𝑡 𝑇 𝑖=2 𝑖=1 𝑏 𝑡−𝑇 (38) for any 𝜀>0provided that 𝑡 is large enough. For any ≥ ( −1) 𝑡 𝑏/𝑥 −𝜀 ∞ 𝛾∈⋂ {𝜉𝑖 ≤𝑇−𝜀}, (30) >𝑥. 𝑖=2 when 𝑡≥𝑇(2𝑇−𝜀)/𝜀, it can be obtained that Thus

𝑁 (𝛾) ∞ 𝑁𝑡 𝑡 𝑓(𝜉 (𝛾) ∧ 𝑇) 𝑏𝑁 (𝛾) 𝑏 𝑓(𝜉𝑖 ∧𝑇) ∑ 𝑖 > 𝑡−𝑇 ⋂ {𝜉 ≤ −𝜀}⊂{∑ >𝑥}. 𝑖 𝑥 𝑡 (39) 𝑖=1 𝑡 𝑡 𝑖=2 𝑖=1 𝑏 ((𝑡−𝑇) / (𝑇−𝜀) −1) ≥ (31) That is 𝑡 𝑁 𝑡 𝑓(𝜉 ∧𝑇) ∞ 𝑏 𝑏 {∑ 𝑖 ≤𝑥}⊂⋃ {𝜉 > −𝜀}, > . 𝑡 𝑖 𝑥 (40) 𝑇 𝑖=1 𝑖=2 Mathematical Problems in Engineering 5

provided that 𝑡 is large enough. Therefore, it can be obtained Note that lim𝑡→∞Ψ𝑡(𝑥) ≥1 Υ (𝑥) by Theorem 8 and that lim𝑡→∞Ψ𝑡(𝑥) ≤2 Υ (𝑥) by Theorem 9. According to the Fatou lemma, we can obtain 𝑁 𝑡 𝑓(𝜉 ∧𝑇) 𝑖 𝑁 Ψ (𝑥) = M {∑ ≤𝑥} +∞ 𝑡 𝑓(𝜉 ∧𝑇) 𝑡→∞lim 𝑡 𝑡→∞lim 𝑖 𝑖=1 𝑡 ∫ 1−Υ (𝑥) 𝑑𝑥 ≤ 𝐸[∑ ] 2 𝑡→∞lim 0 𝑖=1 𝑡 ∞ 𝑏 𝑏 ≤ M {⋃ {𝜉 > −𝜀}}=1−Φ( −𝜀). +∞ 𝑖 = ∫ 1−Ψ (𝑥) 𝑑𝑥 (47) 𝑖=2 𝑥 𝑥 lim 𝑡 𝑡→∞ 0 (41) +∞ ≤ ∫ 1−Υ1 (𝑥) 𝑑𝑥. Letting 𝜀→0,forany𝑥 ≥ 𝑏/𝑇,we have 0 𝑏 So we have lim Ψ𝑡 (𝑥) ≤1−Φ( ). (42) 𝑡→∞ 𝑥 𝑁𝑡 𝑇 𝑓(𝜉𝑖 ∧𝑇) 𝑎 𝑏−𝑎 Φ (𝑥) lim 𝐸[∑ ]= + Φ (𝑇) + ∫ 𝑑𝑥. 𝑡→∞ 𝑡 𝑇 𝑇 𝑥2 The theorem is proved. 𝑖=1 0 (48) Theorem 10. Let 𝜉1,𝜉2,...be a sequence of positive uncertain The theorem is proved. It follows from Theorem 10 that the variables. If 𝜉1 has an uncertainty distribution Φ1 and 𝜉2,𝜉3,... ∗ optimal replacement time 𝑇 is just the replacement time 𝑇 has a common uncertainty distribution Φ, 𝑁𝑡 is an uncertain which satisfies renewal process with uncertain interarrivals 𝜉1 ∧𝑇,𝜉2 ∧𝑇,.... Given that 𝑎 𝑏−𝑎 𝑇 Φ (𝑥) min { + Φ (𝑇) + ∫ 𝑑𝑥} . (49) 𝑇 𝑇 𝑇 𝑥2 𝑏, 𝑖𝑓𝑥<𝑇 0 𝑓 (𝑥) ={ 𝑎, 𝑖𝑓𝑥=𝑇, (43) with 0<𝑎<𝑏,then 4. Number Example

𝑁 𝑡 𝑓(𝜉 ∧𝑇) 𝑎 𝑏−𝑎 𝑇 Φ (𝑥) Let 𝜉1,𝜉2,...be a sequence of positive uncertain variables. If 𝐸[∑ 𝑖 ]= + Φ (𝑇) + ∫ 𝑑𝑥. 𝜉 LOGN(𝑒 ,𝜎 ) 𝑡→∞lim 2 1 has a lognormal uncertainty distribution 1 1 𝑖=1 𝑡 𝑇 𝑇 0 𝑥 where 𝑒1 and 𝜎1 are real numbers with 𝜎>0and 𝜉2,𝜉3,...has (44) a common lognormal uncertainty distribution LOGN(𝑒, 𝜎) where 𝑒 and 𝜎 are real numbers with 𝜎>0,let𝑁𝑡 be an Proof. Let uncertain renewal process with uncertain interarrivals 𝜉1 ∧ 𝑇, 𝜉 ∧𝑇,... 0<𝑇<+∞ 𝑎 2 for any .Giventhat {0 if 𝑥≤ { 𝑇 { 𝑎 𝑏 𝑏, if 𝑥<𝑇 1−Φ(𝑇) <𝑥≤ 𝑓 (𝑥) ={ (50) Υ1 (𝑥) = { if 𝑎, if 𝑥=𝑇, { 𝑇 𝑇 { 𝑏 𝑏 1−Φ( ) 𝑥> , 0<𝑎<𝑏 { 𝑥 if 𝑇 with ,thenitfollowsfromTheorem 10 that the expected cost over infinite time span is 𝑎 (45) 0 𝑥< { if 𝑎 𝑏−𝑎 𝑇 Φ (𝑥) { 𝑇 + Φ (𝑇) + ∫ 𝑑𝑥 { 𝑎 𝑏 𝑇 𝑇 𝑥2 Υ (𝑥) = 1−Φ(𝑇) if ≤𝑥< 0 2 { 𝑇 𝑇 { −1 { 𝑏 𝑏 𝑎 𝑏−𝑎 𝜋(𝑒 − ln 𝑇) 1−Φ( ) if 𝑥≥ . = + (1 + exp ( )) (51) { 𝑥 𝑇 𝑇 𝑇 √3𝜎

Thus 𝑇 1 𝜋 (𝑒− 𝑥) −1 + ∫ (1 + ( ln )) 𝑑𝑥. 2 exp +∞ +∞ 0 𝑥 √3𝜎 ∫ (1 − Υ1 (𝑥))𝑑𝑥=∫ (1 − Υ2 (𝑥))𝑑𝑥 ∗ 0 0 The optimal replacement time 𝑇 is just the replacement time 𝑇 𝑎 𝑏−𝑎 +∞ 𝑏 which satisfies = + Φ (𝑇) + ∫ Φ( )𝑑𝑥 −1 𝑇 𝑇 𝑥 𝑎 𝑏−𝑎 𝜋(𝑒 − ln 𝑇) 𝑏/𝑇 min { + (1 + exp ( )) 𝑇 𝑇 𝑇 √3𝜎 𝑎 𝑏−𝑎 𝑇 Φ (𝑥) = + Φ (𝑇) +𝑏∫ 𝑑𝑥. (52) 𝑇 𝑇 𝑥2 𝑇 1 𝜋(𝑒 − 𝑥) −1 0 + ∫ (1 + ( ln )) 𝑑𝑥} . 2 exp (46) 0 𝑥 √3𝜎 6 Mathematical Problems in Engineering

3 [3] R. L. Scheaffer, “Optimum age replacement policies with an 2.9 inceasing cost factor,” Technometrics,vol.13,no.1,pp.139–144, 1971. 2.8 [4] R. Cleroux, S. Dubuc, and C. Tilquin, “The age replacement 2.7 problem with minimal repair and random repair costs,” Opera- tions Research,vol.27,no.6,pp.1158–1167,1979. 2.6 [5] R. Cleroux and M. Hanscom, “Age replacement with adjustment 2.5 and depreciation costs and interest charges,” Technometrics,vol. Cost 2.4 16, pp. 235–239, 1974. [6] P. J. Boland and F. Proschan, “Periodic replacement with 2.3 increasing minimal repair costs at failure,” Operations Research, 2.2 vol.30,no.6,pp.1183–1189,1982. [7] J. P. Jhang and S. H. Sheu, “Opportunity-based age replacement 2.1 policy with minimal repair,” Reliability Engineering and System 2 Safety,vol.64,no.3,pp.339–344,1999. 0.2 0.4 0.6 0.8 0 1 [8] T. Nakagawa, Maintenance Theory of Reliability, Springer, Lon- Time T don, UK, 2005. Figure 1: The expected cost function curve under the changes of [9] D. Kahneman and A. Tversky, “Prospect theory: an analysis of time 𝑇. decisions under risk,” Econometrica, vol. 47, no. 2, pp. 263–291, 1979. [10] B. Liu, Uncertainty Theory, Springer, Berlin, Germany, 2007. In particular, let 𝑒=1, 𝜎=2, 𝑏 = 0.2,and𝑎 = 0.1;we [11] B. Liu, Uncertainty Theory, Springer, Berlin, Germany, 3rd edition, 2010. give the changing trend of the expected cost function with time 𝑇 in Figure 1.ItcanbeseenobviouslyfromFigure 1 [12] K. Yao and D. A. Ralescu, “Age replacement policy in uncertain environment,” Iranian Journal of Fuzzy Systems,vol.10,no.2, that the expected cost is monotone increasing firstly and then pp.29–39,2013,http://orsc.edu.cn/online/110906.pdf. monotone decreasing in time 𝑇.Wecanobtaintheoptimal ∗ replacement time 𝑇 = 0.2467 and the minimum cost is [13] B. Liu, “Fuzzy process, hybrid process and uncertain process,” Journal of Uncertain Systems,vol.2,no.1,pp.3–16,2008. 2.1117.

5. Conclusions This paper first studied the delayed age replacement policy in uncertain environment. It gave the expected costs in infinite time span and found the optimal replacement time which minimizes the expected cost. The optimal time to replace the unit was irrelevant to the uncertain distribution of the first unit. In addition, a number example was gave.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments This work was supported by the Young Scientists Fund of the National Natural Science Foundation of China (61403395), the Natural Science Foundation of Tianjin Grant 13JCY- BJC39000, and Special Fund of the Civil Aviation University of China of the Fundamental Research Funds for the Central Universities under Grant no. 3122013D004.

References

[1] R. E. Barlow and F.Proschan, Mathematical Theory of Reliability, JohnWiley&Sons,NewYork,NY,USA,1965. [2] B. Fox, “Age replacement with discounting,” Operations Re- search,vol.14,no.3,pp.533–537,1966. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 813565, 11 pages http://dx.doi.org/10.1155/2015/813565

Research Article Minimizing the Discrepancy between Simulated and Historical Failures in Turbine Engines: A Simulation-Based Optimization Method

Ahmed Kibria, Krystel K. Castillo-Villar, and Harry Millwater

Mechanical Engineering Department, The University of Texas at San Antonio, San Antonio, TX 78249, USA

Correspondence should be addressed to Krystel K. Castillo-Villar; [email protected]

Received 4 July 2014; Revised 30 December 2014; Accepted 31 December 2014

Academic Editor: Chia-Cheng Tsai

Copyright © 2015 Ahmed Kibria et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The reliability modeling of a module in a turbine engine requires knowledge of its failure rate, which can be estimated by identifying statistical distributions describing the percentage of failure per component within the turbine module. The correct definition of the failure statistical behavior per component is highly dependent on the engineer skills and may present significant discrepancies with respect to the historical data. There is no formal methodology to approach this problem and a large number of labor hours are spent trying to reduce the discrepancy by manually adjusting the distribution’s parameters. This paper addresses this problem and provides a simulation-based optimization method for the minimization of the discrepancy between the simulated and the historical percentage of failures for turbine engine components. The proposed methodology optimizes the parameter values of the component’s failure statistical distributions within the component’s likelihood confidence bounds. A complete testing of the proposed method is performed on a turbine engine case study. The method can be considered as a decision-making tool for maintenance, repair, and overhaul companies and will potentially reduce the cost of labor associated to finding the appropriate value of the distribution parameters for each component/failure mode in the model and increase the accuracy in the prediction of the mean time to failures (MTTF).

1. Introduction systems a difficult task. Unfortunately, epistemic uncertainty is a common and unavoidable characteristic among real- There are several failure modes in gas turbine engines. Typ- worldsystems.Thedevelopmentofsimulationasmeansto ically, the failure rates of different components are recorded evaluate stochastic systems enhances the ability to obtain per- throughoutthelifeoftheengine.Uponobtainingthefailure formance measure estimates under several given conditions. rates of such components, the first step is to conduct a Furthermore, these performance measures are much more statistical distribution fitting to the initial data per component accurate compared to the estimations of analytical tech- or failure mode. The mean percentage of failures computed niques, which often make crude assumptions about the while considering the fitted statistical distribution may not system’s condition and operation. besimilartothehistoricaldataduetothelackofquantityor Computer simulations are highly effective in answering quality of the data. Therefore, a common practice is to per- evaluative questions concerning a stochastic system. How- form some adjustments in the distribution parameters based ever, it is often necessary to determine the values of the on human intelligence and the experience of the engineers. model’s decision parameters such that a performance mea- Currently, there is no formal methodology to approach this sure is maximized or minimized. This type of problems can complex problem. be tackled by simulation-based optimization methods [1]. The reliability modeling of turbine engines is a complex This paper aims to develop a simulation-based optimiza- stochastic system. The complexities that arise when random- tion method to reduce the discrepancy between simulated ness is embedded within a system make the analysis of these and historical failures, which will not only significantly 2 Mathematical Problems in Engineering

Component Component Component ··· Component 1 2 3 n

Figure 1: Series system for the mock engine model.

reducethecostoflaborhoursassociatedtoperformingthis percentages of failures, which is expressed as the sum of activity manually but will also find an appropriate value of the squares of errors (SSE). Nevertheless, other combinations of distribution parameters for each turbine component/failure feasible distribution’s parameters values may lead to lower mode so that the simulation’s outcome better agrees with the SSE values. (i.e., other high quality solutions could exist in real-life data. nearby neighborhoods). These neighborhoods are explored This research was done in collaboration with an aerospace in the next phase of the method. industry’s maintenance, repair, and overhaul company In Phase II, a SA-based search procedure is used to referred to as ABC Company. A 54-component turbine explore the neighborhood within the feasible region of the engine is considered as a case study. For confidentiality distributions’ parameters in order to locate solutions of reasons, the names of the components, distributions, and potentially higher quality. The objective is to minimize the distributions’ parameters are not mentioned. SSE obtained from Phase I. The SA-based procedure works The structure of the paper is organized as follows. well for Phase II because it has the ability to move away from Section 2 presents the proposed model and its notation. local minima in contrast with gradient methods that cannot Section 3 discusses the details of the proposed methodology. overcome local minima. The SA neighborhood function is In Section 4, the case study is presented along with a design of defined based on the bounds of the distribution parameters of experiments (DOE) to tune the parameters of the optimiza- each component. The following section discusses the imple- tion algorithm. Section 5 shows the results of the case study. mentation in further detail. Finally, concluding remarks and future work are presented in Section 6. 3.1. Monte Carlo Simulation to Obtain the Failure Time of Each Component. A Monte Carlo simulation is often used to 2. Turbine Engine Model obtain not only the simulated failure times per component, but also the distribution of the statistical estimates of the The engine model is developed using the following assump- system’s TTFs: MTTF and STTF. tions: (1) the component events are independent of each At this stage of the proposed methodology, the distribu- other, (2) the component failure rates interact in a series tion and the distributions’ parameters values per component fashion (see Figure 1), and (3) some of the components (obtained from initial data) are considered as primary inputs. mayhaveaccumulativetime(e.g.,anitemthathasbeen The data set for each component contains suspensions (cen- repaired will recover its previous cumulative time). The sored data) which represent the units that have not failed model considers the lowest TTF associated to any component by the failure mode in question. They may have failed by asthesystem’stimetofailure(i.e.,whichevercomponentwill different failure modes or not failed at all [2]. havethelowestTTFateachrunofthesimulationwillcause Inthecaseoflifedata,thesedatasetsarecomposedof the system to fail) units that did not fail. For example, if five units were tested 𝑘 and only three had failed by the end of the test, it would have 𝑔(𝑇 )=min (𝑡1,𝑡2,𝑡3,𝑡4,...,𝑡𝑛). (1) right censored data (or suspension data) for the two units that

𝑘 did not failed. The term right censored implies that the event Equation (1) computes the system’s TTF where 𝑔(𝑇 ) is of interest (i.e., the time to failure) is to the right of our data the system’s TTF and 𝑡𝑖 is the TTF of 𝑖th component. point. In other words, if the units were to keep on operating, the failure would occur to the right on the time scale [3]. 3. Methodology A random number, drawn from a uniform distribution over the interval [0, 1] is used to generate a failure time from The proposed simulation-based optimization method the distribution’s inverse cumulative distribution function searchesforahighqualitysolution(i.e.,thevalueofthe (CDF). For each component of the model, the formula for the distribution’s parameters of the components within their CDF corresponding to the distribution of each component is feasible regions) with the use of the simulated annealing usedandthensolvedfor𝑡 (the failure time). Table 1 displays (SA) metaheuristic. The proposed method is broken down the equations to generate failure times (𝑡). into two phases. For the normal or lognormal distributions, there are no Phase I consists of a Monte Carlo simulation to obtain closed-form expressions but there are good mathematical the simulated percentage of failure per component, given approximations for that purpose. The approximation utilized an initial set of distribution parameters. The simulated in this paper is the one developed by Weisstein [4], which percentages of failures from Phase I are used to compute the considers the error function (erfc). The definition for the erfc initial discrepancy between the simulated and the historical can be found in the Appendix. The inverse CDF equations, Mathematical Problems in Engineering 3

Table 1: Inverse CDF equations for different distributions in the case study.

Distribution Inverse CDF 𝑇−𝜇 Normal [−√2𝜎erfcinv (2 (1 − ((1−𝐹(𝑡)) (1 − (0.5 erfc (− ( )))))))] + 𝜇 − 𝑇 √2𝜎 [−(√2𝜁 (2((−(1−𝐹(𝑡))(1−(0.5 erfc(−((ln 𝑇−𝜆)/√2𝜁)))))+1)))+𝜆] Log normal 𝑒 erfcinv −𝑇 1/𝛽 −(𝑇/𝛼)𝛽 Weibull (2 parameters) 𝛼((−ln ((1−𝐹(𝑡)) 𝑒 )) )−𝑇 1 𝜇 [ ] Exponential ln 1−𝐹(𝑡)

shown in Table 1,includecumulatedtime(𝑇); that is, as cooling. The SA algorithm simulates the energy changes in different individual components are replaced, the different a system subjected to a cooling process until it converges to operating times can be adjusted for the components whose an equilibrium state (steady frozen state). This scheme was reliability has been restored (through either refurbishment or developed in 1953 by Metropolis et al. [10]. replacement). Components that are not replaced are returned The SA-based procedure has been successfully imple- to service (i.e., their operating times are not different than mented to address problems in manufacturing enterprises when they were removed from service). This reliability model such as [11, 12]. The interested reader is referred to the will be used to compute the system’sconditional probability of works of Eglese [13], Aarts and Lenstra [14] and Suman and failure once it has been repaired. The inverse CDF equations Kumar [15], for a comprehensive description of the simulated are derived from the conditional reliability equations. If there annealing metaheuristic. isnocumulatedtime(e.g.,𝑇=0), then the inverse CDF equations would be transformed into the unconditional form. 3.2.1. Optimization Problem Definition, Objective, and Con- The system’s TTF is computed considering the compo- straints. After all the iterations of the Monte Carlo simula- nent with the lowest TTF as the system failure rate (1).Thus,a tion, the percentage of failures is obtained for each compo- single iteration of the loop computes a single system TTF for nent and it is compared to the historical data to obtain the the model. Statistical estimates (mean and standard deviation squared error. The discrepancy or the squared error between of the system TTFs MTTF and STTF, resp.) as well as the thesimulatedandhistoricalpercentageoffailureisobtained simulated percentages of failures for each component are for each component to conform the sum of squared errors obtained after performing all the iterations of the Monte (SSE). Carlo simulation. The objective function for the optimization process is shown in (2) and aims to minimize the sum of squared 3.2. Simulated Annealing-Based Optimization Procedure. The errors (SSE) of the system’s percentage of failures subject to simulated annealing (SA) algorithm is a well-known local constraints which are the confidence bounds (contour plots) search metaheuristic used to address discrete, continuous, on the distribution’s parameters: and multiobjective optimization problems. Its ease of imple- 𝑛 mentation, convergence properties, and ability to escape local 2 optima have made it a popular technique during the past two min [SSE = ∑ (𝑥̂𝑖 −𝑥𝑖) ], (2) decades. The application of the SA algorithm to optimization 𝑖=1 problems emerges from the work of Kirkpatrick et al. [5]and 𝑥̂ 𝑥 ˇ where 𝑖 is the simulated percentage of failures, 𝑖 is the Cerny[´ 6]. In these pioneering works, the SA was applied to historical percentage of failures, and 𝑛 is the number of graph partitioning and very-large-scale integration design. components. In the 1980s, the SA algorithm had a major impact on the The simulation-based optimization method will propose field of heuristic search for its simplicity and efficiency in newvaluesofthedistributionparameters(𝑃𝑖𝑗 ), where 𝑃𝑖𝑗 solving combinatorial optimization problems. Moreover, the is the 𝑖th distribution parameter of 𝑗th component within SA algorithm has been extended to deal with continuous their contour plots (see Figure 2). According to Wasserman optimization problems [7–9]. [16], the likelihood contour plot can be drawn satisfying the The SA algorithm is based on the principles of statistical following relationship: mechanics whereby the annealing process requires heating andthenaslowcoolingofasubstancetoobtainastrong 1 𝐿(𝑃 )= 𝐿∗ − 𝜒2 , crystalline structure. The strength of the structure depends ln 𝑖𝑗 ln 2 1,𝑎 (3) on the rate of cooling of the substance. If the initial tem- ∗ perature is not sufficiently high or a fast cooling is applied, where ln 𝐿 is the log-likelihood of the 𝑗th component at the imperfections (metastable states) are obtained. In this case, estimated distribution parameters using MLE, ln 𝐿(𝑃𝑖𝑗 ) is the the cooling solid will not attain thermal equilibrium at each log-likelihood of the 𝑗th component for the perturbed 𝑖th temperature. Strong crystals are grown from careful and slow distribution parameter, and 𝑎 =1− confidence level. 4 Mathematical Problems in Engineering

Upper bound of 𝛽 Contour plot “mlecustom” function of MATLAB. The general form of the 1.100 likelihood function considering censored samples is [2]

𝑟 𝑘 0.880 𝐿=∏𝑓(𝑥𝑖) ∏ (1 − 𝐹 𝑗(𝑇 )) , (5) 𝑖=1 𝑗=1 Lower bound of 𝛽 𝑟 𝑘 𝛽 where isthenumberofunitsruntofailure, is the number of unfailed units, 𝑥1,𝑥2,𝑥3,...,𝑥𝑟 aretheknownfailure 0.440 times, and 𝑇1,𝑇2,𝑇3,...,𝑇𝑘 are the operating times on each unfailed unit. 0.220 For example, if the time to failure distribution is Weibull- Upper bound of 𝛼 distributed, the modified likelihood function for censored Lower bound of 𝛼 0 samples would be 0 612182430 𝑟 𝛽−1 𝑘 4 𝛽 𝑥 𝛽 𝛽 𝛼(hr) 10 𝑖 −(𝑥𝑖/𝛼) −(𝑇𝑗/𝛼) 𝐿=∏ ( )( ) 𝑒 ∏𝑒 . (6) 𝑖=1 𝛼 𝛼 𝑗=1 Figure 2: Feasible region (contour plot) for the distribution parameters of one component. The “mlecustom” function of MATLAB was used to computetheboundsofthedistribution’sparametersusingthe normal approximation method. The normal approximation In case of unfeasibility, that is, the distribution’s parame- method for confidence interval estimation is used in most ters at the new state not satisfying (3),oneofthedistribution commercial statistical packages because of the relative easi- parameters is set to its nearest bound (upper or lower bound, ness for the bound’s computation. However, the performance whichever is closer) and solved for the other distribution ofsuchprocedurescouldbepoorwhenthesamplesizeisnot parameter so that they satisfy (3).Equation(4) shows the large, or when heavy censoring is considered [17]. equation of likelihood contour plot for the Weibull two- Ononehand,sincetheinputdatasetcontainsheavycen- parameter distribution as an example: soring, the computation of confidence bounds on parameters 𝐿(𝛼( ),𝛽) using the normal approximation is not recommended. On the ln lower bound or upper bound other hand, Fisher matrix bounds tend to be more optimistic 1 (4) than the nonparametric rank based bounds. This may be a = 𝐿∗ − 𝜒2 . ln 2 1,𝑎 concern, particularly when dealing with small sample sizes. Fisher matrix bounds are too optimistic when dealing with The shaded region in Figure 2 shows an example of small sample sizes and, usually, it is preferred to use other the parameters’ feasible region for the simulation-based techniques for calculating confidence bounds, such as the optimization method. likelihood ratio bounds [2]. Hence, the likelihood ratio-based A schematic diagram of the SA-based procedure, used to confidence bounds estimation method is preferred. determine the best value of the distributions’ parameters, is Without loss of generality, the use of the likelihood presented in Figure 3. ratio statistic for developing confidence intervals about a Obtaining the confidence bounds of the distribution’s parameterofinterest(𝛾) can be described. The likelihood parameters is an integral part of the proposed methodology. ratio (LR) test statistic provides a formal framework for There are multiple methods for obtaining the bounds of these testing the hypothesis: 𝐻0: 𝛾=𝛾0 and 𝐻1: 𝛾 =𝛾̸ 0. parameters.Thefollowingsectionprovidesthenecessary As its name implies, the LR statistic test is a ratio of background behind the selection of the likelihood ratio based likelihood functions. However, it is more convenient to work on the confidence bounds as the preferred method as well with the log form, which is computed as the difference as the steps for obtaining the bounds of the distributions’ between two log-likelihood expressions. Specifically, 𝐻0 is parameters. rejected at some level of confidence (1−𝑎), if 2( 𝐿∗ − 𝐿∗ (𝛾 )) > 𝜒2 , 3.2.2. Confidence Bounds on the Distribution Parameters. ln ln 0 1,𝑎 (7) Our data set contains suspensions (censored data); therefore, 𝐿∗ the procedure for computing confidence bounds includes where ln is the log-likelihood function evaluated at the 𝐿∗(𝛽,̂ 𝛾)̂ 𝐿∗(𝛾 )= censored data in the parameter estimation. In this section, we maximum likelihood (ML) point ln and ln 0 ̂ present the methodology to obtain the likelihood ratio-based max𝛽 ln 𝐿(𝛾0,𝛽).If𝛽(𝛾0) isdenotedbythesolutionto confidence bounds of the distribution’s parameters. max𝛽 ln 𝐿(𝛾0,𝛽),then In the general case, the distribution parameters are ∗ ̂ estimated using the maximum likelihood estimation (MLE) ln 𝐿 (𝛾0)=ln 𝐿(𝛾0, 𝛽(𝛾0)) . (8) method with a modified likelihood function for censored data. Using these estimated parameters, confidence bounds An asymptotically correct confidence interval or region can be calculated. The parameters are estimated using on 𝛾 consists of all the values of 𝛾 for which the null Mathematical Problems in Engineering 5

Preprocessing: (a) Obtain the distribution and distribution parameters for each component considering censored data Neighborhood function: (b) Obtain the historical percentage of failures per component (a) Propose new distribution parameters using a Monte Carlo simulation neighborhood step (a) Generate failure times (b) Compute the system’s time to failure (TTF) (b) Verify feasibility condition (c) Compute the simulated percentages of failures (3) (i.e., parameters per component ∗ within the contour plot) Termination criteria determined by - Maximum trials (trails) Calculation of the new - Accepted trials (trails) Single-objective function: system’s temperature - Markov chains without Compute the sum of square errors (SSE) (2) chains SSE TS improvement ( ) accepted before TS = SSElast accepted Metropolis criterion: new parameters are accepted if No No No No No − + /T Verify U(0, 1) < e SSEnew SSEbefore 𝑆 Verify termination SSE ≤ SSE SSE ≤ SSE Termination∗ best last criteriaCriteria* Last YesYes New state (set of Yes Yes Yes = parameters) SSEbest SSE EndEd is accepted

Figure 3: Schematic diagram of the SA-based optimization procedure.

hypothesis, 𝛾=𝛾0, is not rejected at some stated level of Step 3. Use the “mlecustom()” function in MATLAB to significance1−𝑎 ( ). In other words, it consists of all values estimate the distribution’s parameters values. of 𝛾0 satisfying Step 4. Compute the output value of the likelihood function 1 (i.e., the modified likelihood function for a component with 𝐿∗ (𝛾 )≤ 𝐿∗ − 𝜒2 . ln 0 ln 2 1,𝑎 (9) censored data points) for the parameter estimates obtained in Step 3 using (5). The LR confidence intervals can be graphically identified with the use of the likelihood contours, which consist of Step 5. Obtain the graphical estimate of confidence intervals all the values of 𝛾 and 𝛽 for which ln 𝐿(𝛾, 𝛽) is a constant. of the parameters satisfying (10) from the likelihood contour Specifically, the contours constructed are solutions to plot.

1 Step 6. Obtain the chi square statistic value and substitute 𝐿 (𝛾,) 𝛽 = 𝐿∗ − 𝜒2 . that value in (11) [19] for the specified confidence level: ln ln 2 1,𝑎 (10)

∗ ∗ 2 𝐿 (𝜃) 2 Solutions to ln 𝐿 (𝛾0)=ln 𝐿 −(1/2)𝜒 1,𝑎 will lie on these −2 ln ( )≥𝜒 1−𝑎;𝑘, (11) 𝐿(𝜃)̂ contours. For the Weibull, lognormal, and normal distributions, the confidence intervals can be graphically estimated by where 𝐿(𝜃) is the likelihood function for the unknown drawing lines that are both perpendicular to the coordinate ̂ axis of interest and tangent to the likelihood contour [18]. For parameter vector 𝜃, 𝐿(𝜃) is the likelihood function calculated 𝜃 𝜒2 one-parameter distributions (e.g., exponential), no contour at the estimated vector ,and 𝑎;𝑘 is the chi-squared statistic plot is generated and the confidence bound on the parameter with probability 1−𝑎and 𝑘 degrees of freedom, where 𝑘 is will be considered. The summary of steps to obtain the LR the number of quantities jointly estimated. basedboundsonparametersfollows. Step 7. Using the graphical estimates of confidence intervals Step 1. Load the experimental data and define separate sets on the parameters as initial guess, two nonlinear optimization of information containing the failure and the censored data problemsaresolvedtoobtainaccurateLRlimits.Sincethere points from the input data set. are two unknowns (e.g., Weibull’s shape and scale parameters) and only one equation, use an iterative method to obtain Step 2. Define the custom probability density function (PDF) the values of the parameters (i.e., for given values of scale, and cumulative density function (CDF) equations based on obtain the maximum and minimum value of shape parameter the distribution of the component. that satisfy (11)) and vice versa. The “fsolve()” function in 6 Mathematical Problems in Engineering

Initial data Input Distribution xi parameters

Realizations Generation for F(t) of failure times, t

Computation of system’s TTF using

MC simulation MC the system equation

Simulated percentage of failures to obtain discrepancy (squared error) between simulated and historical data Optimization Objective: minimize SSE

Set new value of distribution parameters within confidence bounds

Figure 4: Schematic flowchart of the proposed method.

MATLAB was used to compute solution iteratively. The Table 2: Distribution fittings. default algorithm for “fsolve()” is the “Trust-Region Dogleg,” whichwasdevelopedbyPowell[20, 21]. Distribution Parameters Component number 1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,15,16,19,20,23,24, 3.3. Summary of Proposed Solution Procedure. The following 25, 26, 27, 28, 29, 30, 31, steps summarize the proposed methodology. Weibull (2P) Shape (𝛽), scale (𝛼) 32,33,34,35,36,37,38, 39, 40, 41, 42, 43, 45, 46, Step 1. The initial data, consisting of failure times and 47,48,49,50,51,52,53, operating times of unfailed units, is provided. The input data 54 is used to determine the distribution behavior of the com- Log normal log-𝜇 (𝜆), log-𝜎 (𝜁) 2, 3, 17, 18, 22, 44 ponents’ failure rate and to determine the initial distribution Mean (𝜇), standard Normal 4 parameters values for each component using Weibull++. deviation (𝜎) Exponential Mean (𝜇)21 Step 2. The components’ distributions and their parameters areusedtosimulatetheexpectedfailuretime(𝑡). For each 𝐹(𝑡) 0≤ component, a random number, that is, in the range 4. Case Study 𝐹(𝑡) ≤1, is generated using uniform distribution 𝑈(0, 1). 𝐹(𝑡) A failure time is generated corresponding to from the 4.1. Distribution and Parameter Fittings. The turbine engine inverse cumulative distribution function (CDF) equation for model consists of 54 components/failure modes. Table 2 each component. Finally, the system’s time to failure (TTF) is summarizes the parameters corresponding to the distribution computed using the system’s equation (1). of each component used in the analysis. To select the appropriate distribution and parameter values, a weighted Step 3. The simulated percentages of failure are obtained goodness of fit measure using Weibull++ software along with from the Monte Carlo simulation for all the components engineering experience was utilized. A fixed cumulative time in the system. The discrepancies (e.g., SSE) between the of 1850 hours is used for the 18th component of the model. simulated and historical percentages of failures are computed.

Step 4. The simulated annealing algorithm minimizes the 4.2. The SA’s Parameters Tuning through Statistical Design of sum of square errors (SSE) subject to constraints (e.g., the Experiments (DOE). A statistical design of experiments can confidence bounds on distribution parameters). be defined as a series of tests in which purposeful changes are made to the input variables of a process or system so A schematic diagram of the proposed method is shown in thatchangesintheoutputresponsecanbeobservedand Figure 4. quantified [22]. Mathematical Problems in Engineering 7

Table 3: Factors and levels during the DOE development. Table4:Comparisonoftheresponses(beforeandafterthe optimization). Levels Factor name 12 3 Sumofsquarederrors Mean of squared (SSE) errors (MSE) Trials 50 100 200 Chains 12 4Before optimization 50.6541 0.9380 Alpha 0.9 0.95 0.98 After optimization 35.1834 0.6515

Main effects plot for SSE SSE). After a careful inspection of the DOE results, the best Means combination of SA’s parameter values was found to be Alpha Trials Chains 44.0 (i) cooling rate (Alpha): 0.98,

43.5 (ii) number of accepted trials (Trials): 200, (iii) number of Markov chains (Chains): 4. 43.0 The initial systems’ temperature was fixed to 10. In order Mean of SSE of Mean 42.5 to get this value, the initial temperature is initialized at a very low value and an experimental run is performed where the 42.0 temperature is raised by a ratio until a minimum acceptance 0.90 0.95 0.98 50 100 200 1 2 4 ratio defined by the practitioner (0.8) is met; the acceptance ratioiscomputedasthenumberofacceptedtrials(Trials) Figure 5: Main effects plot for SSE. The gray background represents divided by the trials. terms that were not significant in the ANOVA. A sensitivity analysis was conducted to determine the best value for the Monte Carlo sample (MC is not an SA algorithmic parameter but it is a parameter in the overall solution procedure). It was observed that the SSE becomes In order to tune the parameters of the optimization algo- stable for Monte Carlo sample size of 10,000 and larger values; rithm, a design of experiments is performed to understand that is, the SSE is not sensible to changes in the Monte Carlo the cause and effect relationship among the SA’s parameters sample size after a value of 10,000 samples. On the other hand, and the output response. if the sample size for the Monte Carlo loop is below 10,000, ThefactorsthatwereselectedduringtheDOEdevelop- the SSE considerably increases for this particular problem. (1) ment are the number of accepted trials before the system’s Hence,theselectednumberofMonteCarlosampleswasfixed (2) temperature is decreased (Trials), thenumberofMarkov at 10,000. chains (Chains), and (3) therateofcooling(Alpha). Table 3 shows the selected factors and their levels. The responses are (1) the sum of squared errors (SSE), (2) 5.2. Test Case Results. The evaluation of the test case was the mean of squared errors (MSE), and (3) the average CPU performed based on the outcomes of the previously presented time. DOE analysis. After conducting a computational evaluation of the test case, the value of the SSE was found to be 35.1834 having an average CPU time of 55,239 seconds. 5. Results and Discussion Table 4 compares the values of the SSE before and after the optimization. 5.1. DOE Results. The DOE analysis was performed using The SA-based procedure was executed five times con- 3 Minitab. A total number of 3 =27computational experi- sidering the best combination of algorithmic parameters. ments were performed while considering a 95% significance Figure 6 shows the average performance of the solution level. The ANOVA model adequacy verification did not procedure. exhibit issues with regard to (1) the residual’s normality Table 5 shows the discrepancies between the simulated behavior, (2) the residual’s independency, and (3) the resid- and historical percentages of failures for the components ual’s constant variance assumptions. before and after the optimization. From the ANOVAresults, it was observed that the cooling Figure 7 showsthehistogramofpercentageoffailuresfor rate (Alpha) was not a significant factor and it was fixed to thehistoricaldata,beforeandaftertheoptimization. 0.98. The significant factors were number of accepted trials In order to observe the shifting of the distribution’s (Trials)andthenumberofMarkovchains(Chains). parameters at the different evaluations of the DOE, contour For the sake of brevity, only selected plots are presented. plots were constructed for each component. Figure 8 shows Figure 5 shows the main effects plot considering SSE as a the shifting of the distribution’s parameters at the different response. evaluations of the DOE for the 45th component and Figure 9 From Figure 5, it can be observed that both factors (e.g., shows both the original (fitted) and best found combination Trials and Chains)havegreatimpactontheresponse(i.e., of distribution parameters values for 45th component. 8 Mathematical Problems in Engineering

Table 5: Comparison of the responses (before and after the optimization).

Before After Historical Simulated Simulated Component number percentage [ ] [ ] of failures percentage of Discrepancy % Squared error percentage of Discrepancy % Squared error failure failure 1 0.86 0.5211 −0.3389 0.1149 0.46 −0.4 0.1600 2 1.82 1.5673 −0.2527 0.0639 1.66 −0.16 0.0256 3 3.53 3.6 0.07 0.0049 3.76 0.23 0.0529 4 0.88 0.5947 −0.2853 0.0814 0.63 −0.25 0.0625 5 2.4 2.5271 0.1271 0.0162 2.37 −0.03 0.0009 6 1.31 1.2096 −0.1004 0.0101 1.22 −0.09 0.0081 7 0.76 0.6553 −0.1047 0.0110 0.78 0.02 0.0004 8 0.4 3.5392 3.1392 9.8546 3.04 2.64 6.9696 9 1.04 0.1024 −0.9376 0.8791 0.09 −0.95 0.9025 10 5.35 5.0636 −0.2864 0.0820 5.09 −0.26 0.0676 11 4.56 4.21 −0.35 0.1225 4.26 −0.3 0.0900 12 8.32 7.9168 −0.4032 0.1626 7. 8 7 −0.45 0.2025 13 2.5 2.0153 −0.4847 0.2349 2.38 −0.12 0.0144 14 1.39 1.1423 −0.2477 0.0614 1.18 −0.21 0.0441 15 2.8 2.5067 −0.2933 0.0860 2.45 −0.35 0.1225 16 0.3 0.259 −0.041 0.0017 0.27 −0.03 0.0009 17 0.51 0.499 −0.011 0.0001 0.47 −0.04 0.0016 18 0.91 1.8601 0.9501 0.9027 1.81 0.9 0.8100 19 1.34 0.9799 −0.3601 0.1297 1.24 −0.1 0.0100 20 0.68 0.4956 −0.1844 0.0340 0.68 0 0.0000 21 0.45 0.4621 0.0121 0.0001 0.53 0.08 0.0064 22 0.51 0.4554 −0.0546 0.0030 0.4 −0.11 0.0121 23 0.25 0.1628 −0.0872 0.0076 0.17 −0.08 0.0064 24 11.07 6.3014 −4.7686 22.7395 7. 0 3 −4.04 16.3216 25 4.03 4.1075 0.0775 0.0060 4.26 0.23 0.0529 26 0.55 0.0571 −0.4929 0.2430 0.05 −0.5 0.2500 27 0.7 0.6549 −0.0451 0.0020 0.72 0.02 0.0004 28 5.8 4.4421 −1.3579 1.8439 4.31 −1.49 2.2201 29 0.38 0.2823 −0.0977 0.0095 0.22 −0.16 0.0256 30 3.32 2.1621 −1.1579 1.3407 2.35 −0.97 0.9409 31 0.5 0.4702 −0.0298 0.0009 0.43 −0.07 0.0049 32 0.99 0.7867 −0.2033 0.0413 0.91 −0.08 0.0064 33 0.45 0.3763 −0.0737 0.0054 0.33 −0.12 0.0144 34 0.4 0.3421 −0.0579 0.0034 0.38 −0.02 0.0004 35 0.63 0.822 0.192 0.0369 0.81 0.18 0.0324 36 0.85 1.0452 0.1952 0.0381 0.98 0.13 0.0169 37 0.4 0.561 0.161 0.0259 0.47 0.07 0.0049 38 0.38 0.5465 0.1665 0.0277 0.49 0.11 0.0121 39 0.23 0.3274 0.0974 0.0095 0.36 0.13 0.0169 40 6.02 6.3466 0.3266 0.1067 6.39 0.37 0.1369 41 2.98 4.0667 1.0867 1.1809 3.88 0.9 0.8100 42 0.61 0.9346 0.3246 0.1054 0.93 0.32 0.1024 43 0.6 0.8256 0.2256 0.0509 0.91 0.31 0.0961 44 0.76 0.2627 −0.4973 0.2473 0.27 −0.49 0.2401 Mathematical Problems in Engineering 9

Table 5: Continued. Before After Historical Simulated Simulated Component number percentage [ ] [ ] of failures percentage of Discrepancy % Squared error percentage of Discrepancy % Squared error failure failure 45 6.28 9.0659 2.7859 7.7612 7. 8 4 1.56 2.4336 46 1.23 1.8439 0.6139 0.3769 1.55 0.32 0.1024 47 0.85 1.1438 0.2938 0.0863 1.1 0.25 0.0625 48 1.66 2.1355 0.4755 0.2261 2.37 0.71 0.5041 49 1.29 1.8673 0.5773 0.3333 1.76 0.47 0.2209 50 0.71 0.9715 0.2615 0.0684 1.01 0.3 0.0900 51 2.11 2.9768 0.8668 0.7513 2.92 0.81 0.6561 52 0.38 0.558 0.178 0.0317 0.68 0.3 0.0900 53 0.5 0.7342 0.2342 0.0548 0.82 0.32 0.1024 54 0.45 0.6368 0.1868 0.0349 0.66 0.21 0.0441 Total 50.6541 Total 35.1834

40 Objective value (SSE) value Objective 35 0 0.5 1 1.5 2 2.5 4 Objective function evaluations ×10

Average + Std Dev Average − Std Dev Average

Figure 6: The average performance of the SA-based procedure.

Failures (%) Failures 4

0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 Component number

Historical percentage of failures Simulated percentage of failures (before) Simulated percentage of failures (after)

Figure 7: Histogram of percentages of failures before and after optimization. 10 Mathematical Problems in Engineering

1.5 parameters. The results showed a 30% reduction of the SSE (e.g., from 50.6541 to 35.1834) for the engine model. The average CPU time was approximately 15 hours mainly due to the calculations involved in the likelihood function. 1.45 Alternativeneighborhoodandfeasibilityfunctionscanbe investigated by studying the trends in the shifting parameters’ values per component. For instance, it was observed that 1.4 the distribution’s parameters shifted around the edge of the )

𝛽 contour plot for some components. The proposed simulation-based optimization method can

Shape ( Shape serve as a decision-making tool for maintenance, repair, 1.35 and overhaul companies and will potentially reduce the cost of labor associated to finding the appropriate value of the distribution’s parameters for each component/failure mode in 1.3 themodelandincreasetheaccuracyinthepredictionofthe mean time to failures (MTTF). Future research lines involve parallelization of the algorithm to solve larger models (e.g., thousands of components) 1.25 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 andcomparingtheperformanceofthesimulatedannealing 4 Scale (𝛼) ×10 with other metaheuristics such as Evolutionary Algorithms, Tabu Search, and Particle Swarm Optimization, among oth- Distribution’s parameters for different DOE evaluations ers. Distribution’s parameter using the tuned SA’s parameters

Figure 8: Distribution parameters for the 45th component for Appendix different evaluation cases.

1.5 Error Function. The error function erf(𝑋) is twice the integral of the Gaussian distribution with 0 mean and variance of 1/2. The equation of error function is 1.45 𝑥 2 −𝑡2 erf (𝑥) = ∫ 𝑒 𝑑𝑡. (A.1) √𝜋 0

1.4 ) (𝑋) 𝛽 The complementary error function or erfc is the complement of the error function; that is, erfc(𝑋) = 1 − (𝑋) Shape ( Shape erf 1.35 ∝ 2 −𝑡2 erfc (𝑥) = ∫ 𝑒 𝑑𝑡. (A.2) √𝜋 𝑥 1.3 Nomenclature

1.25 MTTF:Meantimetofailureofthesystem 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 STTF: Standard deviation of time to failure of the 4 Scale (𝛼) ×10 system TTF: Time to failure of the system Original (fitted) distribution’s parameters values 𝐾 𝑔(𝑇 ):SystemTTF Best found distribution’s parameters values 𝐾 𝑇 :Vectorof𝐾 variables Figure 9: Original distribution’s parameters values and best found 𝑡:Failuretime combination of distribution’s parameters values for the 45th compo- 𝐹(𝑡): Cumulative distribution function of nent. failure time (𝑡) 𝑛: Number of components in the engine 6. Conclusions and Future Work model 𝛼: Scale parameter of a Weibull distribution This paper presented a simulated annealing-based optimiza- 𝛽: Shape parameter of a Weibull distribution tion method to minimize the discrepancy of historical and 𝜆:Meanoflog(𝑥) of a lognormal distribution simulated percentages of failures in a turbine engine model. 𝜁: Standard deviation of log(𝑥) of a A DOE was performed for the tuning of the algorithmic lognormal distribution Mathematical Problems in Engineering 11

𝜇: Mean of a normal distribution or an [12]K.K.Castillo-Villar,N.R.Smith,andJ.F.Herbert,“Designand exponential distribution optimization of capacitated supply chain networks including 𝜎: Standard deviation of a normal quality measures,” Mathematical Problems in Engineering,vol. distribution. 2014,ArticleID218913,17pages,2014. [13] R. W. Eglese, “Simulated annealing: a tool for operational research,” European Journal of Operational Research,vol.46,no. Conflict of Interests 3,pp.271–281,1990. The authors declare that there is no conflict of interests [14] E. E. H. Aarts and J. K. Lenstra, LocalSearchinCombinatorial regarding the publication of this paper. Optimization, Princeton University Press, Princeton, NJ, USA, 1997. [15] B. Suman and P. Kumar, “A survey of simulated annealing as a Acknowledgments tool for single and multiobjective optimization,” Journal of the Operational Research Society,vol.57,no.10,pp.1143–1160,2006. This research was supported by the Air Force Research [16] G. Wasserman, Reliability Verification, Testing, and Analysis in Laboratory through General Dynamics Information Tech- Engineering Design,vol.153,CRCPress,2002. nology, TaskOrder F5702-11-04-SC63-01 (PrimeContract no. [17]Y.Hong,L.A.Escobar,andW.Q.Meeker,“Coverageproba- FA8650-11-D-5702/0004). This support is gratefully acknowl- bilities of simultaneous confidence bands and regions for log- edged. The authors wish to acknowledge Carlos Torres, Alan location-scale distributions,” Statistics and Probability Letters, Lesmerises, and Eric Vazquez of StandardAero for their tech- vol.80,no.7-8,pp.733–738,2010. nical assistance (Distribution A. approved for public release, [18] J. F. Lawless, Statistical Models and Methods for Lifetime Data, distribution unlimited; Case no. 88ABW-2014-4202). vol. 362, John Wiley & Sons, 2011. [19] M. Kova´covˇ a,´ “Reliability likelihood ratio confidence bounds,” References Aplimat—Journal of Applied Mathematics,vol.2,no.2,pp.217– 225, 2009. [1] S. L. Rosen and C. M. Harmonosky, “An improved simulated [20] T. J. Dekker, “A floating-point technique for extending the annealing simulation optimization method for discrete parame- available precision,” Numerische Mathematik,vol.18,no.3,pp. ter stochastic systems,” Computers and Operations Research,vol. 224–242, 1971. 32,no.2,pp.343–358,2005. [21] M. J. Powell, “A FORTRAN subroutine for solving systems [2] R. B. Abernethy, The New Weibull Handbook,Dr.RobertB. of nonlinear algebraic equations,” Tech. Rep. AERE-R-5947, Abernethy, 2006. Atomic Energy Research Establishment, Harwell, UK, 1968. [3] Life Data Classification, “ReliaWiki Web Resource,” 2012, [22] D. C. Montgomery, Design and Analysis of Experiments,John http://reliawiki.org/index.php/Life Data Classification. Wiley & Sons, New York, NY, USA, 2008. [4] E. W. Weisstein, Log Normal Distribution, MathWorld—A Wolfram Web Resource, 2006, http://mathworld.wolfram.com/ LogNormalDistribution.html. [5] S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, “Optimization by simulated annealing,” Science, vol. 220, no. 4598, pp. 671–680, 1983. [6] V. Cernˇ y,´ “Thermodynamical approach to the traveling sales- man problem: an efficient simulation algorithm,” Journal of Optimization Theory and Applications,vol.45,no.1,pp.41–51, 1985. [7] A. Dekkers and E. Aarts, “Global optimization and simulated annealing,” Mathematical Programming, Series B,vol.50,no.1– 3, pp. 367–393, 1991. [8] M. Locatelli, “Simulated annealing algorithms for continuous global optimization: convergence conditions,” Journal of Opti- mization Theory and Applications,vol.104,no.1,pp.121–133, 2000. [9] L. Ozdamar¨ and M. Demirhan, “Experiments with new stochastic global optimization search techniques,” Computers and Operations Research,vol.27,no.9,pp.841–865,2000. [10] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” TheJournalofChemicalPhysics,vol.21,no. 6, pp. 1087–1092, 1953. [11] K. K. Castillo-Villar, N. R. Smith, and J. L. Simontoncy, “The impact of the cost of quality on serial supply-chain network design,” International Journal of Production Research,vol.50,no. 19, pp. 5544–5566, 2012. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 839042, 8 pages http://dx.doi.org/10.1155/2014/839042

Research Article Accelerated Testing with Multiple Failure Modes under Several Temperature Conditions

Zongyue Yu,1,2 Zhiqian Ren,1,2 Junyong Tao,1,2 and Xun Chen1,2

1 Science and Technology on Integrated Logistics Support Laboratory, National University of Defense Technology, Changsha, Hunan 410073, China 2 College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha, Hunan 410073, China

Correspondence should be addressed to Junyong Tao; [email protected]

Received 10 June 2014; Revised 9 August 2014; Accepted 16 September 2014; Published 30 September 2014

Academic Editor: Phil Scarf

Copyright © 2014 Zongyue Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Acomplicateddevicemayhavemultiplefailuremodes,andsomeofthefailuremodesaresensitivetolowtemperatures.Toassessthe reliability of a product with multiple failure modes, this paper presents an accelerated testing in which both of the high temperatures and the low temperatures are applied. Firstly, an acceleration model based on the Arrhenius model but accounting for the influence of both the high temperatures and low temperatures is proposed. Accordingly, an accelerated testing plan including both the high temperatures and low temperatures is designed, and a statistical analysis method is developed. The reliability function of the product with multiple failure modes under variable working conditions is given by the proposed statistical analysis method. Finally, a numerical example is studied to illustrate the proposed accelerated testing. The results show that the proposed accelerated testing is rather efficient.

1. Introduction variable-stress situations. van Dorp and Mazzuchi [6]developed a general Bayes exponential inference model. Khamis With the successive development of engineering and science and Higgins [7] proposed a model known as KH model for technology, high reliability devices usually operate for many step-stress ALT, which is based on a time transformation of years under working conditions. Accelerated testing has been the exponential model. The purpose of statistical analysis is to proposed as a means to predict the performances for highly predict the reliability of product under working conditions reliable products. Product reliability can be obtained by based on the accelerated testing data and the acceleration using the accelerated testing techniques, in which the devices model. Tang et al. [8] obtained MLE for parameters in a are subjected to higher-than-normal stress levels, leading to multicensored accelerated testing. Xiong [9]discussedMLE failure within days or weeks rather than years. By fitting for the exponential step-stress ALT with type II censored. the accelerated failure data to an appropriate model, device Fard and Li [10, 11]andBalakrishnanetal.[12–14]finished reliability under normal use conditions can be estimated a lot of researches about the statistical analysis method of [1, 2]. accelerated testing. Most of work on the accelerated testing The acceleration models and the statistical analysis meth- method assumed that there is a single cause of failure. How- ods have been the main focus of the studies about accelerated ever, a complicated device may fail due to several causes. The testing. The relationship between the stress and the reliability accelerated testing method with multiple failure modes has of device is established by an acceleration model. Common been a new focus of the accelerated testing researches. Kim acceleration models include the inverse power law model and Bai [15] and Craiu and Lee [16] described the situations in and the Arrhenius model [3, 4]. A lot of efforts were made engineering when multiple failure modes occurred. McCool by scholars to develop a new acceleration model. Benavides [17] presents a technique for calculating estimate intervals [5] constructed an acceleration model for step-stress and for Weibull parameters of a primary failure mode when 2 Mathematical Problems in Engineering

e failed resistances

Figure 1: The car control device installed in the temperature test Figure 3: The failed resistances. chamber.

Figure 2: The detection equipment for the car control device. Figure 4: The ship control device installed in the temperature test chamber. a secondary failure mode having the same Weibull shape to control the speed and direction of the car. In the low parameter is acting. Klein and Basu presented a series of temperature test, the car control device is installed in the papers [18, 19] on the analysis of accelerated testing when temperature test chamber (as shown in Figure 1), and the more than one failure mode is acting. There are a number of working state of the device is observed by the detection articles on the analysis of multiple failure data, some of which equipment (as shown in Figure 2). At the request of the device are reviewed in Pascual [20, 21], Liu and Qiu [22], and Xing producer, the testing temperature is 233 K. et al. [23]. High temperatures are widely applied in the existing accelerated testing. Failure modes that are sensitive to high 2.1.2. Failure Mode. It is detected that the speed controlled temperatures can be induced quickly by an existing acceler- by the device (the output signal is 42 km/h) is lower than the ated testing. However, the latest results of the low temper- speed set by the control panel (60 km/h) when the car control ature tests for the car control device and the ship control device is tested 220 hours at the temperature 233 K. According device described in the next section show that some failure to the results of a comprehensive circuit analysis, several modes are sensitive to the low temperatures. When the high resistances values are outside the normal range. The failed temperature accelerated testing is applied to a product with resistances are shown in Figure 3 (as the internal structure multiple failure modes, the failure modes that are sensitive to ofthedevicemayberelatedtothecommercialsecretsof the low temperatures will not be induced and the reliability producer,thepartsunrelatedtothefailuremodearecovered). assessment result will far depart from the actual reliability of the product. To assess the reliability of a product with 2.2. The Low Temperature Test for the Ship Control Device multiple failure modes, this paper presents an accelerated testing in which not only the high temperatures but also the 2.2.1. Testing Procedure. The ship control device is installed low temperatures are applied. An acceleration model with in the temperature test chamber, which is shown in Figure 4. multiple failure modes based on the Arrhenius model is The working state of the device is observed by the detection given, an accelerated testing plan is designed, and a statistical equipment and shown in Figure 5.Attherequestofproducer, analysis method is developed. A numerical example shows the testing temperature is 223 K. that the proposed accelerated testing is efficient. 2.2.2. Failure Mode. It is detected that some working param- 2. The Low Temperature Tests eters of the device are zero when the ship control device is tested 180 hours at the temperature 223 K. According to the 2.1. The Low Temperature Test for the Car Control Device results of a comprehensive circuit analysis, a flip-flop in the communication circuit is out of work. 2.1.1. Testing Procedure. The function of the car control device High temperatures are widely applied in the existing istoreceivesignalsfromthecontrolpanelandoutputsignals accelerated testing methods. However, the results of the Mathematical Problems in Engineering 3

The failed flip-flop

Figure 5: The detection equipment for the ship control device. Figure 6: The failed flip-flop. above low temperature tests show that some failure modes are sensitive to the low temperatures. Obviously, if the existing high temperature accelerated testing is applied to a device with the failure modes that are sensitive to the low 𝜆 temperatures, the reliability estimation result will far deviate from the actual reliability of the device (Figure 6).

3. Basic Assumptions The accelerated testing presented in this paper is based onthe following assumptions. (1) All failure modes are independent of each other.

(2) The failure time of every failure mode is assumed to T follow the exponential distribution. (3) The failure time of every failure mode at different temperatures follows the Arrhenius model. Type-I failure mode Type-II failure mode 4. Acceleration Model Figure 7: Two types of failure mode. Arrhenius model was first used by Svante Arrhenius in his studies of the dissociation of electrolytes, but nowadays it is widely accepted as the right tool to describe the influence High temperatures are widely applied in the existing of temperature on the rates of chemical processes, as well accelerated testing methods, and the failure modes (type-I as many other physical processes such as diffusion, thermal failure mode) that are sensitive to the high temperatures and electrical. Arrhenius model can describe the relationship are induced quickly. The failure rate of type-I failure mode betweenthetemperatureandthemeanlifetimeofproducts grows as the temperature rises, which can be described by as the Arrhenius model and shown as the solid line in Figure 7. However, the latest test results of the car control device 𝐸𝑎 𝜃 (𝑇) =𝐴exp ( ), (1) and the ship control device show that some failure modes 𝑘𝑇 (type-II failure mode) are sensitive to low temperatures. The failure rate of type-II failure mode will fall down as the where 𝜃 isthemeanlifetimeofproductsand𝐴 is a constant temperature rises, which is shown as the dashed line in that depends on the product geometry, the specimen size Figure 7. and fabrication, the test method, and other factors. 𝐸𝑎 is the activation energy of the reaction, usually in electron volts. 𝑘 To a complicated device, both the type-I failure mode and −5 ∘ is Boltzmann’s constant, 8.6171 × 10 electron volts per C. 𝑇 type-II failure mode are likely to occur. When the probability is the absolute temperature in Kelvin, which is equivalent to that any failure mode occurs is statistically independent, the the centigrade temperature plus 273.16 degrees. reliability function of the product can be expressed as When the product lifetime follows the exponential distribution, the failure rate of the product is the reciprocal of the 𝑅 (𝑡) =𝑝{𝜃𝑐 >𝑡} mean lifetime. Consider =𝑝{ {𝑡 ,𝑡 ⋅⋅⋅𝑡 }>𝑡} 1 1 −𝐸𝑎 𝑏 min 1 2 𝑚 𝜆 (𝑇) = = exp ( ) =𝑎exp ( ) , (2) 𝜃 (𝑇) 𝐴 𝑘𝑇 𝑇 𝑚 =𝑝{⋂ {𝑡𝑖 >𝑡}} where 𝑎=1/𝐴,𝑏=−𝐸𝑎/𝑘. 𝑖=1 4 Mathematical Problems in Engineering

𝜆 High temperature 1 2 High temperature . . . . .

Low temperature 1 2 Low temperature . . . . .

t0 Time

T0 T Figure 9: Test profile sketch map. Figure 8: The failure rates of multiple failure modes at different temperatures.

5.2. Statistical Analysis. The failure time is assumed to follow 𝑚 the exponential distribution when the temperature is a constant. The distribution function of exponential distribution is = ∏𝑝{𝑡𝑖 >𝑡} 𝑖=1 expressed by

𝑚 = ∏𝑅 𝑡 , 𝑖 ( ) 𝐹 (𝑡) =1−exp (−𝜆𝑡) . (5) 𝑖=1 (3) The failure density function and the reliability function where 𝜃𝑐 is the lifetime of the product with multiple failure are modes and 𝑡1,𝑡2 ⋅⋅⋅𝑡𝑚 are the failure times of the 𝑚 failure 𝑅 (𝑡) 𝑖 modes, respectively. 𝑖 is the reliability function of the th 𝑓 (𝑡) =𝜆exp (−𝜆𝑡) , failure mode. (6) According to (3), the product with multiple failure modes 𝑅 (𝑡) = exp (−𝜆𝑡) . can be considered as a series system including 𝑚 units. The failure rates of the product at different temperatures can be expressed as Assuming that there are 𝑚 failure modes occurring in the test, the 𝑘th failure mode is observedin 𝑠𝑘 samples at 𝑡(𝑘),𝑡(𝑘) ⋅⋅⋅𝑡(𝑘) 𝑘 𝜆 (𝑇) =𝜆ℎ (𝑇) +𝜆𝑙 (𝑇) =𝜆1 (𝑇) +𝜆2 (𝑇) +⋅⋅⋅+𝜆𝑚 (𝑇) times 1 2 𝑠𝑘 . Then, the probability that the th failure 𝑡(𝑘) 𝑏 𝑏 𝑏 modeis observed in a sample at 𝑗 is =𝑎 ( 1 )+𝑎 ( 2 )+⋅⋅⋅+𝑎 ( 𝑚 ). 1 exp 𝑇 2 exp 𝑇 𝑚 exp 𝑇 (4) 𝑚,𝑖=𝑗̸ (𝑘) (𝑘) (𝑖) (𝑘) 𝑝𝑗𝑘 =𝑓 (𝑡𝑗 )[∏ 𝑅 (𝑡𝑗 )]. (7) To a complicated product with multiple failure modes, 𝑖=1 the type-I failure modes are dominating when the product is under high temperature conditions. Contrarily, the type- The probability density function of the failure times of all II failure modes are primary when the product is under low samples can be represented by temperature conditions. Therefore, the failure rates of the device with multiple failure modes at different temperatures canberepresentedinFigure 8. 𝑚 𝑠𝑘 𝑚,𝑖=𝑗̸ [ (𝑘) (𝑘) (𝑖) (𝑘) 𝐿=∏ ∏ [𝑓 (𝑡𝑗 )[∏ 𝑅 (𝑡𝑗 )]] 5. Accelerated Testing 𝑘=1 [𝑗=1 𝑖=1 (8) 5.1. Design of Accelerated Testing Plan. To estimate the 𝑠 (𝑘) 0 ] parameters of the Arrhenius model, there should be no less ×[𝑅 (𝑡0)] , than four levels of temperatures in the accelerated testing plan ] (the high temperature levels are no less than two and the low temperature levels are no less than two). The type-I censoring 𝑚 (𝑘) (𝑘) 𝑠0 testing plan is applied at every temperature level and 𝑡0 is the where ∏𝑘=1[𝑅 (𝑡0 )] is the probability that there are 𝑠0 censoring time. The test profile is shown as in Figure 9. samples with no failure modes before the censoring time 𝑡0. Mathematical Problems in Engineering 5

Table 1: Failure times of devices. Failure time/h Temp. Failure 1 Failure 2 Failure 3 Failure 4 373K 35,43,53,62,73,89 85,96,115,127 Nosample Nosample 353K 76,83,93,102, 162,178,182 Nosample Nosample 213K Nosample Nosample 106,123,135,139 173,193 193K Nosample Nosample 18,29,37,45,53,69 78,89,95,97

Table 2: Failure rates of different failure modes at different temperature levels.

Failure rate Temp. Failure 1 Failure 2 Failure 3 Failure 4 373 K 7.712𝑒 − 003 5.141𝑒 −003 353 K 2.710𝑒 − 003 2.033𝑒 −003 213 K 2.378𝑒 − 003 1.189𝑒 −003 193 K 9.836𝑒 − 003 6.557𝑒 −003

Equation (8) is the likelihood function of all observations. Obviously, ln 𝜆 and 1/𝑇 are in linear relation. The esti- Taking the logarithm on both sides, we have mations of Arrhenius model parameters can be obtained by utilizing the least squares method. 𝐿 ln Furthermore, the failure rates of a product with multiple failure modes at different temperatures can be acquired by 𝑚 𝑠𝑘 𝑚,𝑖=𝑗̸ [ [ (𝑘) (𝑘) (𝑖) (𝑘) substituting the estimations of Arrhenius model parameters = ln ∏ ∏ [𝑓 (𝑡𝑗 )[∏ 𝑅 (𝑡𝑗 )]] 𝑘=1 𝑗=1 𝑖=1 of every failure mode into (4). [ [ In the existing accelerated testing under temperature conditions, the working temperature is usually supposed 𝑠 ∘ (𝑘) 0 ]] to be a constant (such as 20 C). However, the working ×[𝑅 (𝑡0)] ]] temperatures of the majority of devices are mutative, which can be represented as

𝑚 𝑠𝑘 [ (𝑘) 𝑇=𝑓(𝑡) . = ∑ ∑ [ ln (𝜆𝑘 exp (−𝜆𝑘𝑡𝑗 )) (12) 𝑘=1 [𝑗=1 Therefore, the failure rate of product under working envi- 𝑚,𝑖=𝑗̸ (𝑘) (9) ronment will also change on time, which can be expressed as +[ ∑ ln (exp (−𝜆𝑖𝑡𝑗 ))]] 𝑖=1 𝜆 (𝑇) =𝑔(𝑇) =𝑔(𝑓(𝑡)). (13)

𝑠0 ] + ln [(exp (−𝜆𝑘𝑡0))] Finally, the reliability function of device under working ] conditionscanbeobtainedby

𝑡 𝑡 𝑚 𝑠𝑘 −∫ 𝜆(𝑇)𝑑𝑡 −∫ 𝑔(𝑓(𝑡))𝑑𝑡 −𝐸(𝜆)𝑡 [ [[ 𝑘 ] ] 𝑅 (𝑡) =𝑒 0 =𝑒 0 =𝑒 . (14) = ∑ 𝑠𝑘 ln 𝜆𝑘 −𝜆𝑘 ∑ [𝑡𝑗 ] +𝑠0𝑡0 𝑘=1 [ [[𝑗=1 ] ] 6. Illustrative Example 𝑠𝑘 𝑚,𝑖=𝑗̸ 𝑘 ] +∑ [[ ∑ (−𝜆𝑖𝑡𝑗 )]] . To apply the proposed accelerated testing to products, a large 𝑗=1 𝑖=1 ] number of samples will be needed. However, there were insufficient data relating to the car and ship control devices, The estimation of 𝜆𝑘 is obtained by maximizing the so instead we present a numerical study where parameter function (9). Consider valuesaremotivatedbythesesystems.Theproductsaretested 𝑠 at four temperature levels that are 373K, 353K, 213K, and 𝜆̂ = 𝑘 . 𝑘 𝑚 𝑠 (𝑘) 193K.Thesamplesizeateverytemperaturelevelistenand [∑ ∑ 𝑘 𝑡 ]+𝑠 𝑡 (10) 𝑘=1 𝑗=1 𝑗 𝑜 𝑜 the censoring time is two hundred hours. The failure times of different samples are shown in Table 1. Equation (2) can be rewritten in logarithmic form, According to (10), the failure rates of different failure 1 modes at different temperatures are estimated and shown in 𝜆= 𝑎+𝑏 . ln ln 𝑇 (11) Table 2. 6 Mathematical Problems in Engineering

Table 3: Arrhenius model parameters of different failure modes.

Failure 1 Failure 2 Failure 3 Failure 4

Parameters 𝑎1 = exp(13.5939) 𝑎2 = exp(11.1040) 𝑎3 = exp(−19.7425) 𝑎4 = exp(−23.2113) estimation 𝑏1 = −6.8852𝑒 + 003 𝑏2 = −6.1077𝑒 + 003 𝑏3 = 2.9183𝑒 + 003 𝑏4 = 3.5095𝑒 + 003

×10−3 0.01 6 0.008 5 0.006 4

3 0.004 Failure rate Failure

Failure rate Failure 2 0.002 1 0 0 200 220 240 260 280 300 320 340 360 200 220 240 260 280 300 320 340 360 Temperature (K) Temperature (K) Figure 11: Failure rates of the product at different temperatures. Failure 1 Failure 3 Failure 2 Failure 4

Figure 10: Failure rates of different failure modes at different of type-I failure modes and type-II failure modes can be temperatures. obtained by the proposed accelerated testing. To the existing accelerated testing, only the high temperatures are applied. The type-I failure modes (failure mode 1 and failure mode According to (11), the estimations of Arrhenius model 2) are induced quickly, but the type-II failure modes rarely parameters are obtained by utilizing the least squares method occur. Therefore, the failure rate that is estimated by the and represented in Table 3. existing high temperature accelerated testing is The failure rates of different failure modes at different 𝜆 =𝑔 𝑇 =𝜆 +𝜆 . temperatures are calculated by replacing the parameters in ℎ ℎ ( ) 1 2 (16) (2) with the estimations that are shown in Table 3.Thefailure rates of different failure modes at different temperatures are The estimation of failure rate based on the high temper- shown in Figure 10. ature accelerated testing is inaccurate due to the neglect of Furthermore, the failure rates of device with multiple type-II failure modes, and the relative error is defined as failure modes at different temperatures are acquired by4 ( ). Consider 𝜆−𝜆ℎ 𝜆3 +𝜆4 𝑒= × 100% = × 100%. (17) 𝜆 𝜆1 +𝜆2 +𝜆3 +𝜆4 𝜆=𝑔(𝑇) =𝜆ℎ +𝜆𝑙 =𝜆1 +𝜆2 +𝜆3 +𝜆4 −6.8852𝑒 + 003 Figure 12 plots the relative error at different temperatures. = (13.5939) × ( ) exp exp 𝑇 The curve denotes that the relative error is small under high temperature conditions, while it is large under low −6.1077𝑒 + 003 + exp (11.1040) × exp ( ) temperature conditions. The relative error is less than 10% 𝑇 (15) when the temperatures are higher than 310 K and it is more 2.9182𝑒 + 003 than 90% when the temperatures are lower than 273 K. + exp (−19.7425) × exp ( ) In the existing accelerated testing, the working temper- 𝑇 ∘ ature is usually supposed to be a constant (such as 20 C). 3.5095𝑒 + 003 However, the working temperatures of the majority of devices + exp (−23.2113) × exp ( ). 𝑇 are mutative. Figure 13 shows the working temperature of the product. Figure 11 plots the failure rates of product at different tem- The expression of the working temperature in mathemat- peratures based on (15). The curve denotes the fact that the ics is product is more fail under high temperature environments or low temperature environments. The point indicates the 243 0 ≤ 𝑡 < 4, 20 ≤ 𝑡 < 24, (1.5037𝑒 − 004) { minimal failure rate when the product is at {243 + 20𝑡 4 ≤ 𝑡 < 8, the temperature of 282 K. 𝑇=𝑓(𝑡) = {323 8 ≤ 𝑡 < 16, (18) To a complicated device, both of the type-I failure mode { and the type-II failure mode are likely to occur. The data {323 − 20𝑡 16 ≤ 𝑡 < 20. Mathematical Problems in Engineering 7

100 1

80 0.8

60 0.6 R(t) 40 0.4 Relative error (%) error Relative 20 0.2

0 240 260 280 300 320 340 0 0 500 1000 1500 2000 2500 3000 Temperature (K) Time (h) Figure 12: Relative error of high temperature accelerated testing. R(t)

Rh(t)

323 Figure 14: Reliability functions obtained by different accelerated testing methods.

Figure 14, while the reliability function attained through the Temperature high temperature testing method is shown as the dotted 243 line in Figure 14. The reliability life 𝜃0.6 assessed by the proposed method is 851 hours, while it is assessed by the 0 4 8 12162024high temperature testing method is 1521 hour. Obviously, the reliability life is overrated by the high temperature testing Time (h) method due to the neglect of type-II failure modes. Figure 13: The working temperature. 7. Conclusion The mean failure rate of all failure modes is calculated by By focusing on the reliability assessment of a product with (15)and(18). Consider multiple failure modes, this paper presents an accelerated

𝑡 24 testing in which not only the high temperatures but also the ∫ 𝑔(𝑓(𝑡))𝑑𝑡 ∫ 𝑔(𝑓(𝑡))𝑑𝑡 low temperatures are applied. An acceleration model with 𝐸 (𝜆) = 0 = 0 = 6.0000𝑒 − 004. 𝑡 24 multiple failures based on the Arrhenius model is given. The (19) corresponding accelerated testing plan is designed, and the statistical analysis method is developed. The mean failure rate of type-I failure modes is calculated Thefailurerateofacomplicateddevicewithboththe by (16)and(18): type-I failure modes and the type-II failure modes can be obtained by the proposed accelerated testing (as shown in 𝑡 24 ∫ 𝑔 (𝑓 (𝑡))𝑑𝑡 ∫ 𝑔 (𝑓 (𝑡))𝑑𝑡 Figure 11). The device is more fragile under high temperature 𝐸(𝜆 )= 0 ℎ = 0 ℎ ℎ 𝑡 24 conditions or low temperature conditions than the room (20) temperature. The failure rate is minimal (1.5037𝑒−004) when = 3.3568𝑒 − 004. the product is in the favourable temperature (282 K). For the existing accelerated testing, only the high tem- According to the mean failure rate of all failure modes, the peratures are applied. The reliability estimation of the high reliability function of the product obtained by the proposed temperature accelerated testing is inaccurate due to the accelerated testing is neglect of type-II failure modes (the relative error is shown in Figure 12). In comparison to the high temperature accelerated 𝑅 (𝑡) = exp (−6.0000𝑒 − 004𝑡) . (21) testing,theproposedacceleratedtestingcouldinduceboth the type-I and type-II failure modes, and the reliability According to the mean failure rate of type-I failure assessed by the proposed accelerated testing is more close to modes, the reliability function obtained by the existing high the actual reliability of the product. temperature accelerated testing is

𝑅ℎ (𝑡) = exp (−3.3568𝑒 − 004𝑡) . (22) Conflict of Interests The reliability function obtained by the accelerated test- The authors declare that there is no conflict of interests ingproposedinthispaperisshownasthesolidlinein regarding the publication of this paper. 8 Mathematical Problems in Engineering

Acknowledgment [18] J. P. Klein and A. P. Basu, “Weibull accelerated life tests when there are competing causes of failure,” Communications in This study was supported by the National Natural Science Statistics A: Theory and Methods, vol. 10, no. 20, pp. 2073–2100, Foundation of China (50905181). 1981. [19] J. P.Klein and A. P.Basu, “Accelerated life testing under compet- References ing exponential failure distributions,” IAPQR Transactions,vol. 7,no.1,pp.1–20,1982. [1] M. D. Turner, “Apractical application of quantitative accelerated [20] F. Pascual, “Accelerated life test planning with independent life testing in power systems engineering,” IEEE Transactions on Weibull competing risks,” IEEE Transactions on Reliability,vol. Reliability,vol.59,no.1,pp.91–101,2010. 57, no. 3, pp. 435–444, 2008. [2] G. B. Yang, “Accelerated life test plans for predicting warranty [21] F. Pascual, “Accelerated life test planning with independent cost,” IEEE Transactions on Reliability,vol.59,no.4,pp.628– lognormal competing risks,” Journal of Statistical Planning and 634, 2010. Inference, vol. 140, no. 4, pp. 1089–1100, 2010. [3] W.Nelson, Accelerated Testing: Statistical Models, Test Plans and [22] X. Liu and W. S. Qiu, “Modeling and planning of step-stress Data Analysis, John Wiley & Sons, New York, NY, USA, 1990. accelerated life tests with independent competing risks,” IEEE [4] J. Tao, Z. Yu, Z. Ren, and X. Yi, “Study of an adaptive accelerated Transactions on Reliability,vol.60,no.4,pp.712–720,2011. model and a data transfer method based on a reliability [23] L. Xing, C. Wang, and G. Levitin, “Competing failure analysis enhancement test,” Eksploatacja i Niezawodno´s´c,vol.16,no.1, in non-repairable binary systems subject to functional depen- pp.128–132,2014. dence,” Journal of Risk and Reliability,vol.226,no.4,pp.406– [5] E. M. Benavides, “Reliability model for step-stress and variable- 416, 2012. stress situations,” IEEE Transactions on Reliability,vol.60,no.1, pp.219–233,2011. [6] J. R. van Dorp and T. A. Mazzuchi, “Ageneral Bayes exponential inference model for accelerated life testing,” Journal of Statistical Planning and Inference,vol.119,no.1,pp.55–74,2004. [7]I.H.KhamisandJ.J.Higgins,“Anewmodelforstep-stress testing,” IEEE Transactions on Reliability,vol.47,no.2,pp.131– 134, 1998. [8]L.C.Tang,Y.S.Sun,T.N.Goh,andH.L.Ong,“Analysis of step-stress accelerated-life-test data: a new approach,” IEEE Transactions on Reliability,vol.45,no.1,pp.69–74,1996. [9] C. Xiong, “Inferences on a simple step-stress model with type- II censored exponential data,” IEEE Transactions on Reliability, vol.47,no.2,pp.142–146,1998. [10] N. Fard and C. Li, “Optimal simple step stress accelerated life test design for reliability prediction,” Journal of Statistical Planning and Inference,vol.139,no.5,pp.1799–1808,2009. [11] N. Fard and C. Li, “Optimum bivariate step-stress accelerated life test for censored data,” IEEE Transactions on Reliability,vol. 56,no.1,pp.77–84,2007. [12] N. Balakrishnan and D. Han, “Optimal step-stress testing for progressively Type-I censored data from exponential distribution,” Journal of Statistical Planning and Inference,vol.139,no. 5, pp. 1782–1798, 2009. [13] N. Balakrishnan, D. Kundu, H. K. T. Ng, and N. Kannan, “Point and interval estimation for a simple step-stress model with type- II censoring,” Journal of Quality Technology,vol.39,no.1,pp. 35–47, 2007. [14] N. Balakrishnan, L. Zhang, and Q. Xie, “Inference for a simple step-stress model with type-I censoring and lognormally distributed lifetimes,” Communications in Statistics: Theory and Methods, vol. 38, no. 8–10, pp. 1690–1709, 2009. [15] C. M. Kim and D. S. Bai, “Analyses of accelerated life test data under two failure modes,” International Journal of Reliability, Quality and Safety Engineering,vol.9,no.2,pp.111–125,2002. [16] R. V. Craiu and T. C. M. Lee, “Model selection for the competing-risks model with and without masking,” Technomet- rics,vol.47,no.4,pp.457–467,2005. [17] J. I. McCool, “Competing risk and multiple comparison analysis for bearing fatigue tests,” ASLE Transactions,vol.21,no.4,pp. 271–284, 1978. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 142169, 8 pages http://dx.doi.org/10.1155/2014/142169

Research Article Reliability Analysis of the Proportional Mean Residual Life Order

M. Kayid,1,2 S. Izadkhah,3 and H. Alhalees1

1 Department of Statistics and Operations Research, College of Science, King Saud University, Riyadh 11451, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Suez University, Suez 41522, Egypt 3 School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad 91779, Iran

Correspondence should be addressed to M. Kayid; el [email protected]

Received 4 April 2014; Accepted 2 August 2014; Published 28 August 2014

Academic Editor: Shaomin Wu

Copyright © 2014 M. Kayid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The concept of mean residual life plays an important role in reliability and life testing. In this paper, we introduce and study anew stochastic order called proportional mean residual life order. Several characterizations and preservation properties of the new order under some reliability operations are discussed. As a consequence, a new class of life distributions is introduced on the basis of the anti-star-shaped property of the mean residual life function. We study some reliability properties and some characterizations of this class and provide some examples of interest in reliability.

1. Introduction failed. Another useful reliability measure is the hazard rate (HR) function of 𝑋 which is given by Stochastic orders have shown that they are very useful in applied probability, statistics, reliability, operation research, 𝑓 (𝑡) 𝑟𝑋 (𝑡) = ,𝑡≥0. (2) economics, and other related fields. Various types of stochas- 𝐹 (𝑡) tic orders and associated properties have been developed rapidly over the years. Let 𝑋 be a nonnegative random vari- The HR function is particularly useful in determining the able which denotes the lifetime of a system with distribution appropriate failure distributions utilizing qualitative information about the mechanism of failure and for describing the function 𝐹, survival function 𝐹=1−𝐹, and density function way in which the chance of experiencing the event changes 𝑓. The conditional random variable 𝑋𝑡 =(𝑋−𝑡|𝑋>𝑡),𝑡≥ with time. In replacement and repair strategies, although 0, is known as the residual life of the system after 𝑡 given that the shape of the HR function plays an important role, the it has already survived up to 𝑡. The mean residual life (MRL) MRL function is found to be more relevant than the HR function of 𝑋 is the expectation of 𝑋𝑡,whichisgivenby function because the former summarizes the entire residual life function whereas the latter involves only the risk of ∞ 𝐹 (𝑢) instantaneous failure at some time 𝑡. For an exhaustive mono- {∫ 𝑑𝑢, 𝑡 >0, 𝜇 (𝑡) = graph on the MRL and HR functions and their reliability 𝑋 { 𝑡 𝐹 (𝑡) (1) 0, 𝑡 ≤ 0. analysis, we refer the readers to Ramos-Romero and Sordo- { D´ıaz [1], Belzunce et al. [2], and Lai and Xie [3]. Based on the MRL function, a well-known MRL order has been introduced The MRL function is an important characteristic in andstudiedintheliterature.GuptaandKirmani[4]and various fields such as reliability engineering, survival analysis, Alzaid [5] were among the first who proposed the MRL order. and actuarial studies. It has been extensively studied in the Over the years, many authors have investigated reliability literature especially for binary systems, that is, when there are properties and applications of the MRL order in reliability only two possible states for the system as either working or and survival analysis (cf. Shaked and Shanthikumar [6]and 2 Mathematical Problems in Engineering

𝑋≤ 𝑌) Muller¨ and Stoyan [7]). On the other hand, the proportional (i) HR order (denoted as HR if stochastic orders are considered in the literature to generalize some existing notions of stochastic comparisons of random 𝐺 (𝑡) + is increasing in 𝑡∈R , (3) variables. Proportional stochastic orders as extended versions 𝐹 (𝑡) of the existing common stochastic orders in the literature were studied by some researchers such as Ramos-Romero and (ii) reversed hazard (RH) order (denoted as 𝑋≤ 𝑌) if Sordo-D´ıaz [1]andBelzunceetal.[2]. Recently, Nanda et al. RH [8] gave an effective review of the different partial ordering 𝑓 (𝑥) 𝑔 (𝑥) ≤ ,∀𝑥∈R+, results related to the MRL order and studied some reliability 𝐹 (𝑥) 𝐺 (𝑥) (4) models in terms of the MRL function. The purpose of this paper is to propose a new stochastic which denotes the reversed hazard (RH) rate order, order called proportional mean residual life (PMRL) order 𝑋≤ 𝑌 which extends the MRL order to a more general setting. (iii) MRL order (denoted as MRL )if Some implications, characterization properties, and preser- ∞ ∫ 𝐺 (𝑢) 𝑑𝑢 vation results under weighted distributions of this new order 𝑡 + ∞ is increasing in 𝑡∈R . (5) including its relationships with other well-known orders are ∫ 𝐹 (𝑢) 𝑑𝑢 derived. In addition, two characterizations of this order based 𝑡 on residual life at random time and the excess lifetime in renewal processes are obtained. As a consequence, a new Definition 2 (Lai and Xie [3]). The nonnegative random class of lifetime distributions, namely, anti-star-shaped mean variable 𝑋 is said to have a decreasing mean residual life residual life (ASMRL) class of life distribution, which is (DMRL) whenever the MRL of 𝑋 is decreasing. closely related to the concept of the PMRL order, is introduced and studied. A number of useful implications, char- Definition 3 (Lariviere and Porteus [9]). The nonnegative 𝑋 acterizations, and examples for this class of life distributions random variable is said to have an increasing generalized are discussed along with some reliability applications. The failure rate (IGFR) whenever the generalized failure rate 𝛽 𝑋 𝛽 (𝑥) = 𝑥𝑟 (𝑥) paper is organized as follows. The precise definitions of some function 𝑋 of which is given by 𝑋 𝑋 is 𝑥≥0 stochastic orders as well as some classes of life distributions increasing in . 𝑋 whichwillbeusedinthesequelaregiveninSection 2.In Note that, in view of a result in Lariviere [10], has 𝑥𝑋≤ 𝑋 𝑥 ∈ (0, 1] that section, the PMRL order is introduced and studied. IGFR property if and only if HR ,forall ,or 𝑋≤ 𝑥𝑋 𝑥≥1 Several characterizations and preservation properties of this equivalently if HR ,forany . new order under some reliability operations are discussed. Definition 4 (Karlin [11]). A nonnegative measurable func- In addition, to illustrate the concepts, some applications in tion ℎ(𝑥, 𝑦) is said to be totally positive of order 2 (TP2)in the context of reliability theory are included. In Section 3,the 𝑥∈R and 𝑦∈R,whenever ASMRL class of life distributions is introduced and studied. Finally, in Section 4, we give a brief conclusion and some 󵄨 󵄨 󵄨ℎ(𝑥1,𝑦1)ℎ(𝑥1,𝑦2)󵄨 remarks of the current research and its future. 󵄨 󵄨 ≥0, ∀𝑥1 ≤𝑥2,𝑦1 ≤𝑦2. (6) 󵄨ℎ(𝑥2,𝑦1)(𝑥2,𝑦2) 󵄨 Throughout this paper, the term increasing is used instead of monotone nondecreasing and the term decreasing is used Definition 5 (Shaked and Shanthikumar [6]). A nonnegative + instead of monotone nonincreasing. Let us consider two function 𝜓 is said to be anti-star-shaped on a set 𝐴⊆R random variables 𝑋 and 𝑌 having distribution functions if 𝜓(𝛼𝑥) ≥ ,forall𝛼𝜓(𝑥) 𝑥 on 𝐴 and for every 𝛼∈ 𝐹 and 𝐺, respectively, and denote by 𝐹(𝑓) and 𝐺(𝑔) their [0, 1].Equivalently,𝜓 is anti-star-shaped on 𝐴 if 𝜓(𝑥)/𝑥 is respective survival (density) functions. We also assume that nonincreasing in 𝑥∈𝐴. all random variables under consideration are absolutely Below, we present the definition of the proportional continuous and have 0 as the common left endpoint of their hazard rate (PHR) order and its related proportional aging supports, and all expectations are implicitly assumed to be class. finite whenever they appear. In addition, we use the notations + 𝑠𝑡 𝑋 𝑌 R =(−∞,∞),R =(0,∞),= denotes the equality in Definition 6 (see Belzunce et al. [2]). Let and be two nonnegative continuous random variables. It is said that distribution, and 𝑋V is the weighted version of 𝑋 according V to the weight . (i) 𝑋 is smaller than 𝑌 in the PHR order (denoted as 𝑋≤ 𝑌 𝑥𝑋≤ 𝑌 𝑥 ∈ (0, 1] P-HR )if HR ,forall , 2. Proportional Mean Residual Life Order (ii) 𝑋 is increasing proportional hazard rate (IPHR), if 𝑥𝑋≤ 𝑋 𝑥 ∈ (0, 1] HR ,forall . For ease of reference, before stating our main results, let us recall some stochastic orders, classes of life distributions, and Consider the situation wherein 𝑋 denotes the risk that dependence concepts which will be used in the sequel. the direct insurer faces and 𝜙 the corresponding reinsurance contract. One important reinsurance agreement is quota- Definition 1. The random variable 𝑋 is said to be smaller than share treaty defined as 𝜙(𝑋) =,for 𝑥𝑋 𝑥 ∈ (0, 1].Therandom 𝑌 in the variable 𝑌 denotes the risk that an independent insurer Mathematical Problems in Engineering 3 faces. Insurers sometimes seek a quota-share treaty when function equal to the reciprocal of the HR function. Some they require financial support from their reinsurers, thus of the well-known and important distributions in statistics maintaining an adequate relation between net income and and applied probability may be expressed as weighted dis- capital reserves. Motivated by this, we propose the following tributions such as truncated distributions, the equilibrium newstochasticorder. renewal distribution, distributions of order statistics, and distributions arisen in proportional hazards and proportional Definition 7. Let 𝑋 and 𝑌 be two nonnegative random reversed hazards models. Recently, Izadkhah et al. [17]have variables. The random variable 𝑋 is smaller than 𝑌 in the considered the preservation property of the MRL order 𝑋≤ 𝑌 𝑥𝑋≤ 𝑌 PMRL order (denoted as P-MRL ), if MRL ,forall under weighted distributions. Here we develop a similar 𝑥 ∈ (0, 1]. preservation property for the PMRL order under weighted distributions. For two weight functions 𝑤1 and 𝑤2, assume 𝑋≤ 𝑌 ⇔ 𝑥𝑋≤ 𝑌 𝑥∈ Remark 8. Note that P-MRL MRL ,forall that 𝑋𝑤 and 𝑌𝑤 denote the weighted versions of the random (0, 1] 𝑥=1 𝑋≤ 𝑌 1 2 ,thus;with we have MRL ,thatis;thePMRL variables of 𝑋 and 𝑌, respectively, with respective density order is stronger than the MRL order. functions The first results of this section provide an equivalent condition for the PMRL order. 𝑤1 (𝑥) 𝑓 (𝑥) 𝑤2 (𝑥) 𝑔 (𝑥) 𝑓1 (𝑥) = ,𝑔1 (𝑥) = , (9) ]1 ]2 Theorem 9. The following assertions are equivalent: where 0<]1 =𝐸(𝑤1(𝑋)) < ∞ and 0<]2 =𝐸(𝑤2(𝑌)) <. ∞ 𝑋≤ 𝑌 (i) P-MRL ; Let 𝐵1(𝑥) = 𝐸(𝑤1(𝑋) | 𝑋 > 𝑥) and 𝐵2(𝑥) = 𝐸(𝑤2(𝑌) | 𝑌 > 𝑥𝜇 (𝑡) ≤ 𝜇 (𝑥𝑡) 𝑡≥0 𝑥 ∈ (0, 1] 𝑥) 𝑋 𝑌 (ii) 𝑋 𝑌 ,forall ,andeach ; .Thensurvivalfunctionsof 𝑤1 and 𝑤2 are, respectively, ∞ ∞ ∫ 𝐺(𝑢)𝑑𝑢/ ∫ 𝐹(𝑢)𝑑𝑢 𝑡 𝑥∈ given by (iii) 𝑥𝑡 𝑡 is increasing in ,forall (0, 1]. 𝐵 (𝑥) 𝐹 (𝑥) 𝐵 (𝑥) 𝐺 (𝑥) 𝐹 (𝑥) = 1 , 𝐺 (𝑥) = 2 . (10) 1 ] 1 ] Proof. First, we prove that (i) and (ii) are equivalent. Note that 1 2 the MRL of 𝑥𝑋 as a function of 𝑠 is given by 𝑥𝜇𝑋(𝑠/𝑥),forall First, we consider the following useful lemma which is 𝑠≥0and for any 𝑥 ∈ (0, 1].Now,wehave𝑋≤ 𝑌 if, for P-MRL straightforward and hence the proof is omitted. all 𝑥 ∈ (0, 1],itholdsthat Lemma 10. 𝑋 𝑥𝑋≤ 𝑌 Let be a nonnegative absolutely continuous MRL random variable. Then, for any weight function 𝑤1, 𝑠 ⇐⇒ 𝑥 𝜇 𝑋 ( )≤𝜇𝑌 (𝑠) ,∀𝑠≥0, 𝑠𝑡 𝑥 (𝑥𝑋)V =𝑥𝑋𝑤 , ∀𝑥>0, (11) (7) 1 ⇐⇒ 𝑥 𝜇 𝑋 (𝑡) ≤𝜇𝑌 (𝑥𝑡) ,∀𝑡≥0, where V is a weight function of the form V(𝑡) =1 𝑤 (𝑡/𝑥). 𝑠 𝑡= . Theorem 11. 𝐵 where 𝑥 Let 2 be an increasing function and let 𝐵2(𝑠𝑥)/𝐵1(𝑠) increase in 𝑠≥0,forall𝑥 ∈ (0, 1].Then To prove that (ii) and (iii) are equivalent, we have 𝑋≤ 𝑌󳨐⇒𝑋 ≤ 𝑌 . P-MRL 𝑤1 P-MRL 𝑤2 (12) ∞ 𝑑 ∫ 𝐺 (𝑢) 𝑑𝑢 𝑥𝑡 𝑥 ∈ (0, 1] 𝑋≤ 𝑌 ∞ Proof. Let be fixed. Then, P-MRL gives 𝑑𝑡 ∫ 𝐹 (𝑢) 𝑑𝑢 𝑥𝑋≤ 𝑌 𝐵 𝑡 MRL . We know by assumption that 2 is increasing and ∞ ∞ the ratio 𝐹 (𝑡) ∫ 𝐺 (𝑢) 𝑑𝑢 −𝑥𝐺 (𝑥𝑡) ∫ 𝐹 (𝑢) 𝑑𝑢 = 𝑥𝑡 𝑡 𝐸(𝑤 (𝑌) |𝑌>𝑡) ∞ 2 (8) 2 [∫ 𝐹 (𝑢) 𝑑𝑢] 𝑡 𝐸 (V (𝑥𝑋) |𝑥𝑋>𝑡) 𝐹 (𝑡) 𝐺 (𝑥𝑡) 𝐸(𝑤 (𝑌) |𝑌>𝑡) = ×[𝜇 (𝑥𝑡) −𝑥𝜇 (𝑡)]. 2 ∞ 2 𝑌 𝑋 = (13) [∫ 𝐹(𝑢)𝑑𝑢] 𝐸(𝑤1 (𝑋) |𝑋>𝑡/𝑥) 𝑡 𝐸(𝑤 (𝑌) |𝑌>𝑠𝑥) 𝐵 (𝑠𝑥) It is obvious that the last term is nonnegative if and only if = 2 = 2 𝐸(𝑤 (𝑋) |𝑋>𝑠) 𝐵 (𝑠) 𝑥𝜇𝑋(𝑡) ≤𝑌 𝜇 (𝑥𝑡),forall𝑡≥0and for any 𝑥∈(0,1]. 1 1 In the context of reliability engineering and survival is increasing in 𝑡≥0,when𝑠=𝑡/𝑥. In view of Theorem 2 in (𝑥𝑋) ≤ 𝑌 analysis, weighted distributions are of tremendous practical Izadkhah et al. [17], we conclude that V MRL 𝑤2 .Because importance (cf. Jain et al. [12], Bartoszewicz and Skolimowska of Lemma 10 andbecausetheequalityindistributionof(𝑥𝑋)V [13], Misra et al. [14], Izadkhah and Kayid [15], and Kayid et and 𝑥𝑋𝑤 implies the equality in their MRL functions, it 1 𝑥𝑋 ≤ 𝑌 𝑥 ∈ (0, 1] al. [16]). In renewal theory the residual lifetime has a limiting follows that 𝑤1 MRL 𝑤2 .So,forall we have 𝑥𝑋 ≤ 𝑌 𝑋 ≤ 𝑌 . distribution that is a weighted distribution with the weight 𝑤1 MRL 𝑤2 which means that 𝑤1 P-MRL 𝑤2 4 Mathematical Problems in Engineering

On the other hand, in many reliability engineering prob- According to Barlow and Proschan [22], it holds for all 𝑡≥0 lems, it is interesting to study 𝑋𝑌 =[𝑋−𝑌|𝑋>𝑌],the and 𝑥≥0that residual life of 𝑋 with a random age 𝑌. The residual life at 𝑡 random time (RLRT) represents the actual working time of 𝑃(𝛾(𝑡) >𝑥)=𝐹 (𝑡+𝑥) + ∫ 𝐹 (𝑡+𝑥−𝑧) 𝑑𝑀 (𝑧) . (19) the standby unit if 𝑋 is regarded as the total random life of 0 𝑌 a warm standby unit with its age . For more details about In the literature, several results have been given to RLRT we refer the readers to Yue and Cao [18], Li and Zuo 𝑋 characterize the stochastic orders by the excess lifetime in [19], and Misra et al. [20], among others. Suppose that and a renewal process. Next, we investigate the behavior of the 𝑌 are independent. Then, the survival function of 𝑋𝑌,forany 𝑥≥0 excess lifetime of a renewal process with respect to the PMRL ,isgivenby order. ∞ ∫ 𝐹(𝑥+𝑦)𝑑𝐺(𝑦) Theorem 13. 𝑋 ≤ 𝑋 𝑡≥0 𝛾(𝑡)≤ 𝛾(0) 0 If 𝑡 P-MRL ,forall ,then P-MRL 𝑃(𝑋𝑌 >𝑥)= ∞ . (14) ∫ 𝐹(𝑦)𝑑𝐺(𝑦) for all 𝑡≥0. 0 𝑋 ≤ 𝑋 𝑡≥0 Theorem 12. Let 𝑋 and 𝑌 be two nonnegative random Proof. First note that 𝑡 P-MRL ,forall ,ifandonlyif 𝑋 ≤ 𝑋 𝑌 𝑋 for any 𝑡≥0, 𝑠>0,and𝑥 ∈ (0, 1] variables. 𝑌 P-MRL for any which is independent of ,if ∞ and only if ∞ ∫ 𝐹 (𝑢) 𝑑𝑢 ∫ 𝑥𝐹 𝑡+𝑢 𝑑𝑢 ≤ 𝐹 𝑡+𝑠 𝑠𝑥 . 𝑋 ≤ 𝑋, ∀𝑡 ≥ 0. ( ) ( ) (20) 𝑡 P-MRL (15) 𝑠 𝐹 (𝑠𝑥) 𝑋 ≤ 𝑋 𝑡≥0 Proof. To prove the “if” part, let 𝑡 P-MRL for all .It In view of the identity of (19) and the inequality in (20)we then follows that, for all 𝑠>0and 𝑥 ∈ (0, 1], can get ∞ 𝐹 (𝑡+𝑠) ∞ ∞ ∫ 𝑥𝐹 (𝑡+𝑢) 𝑑𝑢 ≤ ∫ 𝐹 (𝑢) 𝑑𝑢. (16) ∫ 𝑥𝑃 (𝛾 (𝑡) >𝑢)𝑑𝑢 𝑠 𝐹 (𝑠𝑥) 𝑠𝑥 𝑠 ∞ By integrating both sides of (16)withrespectto𝑡 through the = ∫ 𝑥𝐹 (𝑡+𝑢) 𝑑𝑢 measure 𝐺,wehave 𝑠 ∞ ∞ 𝑡 ∞ ∫ ∫ 𝑥𝐹 (𝑡+𝑢) 𝑑𝑢 𝑑𝐺 (𝑡) + ∫ ∫ 𝑥𝐹 (𝑡 − 𝑦 + 𝑢) 𝑑𝑢 𝑑𝑀(𝑦) 0 𝑠 0 𝑠 ∞ ∞ 𝐹 (𝑡+𝑠) ∞ ≤ ∫ 𝑥𝐹 (𝑡+𝑢) 𝑑𝑢 ≤ ∫ [ ∫ 𝐹 (𝑢) 𝑑𝑢] 𝑑𝐺 (𝑡) 𝑠 0 𝐹 (𝑠𝑥) 𝑠𝑥 (17) 𝑡 ∞ ∞ 𝐹(𝑡−𝑦+𝑠) ∞ ∫ 𝐹 (𝑢) 𝑑𝑢 + ∫ [ ∫ 𝐹 (𝑢) 𝑑𝑢] 𝑑𝑀 (𝑦) = ∫ 𝐹 (𝑡+𝑠) 𝑑𝐺 (𝑡) 𝑠𝑥 , 0 𝐹 (𝑠𝑥) 𝑠𝑥 0 𝐹 (𝑠𝑥) ∞ (21) ∀𝑠 ≥ 0, 𝑥 ∈ (0, 1] , = ∫ 𝑥𝐹 (𝑡+𝑢) 𝑑𝑢 𝑠 ∞ which is equivalent to saying that 𝑋𝑌≤ 𝑋,forall𝑌’s that ∫ 𝐹 (𝑢) 𝑑𝑢 P-MRL 𝑠𝑥 are independent of 𝑋. For the “only if ” part, suppose that + [𝑃 (𝛾 (𝑡) >𝑠)−𝐹 (𝑡+𝑠)] 𝑋 ≤ 𝑋 𝑌 𝐹 (𝑠𝑥) 𝑌 P-MRL holds for any nonnegative random variable . 𝑋 ≤ 𝑋 𝑡≥0 𝑌 ∞ Then 𝑡 P-MRL ,forall ,followsbytaking as a ∫ 𝐹 (𝑢) 𝑑𝑥 degenerate random variable. ≤ 𝑠𝑥 [𝐹 (𝑡+𝑠)] 𝐹 (𝑠𝑥) {𝑋𝑛, 𝑛 = 1,2,...} Let be a sequence of mutually ∞ ∫ 𝐹 (𝑢) 𝑑𝑢 independent and identically distributed (i.i.d.) nonnegative 𝑠𝑥 random variables with common distribution function 𝐹.For + [𝑃 (𝛾 (𝑡) >𝑠)−𝐹 (𝑡+𝑠)] 𝑛 𝐹 (𝑠𝑥) 𝑛≥1, denote 𝑆𝑛 =∑𝑖=1 𝑋𝑖 which is the time of the 𝑛th ∞ arrival and 𝑆0 =0,andlet𝑁(𝑡) =Sup{𝑛 :𝑛 𝑆 ≤𝑡}represent ∫ 𝐹 (𝑢) 𝑑𝑢 the number of arrivals during the interval [0, 𝑡].Then,𝑁= = 𝑠𝑥 𝑃(𝛾(𝑡) >𝑠). {𝑁(𝑡), 𝑡 ≥0} is a renewal process with underlying distribution 𝐹 (𝑠𝑥) 𝐹 (see Ross [21]). Let 𝛾(𝑡) be the excess lifetime at time 𝑡≥0; Hence, it holds that, for all 𝑡≥0, 𝑠>0and for any 𝑥 ∈ (0, 1], that is, 𝛾(𝑡)𝑁(𝑡)+1 =𝑆 −𝑡. In this context we denote the renewal ∞ ∞ function by 𝑀(𝑡) = 𝐸[𝑁(𝑡)] which satisfies the following ∫ 𝑥𝑃 (𝛾 (𝑡) >𝑢)𝑑𝑢 ∫ 𝐹 (𝑢) 𝑑𝑢 well-known fundamental renewal equation: 𝑠 ≤ 𝑠𝑥 , (22) 𝑃(𝛾(𝑡) >𝑠) 𝐹 (𝑠𝑥) 𝑡 𝑀 (𝑡) =𝐹(𝑡) + ∫ 𝐹 (𝑡 − 𝑦) 𝑑𝑀 (𝑦) ,𝑡≥0. (18) which means 𝛾(𝑡)≤ 𝛾(0) for all 𝑡≥0. 0 P-MRL Mathematical Problems in Engineering 5

𝑥𝑋≤ 𝑌 𝑥 ∈ (0, 1] 3. Anti-Star-Shaped Mean Residual Life Class Hence it holds that MRL ,forall ,whichmeans 𝑋≤ 𝑌 𝑌 P-MRL .Theproofoftheresultwhen is ASMRL is similar Statisticians and reliability analysts have shown a growing 𝑌 𝑌≤ 𝛼𝑌 by taking the fact that is ASMRL if and only if MRL , interest in modeling survival data using classifications of life 𝛼≥1 𝑋≤ 𝑌 for all , into account. Note also that P-MRL if and distributions by means of various stochastic orders. These 𝑋≤ 𝛼𝑌 𝛼≥1. only if MRL ,forany categories are useful for modeling situations, maintenance, inventory theory, and biometry. In this section, we propose The following counterexample shows that the MRL order a new class of life distributions which is related to the MRL does not generally imply the ASMRL order and hence the function. We study some characterizations, preservations, sufficient condition in Theorem 18 cannot be removed. and applications of this new class. Some examples of interest in the context of reliability engineering and survival analysis 2 Counter Example 1. Let 𝑋 have MRL 𝜇𝑋(𝑡) = (𝑡 − 1/2) ,for arealsopresented. 𝑡∈[0,∞), and let 𝑌 have MRL 𝜇𝑌(𝑡) = 1/6,for𝑡∈[0,1/6], 𝜇 (𝑡) = 3(𝑡 − 1/2)2/2 𝑡 ∈ (1/6, ∞) Definition 14. The lifetime variable 𝑋 is said to have an anti- and 𝑌 ,for . These MRL star-shaped mean residual life (ASMRL), if the MRL function functionsarereadilyshownnottobeASMRL.Wecanalso 𝜇 (𝑡) ≤ 𝜇 (𝑡) 𝑡≥0 𝑋≤ 𝑌 of 𝑋 is anti-star-shaped. see that 𝑋 𝑌 ,forall ;thatis, MRL .Itcanbe 𝑥𝑋≰ 𝑌 𝑥=1/4 It is simply derived that 𝑋∈ASMRL whenever 𝜇𝑋(𝑡)/𝑡 easily checked now that MRL ,for ,whichmeans 𝑋≰ 𝑌 is decreasing in 𝑡>0. Useful description and motivation that P-MRL . for the definition of the ASMRL class which is due to As an obvious conclusion of Theorem 18 above and Nanda et al. [8] are the following. Consider a situation in Theorem 2.9 in Nanda et al. [8], if 𝑋 is DMRL, then 𝑋 is which 𝑋 represents the risk that the direct insurer faces ASMRL. The next result presents another characterization of and 𝜙 the corresponding reinsurance contract. The ASMRL the ASMRL class. class provides that the quota-share treaty related to a risk is less than risk itself in the sense of the MRL order. In Theorem 19. A lifetime random variable 𝑋 is ASMRL if and 𝑍𝑋≤ 𝑋 𝑍 𝑆 = (0, 1] what follows, we focus on the ASMRL class as a weaker only if MRL , for each random variable with 𝑍 , class than the DMRL class to get some basic results. First, which is independent of 𝑋. consider the following characterization property which can be immediately obtained by Theorem 9(ii). Proof. Toprovethe“if”part,notethat𝜇𝑧𝑋(𝑡) = 𝑧𝜇𝑋(𝑡/𝑧), for each 𝑧 ∈ (0, 1] and any 𝑡>0.Take𝑍=𝑧,foreach𝑧∈ Theorem 15. The lifetime random variable 𝑋 is ASMRL if and (0, 1] one at a time, as a degenerate random variable implying 𝑋≤ 𝑋 𝑧𝑋≤ 𝑋 𝑧 ∈ (0, 1] 𝑋∈ only if P-MRL . MRL ,forall ,whichmeans ASMRL. For the “only if” part, assume that 𝑍 has distribution function 𝐺. Theorem 16. 𝑋 The lifetime random variable is ASMRL if and From the assumption and the well-known Fubini theorem, only if for all 𝑥>0,itfollowsthat + 𝜃1𝑋≤ 𝜃2𝑋, for any 𝜃1 ≤𝜃2 ∈ R . (23) ∞ MRL ∫ 𝑃 (𝑍𝑋 >𝑢) 𝑑𝑢 𝑋(𝜃 )=𝜃𝑋 𝑖=1,2 𝜇 (𝑥) = 𝑥 Proof. Denote 𝑖 𝑖 ,for . The MRL function of 𝑍𝑋 𝑃 (𝑍𝑋 >𝑥) 𝑋(𝜃 ) 𝜇 (𝑡) = 𝜃 𝜇 (𝑡/𝜃 ) 𝑡≥0 𝑖 is then given by 𝑋(𝜃𝑖) 𝑖 𝑋 𝑖 ,forall and 𝑖=1,2 𝜃 𝑋≤ 𝜃 𝑋 𝜃 ≤𝜃 ∈ ∞ 1 .In view of the fact that 1 MRL 2 ,forall 1 2 ∫ ∫ 𝐹 (𝑢/𝑧) 𝑑𝐺 (𝑧) 𝑑𝑢 R+ = 𝑥 0 ,ifandonlyif 1 ∫ 𝐹 (𝑥/𝑧) 𝑑𝐺 (𝑧) 𝑡 𝑡 + 0 𝜃1𝜇𝑋 ( )≤𝜃2𝜇𝑋 ( ), for any 𝜃1 ≤𝜃2 ∈ R . 𝜃 𝜃 (24) 1 ∞ 1 2 ∫ ∫ 𝐹 (𝑢/𝑧) 𝑑𝑢 𝑑𝐺 (𝑧) = 0 𝑥 𝑥=𝜃/𝜃 𝑠=𝑡/𝜃 1 By taking 1 2 and 1 the above inequality is ∫ 𝐹 (𝑥/𝑧) 𝑑𝐺 (𝑧) equivalent to saying that 𝑥𝜇𝑋(𝑠) ≤𝑋 𝜇 (𝑥𝑠),forall𝑠≥0and 0 (26) for any 𝑥 ∈ (0, 1].Thismeansthat𝑋 is ASMRL. 1 ∫ 𝑧𝑚 (𝑥/𝑧) 𝐹 (𝑥/𝑧) 𝑑𝐺 (𝑧) = 0 𝑋 1 Remark 17. The result of Theorem 16 indicates that the family ∫ 𝐹 (𝑥/𝑧) 𝑑𝐺 (𝑧) 0 of distributions 𝐹𝜃(𝑥) = 𝐹(𝑥/𝜃),,isstochastically 𝜃>0 𝜃 1 increasing in with respect to the MRL order if and only ∫ 𝑚 (𝑥) 𝐹 (𝑥/𝑧) 𝑑𝐺 (𝑧) if the distribution 𝐹 has an anti-star-shaped MRL function. ≤ 0 𝑋 𝑋≤ 𝑋 1 Another conclusion of Theorem 16 is to say that P-MRL if ∫ 𝐹 (𝑥/𝑧) 𝑑𝐺 (𝑧) 𝑋≤ 𝑥𝑋 𝑥∈[1,∞) 0 and only if MRL ,forall . =𝜇 (𝑥) . Theorem 18. 𝑋≤ 𝑌 𝑋 𝑌 𝑋 If MRL and if either or has an anti-star- shaped MRL function, then 𝑋≤ 𝑌. 𝑍𝑋≤ 𝑋. P-MRL That is, MRL Proof. Let 𝑋≤ 𝑌 and let 𝑋 be ASMRL. Then, we have MRL The following result presents a sufficient condition fora 𝑥𝑋≤ 𝑋≤ 𝑌, ∀𝑥 ∈ (0, 1] . MRL MRL (25) probabilitydistributiontobeASMRL. 6 Mathematical Problems in Engineering

Theorem 20. Let the lifetime random variable 𝑋 be IGFR. Example 22. The generalized Pareto distribution has been Then, 𝑋 is ASMRL. extensively used in reliability studies when robustness is required against heavier tailed or lighter tailed alternatives Proof. Recall that 𝑋 is IGFR if and only if 𝑥𝑟𝑋(𝑥) is increasing to an exponential distribution. Let 𝑋 have generalized Pareto in 𝑥≥0. Because of the identity distribution with survival function ∞ ∞ 𝑡𝐹 (𝑡) = ∫ 𝑢𝑓 (𝑢) 𝑑𝑢 − ∫ 𝐹 (𝑢) 𝑑𝑢 ∀𝑡 >0, (27) (1/𝑎)+1 𝑡 𝑡 𝑏 𝐹 (𝑥) = ( ) , 𝑥≥0,𝑎>0,𝑏>0. (33) we can write, for all 𝑡>0, 𝑎𝑥 +𝑏 ∞ 𝜇 (𝑡) ∫ 𝐹 (𝑢) 𝑑𝑢 𝑋 = 𝑡 𝑟 (𝑥) = (1 + 𝑎)/(𝑎𝑥 +𝑏) 𝑡 𝑡𝐹 (𝑡) The HR function is given by 𝑋 .Thus we get 𝑥𝑟𝑋(𝑥) = (1 + 𝑎)𝑥/(𝑎𝑥 +𝑏) which is increasing in 𝑥 ∞ ∫ 𝐹 (𝑢) 𝑑𝑢 for all parameter values and so Theorem 20 concludes that 𝑋 𝑡 = ∞ ∞ (28) is ASMRL. ∫ 𝑢𝑓 (𝑢) 𝑑𝑢 −∫ 𝐹 (𝑢) 𝑑𝑢 𝑡 𝑡 Example 23. Let 𝑇𝑖 be a lifetime variable having survival 1 𝐹 (𝑡) = 𝐸(𝐺(𝑍 𝑡)), 𝑡 ≥0 𝑍 = ∞ ∞ . function given by 𝑖 𝑖 ,where 𝑖 is a (∫ 𝑢𝑓 (𝑢) 𝑑𝑢/ ∫ 𝐹 (𝑢) 𝑑𝑢) −1 𝐺 𝑡 𝑡 nonnegative random variable and is the survival function 𝑌 𝑖=1,2 𝜇 (𝑡)/𝑡 𝑡>0 of a lifetime variable ,foreach . This is called scale Thus, 𝑋 is decreasing in if and only if the ratio change random effects model in Ling et al.23 [ ]. Noting the ∞ ∫ 𝑢𝑓 (𝑢) 𝑑𝑢 fact that 𝑌≤ 𝑘𝑌,forall𝑘≥1,isequivalenttosayingthat 𝑡 MRL ∞ is increasing in 𝑡>0. (29) 𝑌 is ASMRL, according to Theorem 3.10 of Ling et al. [23]if ∫ 𝐹 (𝑢) 𝑑𝑢 𝑍 ≤ 𝑍 𝑌 𝑇 ≥ 𝑇 𝑡 1 RH 2 and is ASMRL, then 1 MRL 2. Let In the context of reliability theory, shock models are ∞ of great interest. The system is assumed to have an ability 𝜌 (𝑖,) 𝑡 = ∫ 𝜙 (𝑖, 𝑢) 𝜓 (𝑢,) 𝑡 𝑑𝑢 (30) towithstandarandomnumberoftheseshocks,anditis 0 commonly assumed that the number of shocks and the as a function of 𝑖=1,2and of 𝑡>0,where interarrival times of shocks are s-independent. Let 𝑁 denote 𝑋 𝑢𝑓 (𝑢) , 𝑖=2, thenumberofshockssurvivedbythesystem,andlet 𝑗 𝜙 (𝑖, 𝑢) ={ if denote the random interarrival time between the (𝑗 − 1)th 𝐹 (𝑢) , if 𝑖=1, and 𝑗th shocks. Then the lifetime 𝑇 of the system is given (31) 𝑁 by 𝑇=∑𝑗=1 𝑋𝑗. Therefore, shock models are particular 1, if 𝑢>𝑡, 𝜓 (𝑢,) 𝑡 ={ cases of random sums. In particular, if the interarrivals are 0, 𝑢≤𝑡. if assumed to be s-independent and exponentially distributed 𝜆 Note that the ratio given in (29)isincreasingin𝑡>0 (with common parameter ), then the distribution function of 𝑇 canbewrittenas if and only if 𝜌 is TP2 in (𝑖,𝑡) ∈ {1,2} × (0,∞).From the assumption, since 𝑢𝑟𝑋(𝑢) is increasing, then 𝜙 is TP2 in ∞ −𝜆𝑡 𝑘 (𝑖, 𝑢) ∈ {1, 2} × (0, ∞).Alsoitiseasytoseethat𝜓 is TP2 in 𝑒 (𝜆𝑡) (𝑢, 𝑡) ∈ (0, ∞) × (0, ∞) 𝐻 (𝑡) = ∑ 𝑃 ,𝑡≥0, .Byapplyingthegeneralcomposition 𝑘! 𝑘 (34) theorem of Karlin [11]totheequalityof(30), the proof is 𝑘=0 complete. 𝑃 =𝑃[𝑁≤𝑘] 𝑘∈𝑁 𝑃 =1 To demonstrate the usefulness of the ASMRL class in where 𝑘 for all (and 0 ). Shock reliability engineering problems, we consider the following models of this kind, called Poisson shock models, have been examples. studied extensively. For more details, we refer to Fagiuoli and Pellerey [24], Shaked and Wong [25], Belzunce et al. [26], and Example 21. The Weibull distribution is one of the most Kayid and Izadkhah [27]. widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics In the following, we make conditions on the random of other types of distributions, based on the value of the shape number of shocks under which 𝑇 has ASMRL property. First, parameter. Let 𝑋 have the Weibull distribution with survival let us define the discrete version of the ASMRL class. function Definition 24. A discrete distribution 𝑃𝑘 is said to have 𝐹 (𝑥) = (−(𝜆𝑥)𝛼) , 𝑥 ≥ 0, 𝛼 > 0, 𝜆 >0. exp (32) discrete anti-star-shaped mean residual life (D-ASMRL) ∞ 𝛼 𝛼−1 property if ∑ 𝑃𝑗/𝑘𝑃𝑘−1 is nonincreasing in 𝑘∈𝑁. The HR function is given by 𝑟𝑋(𝑥) = 𝛼𝜆 𝑥 .Thuswe 𝑗=𝑘 𝛼 have 𝑥𝑟𝑋(𝑥) = 𝛼(𝜆𝑥) which is increasing in 𝑥>0for Theorem 25. 𝑃 ,𝑘 ∈ 𝑁 𝑇 all parameter values and hence according to Theorem 20𝑋 is If 𝑘 ,in(34) is D-ASMRL, then with 𝐻(𝑡) ASMRL. the sf as given in (34) is ASMRL. Mathematical Problems in Engineering 7

Proof. We may note that, for all 𝑡>0, Example 28. Consider 𝑛 units (not necessarily independent) with lifetimes 𝑇𝑖, 𝑖 = 1,2,...,𝑛. Suppose that the units ∞ 𝑘 1 −𝜆𝑡 (𝜆𝑡) are working in a common operating environment, which 𝑡𝐻 (𝑡) = ∑𝑒 𝑘𝑃𝑘−1, 𝜆 𝑘! is represented by a random vector Θ =(Θ1,Θ2,...,Θ𝑛), 𝑘=0 𝑇 ,𝑇 ,...,𝑇 (35) independent of 1 2 𝑛,andhasaneffectontheunits ∞ 1 ∞ (𝜆𝑡)𝑘 ∞ of the form ∫ 𝐻 (𝑢) 𝑑𝑢 = ∑𝑒−𝜆𝑡 ∑𝑃 . 𝜆 𝑘! 𝑗 𝑇 𝑡 𝑘=0 𝑗=𝑘 𝑖 𝑋𝑖 = , 𝑖=1,2,...,𝑛. (40) Θ𝑖 𝑇 Hence is ASMRL if and only if 𝑛 If Θ has support on (1, ∞) , then the components are ∑∞ 𝑒−𝜆𝑡 ((𝜆𝑡)𝑘/𝑘!)𝑃 𝑘 working in a harsh environment, and if they have support 𝑡𝐻 (𝑡) 𝑘=0 𝑘−1 𝑛 ∞ = (36) on (0, 1) , then the components are working in a gentler ∫ 𝐻 (𝑢) 𝑑𝑢 ∑∞ 𝑒−𝜆𝑡 ((𝜆𝑡)𝑘/𝑘!) ∑∞ 𝑃 𝑡 𝑘=0 𝑗=𝑘 𝑗 environment (see Ma [28]). In a harsh environment let 𝑇𝑗 ∈ ASMRL for some 𝑗=1,2,...,𝑛.Then,Theorem 19 states that, is increasing in 𝑡,orequivalentlyif 𝑍 [0, 1] 𝑍𝑇 ≤ 𝑇 for each with support on ,wemusthave 𝑗 MRL 𝑗. 𝑍=1/Θ 𝑇 /Θ ≤ 𝑇 ∞ 𝑘 Thus, by taking 𝑗,wemusthave 𝑗 𝑗 MRL 𝑗. (𝜆𝑡) −𝜆𝑡 𝑋 ≤ 𝑇 Ψ (𝑖,) 𝑡 = ∑Φ (𝑖, 𝑘) 𝑒 (𝑖,) 𝑡 , Hence, by (40)itstandsthat 𝑗 MRL 𝑗.Withasimilar 𝑘! is TP2 in (37) 𝑘=0 discussion, in a gentler environment if 𝑋𝑗 ∈ ASMRL for some 𝑗=1,2,...,𝑛,thenwemusthave𝑇𝑗≤ 𝑋𝑗. + MRL for 𝑖∈{1,2}and 𝑡∈R ,where In the following we state the preservation property of the 𝑋 ∞ ASMRL class under weighted distribution. Let have density { function 𝑓 and survival function 𝐹.Thefollowingresult {∑𝑃𝑗, if 𝑖=1, Φ (𝑖, 𝑘) = states the preservation of the ASMRL class under weighted {𝑗=𝑘 (38) distributions. The proof is quite similar to that of Theorem 11 {𝑘𝑃𝑘−1, if 𝑖=2. and hence omitted.

By the assumption, Φ(𝑖, 𝑘) is TP2 in (𝑖, 𝑘),for𝑖∈{1,2}and −𝜆𝑡 𝑘 Theorem 29. Let 𝐵 be an increasing function and let 𝑘∈𝑁.Itisalsoevidentthat𝑒 (𝜆𝑡) /𝑘! is TP2 in (𝑘, 𝑡),for + 𝐵(𝑠𝑥)/𝐵(𝑠) increase in 𝑠≥0,forall𝑥 ∈ (0, 1].Then𝑋 is 𝑘∈𝑁and 𝑡∈R . The result now follows from the general ASMRL implying that 𝑋𝑤 is ASMRL. composition theorem of Karlin [11].

Lemma 26. Let 𝑋1,𝑋2,...,𝑋𝑛 be an i.i.d. sample from 4. Conclusion 𝐹 and let 𝑌1,𝑌2,...,𝑌𝑛 be an i.i.d. sample from 𝐺.Then Due to economic consequences and safety issues, it is nec- min{𝑋1,𝑋2,...,𝑋𝑛}≤ min{𝑌1,𝑌2,...,𝑌𝑛} implies MRL essary for the industry to perform systematic studies using 𝑋𝑖≤ 𝑌𝑖, 𝑖=1,2,...,𝑛. MRL reliability concepts. There exist plenty of scenarios where a Example 27. Reliability engineers often need to work statistical comparison of reliability measures is required in with systems having elements connected in series. both reliability engineering and biomedical fields. In this Let 𝑋1,𝑋2,...,𝑋𝑛 be i.i.d. random lifetimes such that paper, we have proposed a new stochastic order based on 𝑇=min{𝑋1,𝑋2,...,𝑋𝑛} has the ASMRL property. Then, the MRL function called proportional mean residual life 𝑘𝑇≤ 𝑇 𝑘 ∈ (0, 1] (PMRL) order. The relationships of this new stochastic order according to Theorem 15, MRL ,forall .This means that with other well-known stochastic orders are discussed. It was shown that the PMRL order enjoys several reliability min {𝑘𝑋1,𝑘𝑋2,...,𝑘𝑋𝑛} properties which provide several applications in reliability (39) and survival analysis. We discussed several characterization ≤ {𝑋 ,𝑋 ,...,𝑋 }, ∀𝑘∈(0, 1] . MRL min 1 2 𝑛 and preservation properties of this new order under some reliability operations. To enhance the study, we proposed a By appealing to Lemma 26,itfollowsthat𝑘𝑋𝑖≤ 𝑋𝑖,𝑖 = MRL new class of life distributions called anti-star-shaped mean 1,2,...,𝑛,forall𝑘 ∈ (0, 1].Thatis,𝑋𝑖, 𝑖 = 1,2,...,𝑛, residual life (ASMRL) class. Several reliability properties of is ASMRL. Hence, the ASMRL property passes from the the new class as well as a number of applications in the lifetime of the series system to the lifetime of its i.i.d. context of reliability and survival analysis are included. Our components. resultsprovidenewconceptsandapplicationsinreliability, Accelerated life models relate the lifetime distribution to statistics, and risk theory. Further properties and applications the explanatory variables (stress, covariates, and regressor). of the new stochastic order and the new proposed class can This distribution can be defined by the survival, cumulative be considered in the future of this research. In particular, distribution, or probability density functions. Nevertheless, thefollowingtopicsareinterestingandstillremainasopen the sense of accelerated life models is best seen if they are problems: formulated in terms of the hazard rate function. In the following example, we state an application of Theorem 16 in (i) closure properties of the PMRL order and the ASMRL accelerated life models. class under convolution and coherent structures, 8 Mathematical Problems in Engineering

(ii) discrete version of the PMRL order and enhancing the [15] S. Izadkhah and M. Kayid, “Reliability analysis of the harmonic obtained results related to the D-ASMRL class, mean inactivity time order,” IEEE Transactions on Reliability, vol. 62, no. 2, pp. 329–337, 2013. (iii) testing exponentiality against the ASMRL class. [16] M. Kayid, I. A. Ahmad, S. Izadkhah, and A. M. Abouammoh, “Further results involving the mean time to failure order, and Conflict of Interests the decreasing mean time to failure class,” IEEE Transactions on Reliability, vol. 62, no. 3, pp. 670–678, 2013. The authors declare that there is no conflict of interests [17] S. Izadkhah, A. H. Rezaei Roknabadi, and G. R. M. Borzadaran, regarding the publication of this paper. “Aspects of the mean residual life order for weighted distributions,” Statistics,vol.48,no.4,pp.851–861,2014. 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Research Article Reliability Analysis of a Cold Standby System with Imperfect Repair and under Poisson Shocks

Yutian Chen, Xianyun Meng, and Shengqiang Chen

Department of Applied Mathematics, Yanshan University, Qinhuangdao 066001, China

Correspondence should be addressed to Yutian Chen; [email protected]

Received 9 November 2013; Revised 23 February 2014; Accepted 22 March 2014; Published 22 April 2014

Academic Editor: Shaomin Wu

Copyright © 2014 Yutian Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers the reliability analysis of a two-component cold standby system with a repairman who may have vacation. The system may fail due to intrinsic factors like aging or deteriorating, or external factors such as Poisson shocks. The arrival time of the shocks follows a Poisson process with the intensity 𝜆>0. Whenever the magnitude of a shock is larger than the prespecified threshold of the operating component, the operating component will fail. The paper assumes that the intrinsic lifetime and the repair time on the component are an extended Poisson process, the magnitude of the shock and the threshold of the operating component are nonnegative random variables, and the vacation time of the repairman obeys the general continuous probability distribution. By using the vector Markov process theory, the supplementary variable method, Laplace transform, and Tauberian theory, the paper derives a number of reliability indices: system availability, system reliability, the rate of occurrence of the system failure, and the mean time to the first failure of the system. Finally, a numerical example is given to validate the derived indices.

1. Introduction cause great impact on thermal shock stress. This phenomenon is called thermal shock. In this case, the system may Reliability has a wide range of applications in the field of failduetotheadverseenvironment. engineering and natural science; its theoretical research has The shock model, one kind of familiar models in the reli- attracted considerable attention in the reliability literature [1]. ability theory, has been extensively studied [4]. Traditionally, In practical engineering applications, a two-component cold there are three classic random shock models focused: extreme standby system model is one of the most important models shock models, cumulative shock models, and 𝛿-shock mod- [2]. A two-component cold standby system is composed of a els [5, 6].Studytheextremeshockmodels:thesystem primary component and a backup component, and the fails when an individual shock is too large. In these papers, backup component is only called upon when the primary they assume that if the magnitude of a shock exceeds the component fails. Cold standby systems are commonly used in thresholdoftheoperatingcomponent,itwillfail.Inthelast noncritical applications. They are important structures in the decades, the main interest of existing research [7]focuseson reliability engineering and have been widely applied in reality one-component system under Poisson shocks, it assumes that [3]. the damage to the system which resulted from a single shock Most of the researchers assume that the system may fail canbeaccumulated,andthesystemfailswhenthedamage due to intrinsic factors such as ageing or deterioration. How- has accumulated to a certain level. But this model does not ever, in practice, external factors can also cause the system apply to all of the problems. Wang and Zhang [8]studya failure. For example, a computer system may fail due to the shock model for a repairable system with two-type failures. It invasion of some virus or an attack from the raider, for the assumes that two kinds of shock in a sequence of random virus and the raider may arrive randomly; they are called shockswillmakethesystemfail:onebasedontheinterarrival stochastic shocks. Another example is the metal materials due time between two consecutive shocks less than a given pos- to a very rapid hot-to-cold or cold-to-hot instantaneous itive value 𝛿 and the other based on the shock magnitude of change; the internal temperature will produce big change and single shock more than a given positive value 𝛾. Under 2 Mathematical Problems in Engineering this assumption, they obtain some reliability indices of the component and a repairman with multiple vacations; besides, shock model such as the system reliability and the mean they investigate the parameters’ effect on the steady-state working time before system failure. Recently, literature [9] availability by numerical comparison and analyze the benefits considers that the operating environment of the component is of the system. random, the working component may be influenced by other Motivated by the above aspects, this paper considers a causes, they extend threshold on the component which is a two-component cold standby system, in which the operating random variable, and this assumption is more realistic. Due component may fail due to the intrinsic factors or external to the variability of the external environment, the influence of factors; besides, the repairman can take vacation. Assume various random factors impact on the reliability of system is that the intrinsic lifetime and the repair time of the com- a common phenomenon; learning and mastering system’s ponents are extended Poisson process and the arrival of the operation and its regularity have practical significance. With shock follows a Poisson process, while the vacation time of the development of science and technology, the high preci- the repairman obeys the general continuous probability dis- sion and the high reliability system research of the system tribution. The paper derives the reliability indices and a num- are taken more seriously, so it makes the shock models have erical example is given to validate the derived indices. broader applications. In the practical engineering applications, due to aging or 2. Definitions and Assumptions deterioration, the system will be older and older, and after failure the system usually cannot be repaired as good as new; Notation it seems more reasonable to assume that the operating time of In this paper, we use the following notation. thesystemafterrepairwillbecomeshorterandshorterwhile the repair time of the system after failure will become longer (1) The time interval from the completion of (𝑛 − 1)th and longer; at last, system will no longer work. The patterns of repair to the completion of the 𝑛th repair of compo- the operating and repair time can be described as geometric nent 𝑖 is called the 𝑛th cycle of component 𝑖,where𝑖= processes just like many authors have studied [10, 11]. The 1, 2; 𝑛=1,2.... geometric process has been applied in reliability analysis and (2) Time to failure that is due to the intrinsic cause is maintenance policy optimization by many authors [12–14]. called the intrinsic lifetime of component 𝑖. But the geometric process cannot be used to describe the bathtub-shaped failure process of system. The bathtub- (3)Timetofailurethatisduetotheextrinsicshocksis shapedfailurereferstothewholelifecycleoftheproduct called the shock lifetime of component 𝑖. from the input to the croak; its reliability changes present (𝑖) 𝑇𝑛 : Operating time of component 𝑖 in the 𝑛th circle certain regularity. Let the product failure rate be the product (𝑖) reliability characteristic values; it is based on the use of time 𝑋𝑛 : Intrinsic lifetime of component 𝑖 in the 𝑛th circle as the abscissa, ordinate for failure rate of a curve. The curve (𝑖) 𝜉 : Operating time of component 𝑖 in the 𝑛th under at both ends is high, middle low, looks like a bathtub, and so 𝑛 Poisson shocks is called the bathtub curve. Hence, to overcome this short- (𝑖) coming, Wu and Clements-Croome [15, 16]introduce 𝑈𝑛 : Operating time of component 𝑖 in the 𝑛th circle extended Poisson process (EPP) that is used to describe the (𝑖) (𝑖) 𝐿 : Cumulative distribution function (cdf) of 𝑋 change of the failure intensity is bathtub-shaped. 𝑛 𝑛 (𝑖) (𝑖) The existing research mostly focuses on the reliability 𝐺𝑛 :cdfof𝑈𝑛 analysis with the behaviors of the systems themselves, but the 𝐻(𝑖) 𝑌(𝑖) reliability analysis for a system with repairman vacation is less 𝑛 :cdfof 𝑛 (𝑖) studied. From the opinion of using resources rationally, the 𝑌𝑛 :Repairtimeofcomponent𝑖 in the 𝑛th circle introduction of repairman vacation makes the repairable sys- 𝑍 𝑛 tem more realistic and reasonable. This is due to the fact that : Vacation time of repairman in the th circle the mostly small and medium-sized enterprises cannot afford 𝑋̂:Magnitudeofeachshock to hire a full-time repairman. During the vacation, the repair- 𝜏𝑖:Thresholdofcomponent𝑖 man can purchase the parts or do other works to increase the system benefits. So, the repairman usually plays three 𝑁(𝑡): State of the system at time 𝑡. roles: one for caring the facility in his idle, one for repairing the failure component, and one for other works in his 2.1. Definition vacation. Under normal circumstance, the repairman needs to periodically check the status of the system in his idle. If he Definition 1 (see [20]). Given random variables 𝑋 and 𝑌,one 𝑋 𝑌 𝑋≥ 𝑌 𝑌 checks out the system fails, he needs to repair it immediately calls that is stochastically larger than , st ,or is 𝑋 𝑌≤ 𝑋 after the end of vacation; otherwise, he will leave the system stochastically smaller than , st ,when again for next vacation. Doshi [17] studies a comprehensive survey on vacation system models. Su and Shi [18]discuss 𝑃 {𝑋>𝑧} ≥𝑃{𝑌>𝑧} ∀ real 𝑧. (1) the reliability of a 𝑛-component series system in which the repairman takes multiple vacations. Yue et al. [19]study Astochasticprocess{𝑋𝑛,𝑛 = 1,2,...} is stochastically 𝑋 ≤ (≥ )𝑋 𝑛 = 1, 2, . . Gaver’s parallel repairable system attended by a cold standby increasing (decreasing) if 𝑛 st st 𝑛+1 for all . Mathematical Problems in Engineering 3

Definition 2 (see [13]). A sequence of nonnegative indepen- the magnitude of the shock exceeds the threshold. The dence random variable {𝑋𝑛,𝑛=1,2,...}is called a geometric threshold of component 𝑖 is 𝜏𝑖 with a distribution function process (GP), if for some 𝑎>0the cumulative distribution Φ𝑖, (𝑖 = 1, 2); every shock is independent. 𝑛−1 function of 𝑋𝑛 is 𝐹(𝑎 𝑥). 𝑎 is called the parameter of the (A4) The intrinsic lifetime and the repair time on com- GP. ponents follow the extend Poisson process, respectively. The distribution of the intrinsic lifetime and the repair time of With Definition,wehavethefollowing. 2 component 𝑖 in the 𝑛th cycle (𝑖=1,2;𝑘=1,2,...)are 𝑎>1 𝑛 = 1, 2, . . {𝑋 ,𝑛 = 1,2,...} (1) If for ,then 𝑛 is 𝐿 𝑥 = 𝐿 [(𝛼 𝑎𝑘−1 +𝛽 )𝑥] 𝑋 > 𝑋 𝑘 ( ) 1 1 stochastically decreasing: 𝑛 st 𝑛+1. 0<𝑎<1 𝑛 = 1, 2, . . {𝑋 ,𝑛 = 1,2,...} 𝑘−1 (2) If for ,then 𝑛 is =1−exp [− (𝛼1𝑎 +𝛽1) 𝜂𝑥] , 𝑋 < 𝑋 stochastically increasing: 𝑛 st 𝑛+1. (2) 𝑘−1 (3) If 𝑎=1for 𝑛 = 1, 2, .,then . {𝑋𝑛,𝑛 = 1,2,...}is a 𝐻𝑘 (𝑦) = 𝐻 2[(𝛼 +𝛽2𝑏 )𝑦] renewal process. 𝑘−1 =1−exp [− (𝛼2 +𝛽2𝑏 )𝜇𝑦], Belowwewillintroduceanewprocesswhichcanbeusedto describe scenarios with complicated failure intensities. where 𝑥, 𝑦, >0 𝛼𝑖 ≥0, 𝛽𝑖 ≥0, 𝛼𝑖 +𝛽𝑖 =1, 𝑎>1,and0<𝑏<1 (𝑖 = 1, 2),respectively. Definition 3 (see [16]). A sequence of nonnegative indepen- (A5) The repairman has single vacation rules as follows. dent random variable {𝑋𝑛,𝑛 = 1,2,...}is called an extended When a component fails with the presence of the repairman, Poisson process (EPP), if some 𝛼𝛽 =0̸ , 𝛼, 𝛽, ≥0 𝑎≥1,and0< it will be repaired immediately. Once the failed component is 𝑏≤1, the cumulative distribution function (cdf) of 𝑋𝑛 is 𝑛−1 𝑛−1 repaired and there is no failure component in system, the 𝐺[(𝛼𝑎 +𝛽𝑏 )𝑥] and 𝐺(𝑥) is an exponential cdf. 𝛼, 𝛽, 𝑎, 𝑏 repairman will take vacation. If one component fails when the are parameters of the process. other is being repaired, the newly failed component must wait With Definition, 3 the following can be obtained. forrepairandthesystemisdown.Iftwocomponentsare waiting for repair when the repairman returns from a vaca- Scenarios tion, the repair rule is “first-in-first-out.” If there is no failure component when the repairman returns from a vacation, he (1) If 𝑎=𝑏=1, then the EPP is an HPP. remainsidleuntilthefirstfailurecomponentappears.Denote 𝑛−1 𝑛−1 𝑛−1 𝑛−1 𝑍 (2) If 𝛼𝑎 =0̸ and 𝛽𝑏 =0(or 𝛼𝑎 =0and 𝛽𝑏 ≠ by the vacation length of the repairman. Its distribution is 0 {𝑋 ,𝑛=1,2,...} ), then 𝑛 is a GP. 𝑡 𝑡 𝑛−1 (3) If 𝛼𝑎 =0,𝑎≯ 1and 𝑏=1,then{𝑋𝑛,𝑛 = 1,2,...} 𝑉 (𝑡) = ∫ 𝑉 (𝑧) 𝑑𝑧 = 1− exp {− ∫ 𝛾 (𝑧) 𝑑𝑧} . (3) can describe the periods from the intrinsic failure 0 0 period to the wear-out time period in a bathtub curve. −1 Denote 𝐸(𝑍) =𝑐 , 𝑉(𝑧) = 1 − 𝑉(𝑧). 𝑎=1 𝛽𝑏𝑛−1 =0̸ 0<𝑏<1 {𝑋 ,𝑛 = 1,2,...} (4) If , , ,then 𝑛 (A6) The failure of the operating component may be candescribetheperiodsfromtheburn-intimeperiod caused by intrinsic factors or external shocks, and the system to the end of intrinsic failure period in a bathtub fails only if both of the components fail. curve. (A7) After repair, both of the components are not as good 𝑛−1 𝑛−1 (5) If 𝛼𝑎 =0̸ , 𝛽𝑏 =0̸ , 𝑎>1, 0<𝑏<1,then{𝑋𝑛, as new. All random variables and processes are statistically 𝑛 = 1, 2, . . .} can describe more complicated failure independent. intensity curves. 3. Model Development 2.2. Assumptions. The following assumptions are assumed to hold in what follows. Basedontheabovemodelassumption(A3),wecangetthe (A1) The system is composed of two components: a switch probability that one shock causes the operating component andarepairman.Attheinitialtime,bothcomponentsare failure in the 𝑛th cycle (𝑛=1,2,...)which is new: component 1 is operating while component 2 is on ̂ cold standby, and the repairman is in idle. Once the oper- 𝑟=𝑃{𝑋𝑛 >𝜏𝑛} ating component fails, the cold standby component will be ∞ (4) switched to the operating state, assuming the switch is perfect ̂ ̂ = ∫ 𝑃{𝜏𝑛 <𝑥|𝑋𝑛 =𝑥}𝑑𝑃{𝑋𝑛 ≤𝑥}. and switch time can be neglected. 0 (A2) The system subjects to shock. The arrival ofthe ̃ Lemma 4 𝜉(𝑖) shocks follow a Poisson process {𝑆(𝑡), 𝑡 ≥0} with the intensity (see [5]). The distribution function of 𝑛 is 𝜆>0. The magnitude of each shock is an independent ̂ −𝑟𝑡 random 𝑋 with distribution function 𝐾(𝑥). 𝑆𝑛 (𝑡) =1−𝑒 ,𝑡>0,(𝑛 = 1, 2, .) . ; (5) (A3) When a shock arrives, it only affects the operating −𝑟𝜆𝑡 component and the operating component will fail only when hence, 𝑆𝑛(𝑡) = 1 − 𝑃(𝑋𝑛 >𝑡)=1−𝑒 . 4 Mathematical Problems in Engineering

Lemma 5 (see [6]). The operating time of the component 𝑖 in Thestateprobabilitiesofthesystemattime𝑡 are defined (𝑖) (𝑖) the 𝑛th cycle is 𝑈𝑛 ; its distribution function is 𝐺𝑛 (𝑥);then by

(𝑖) 𝑘−1 𝑝𝑖,𝑘,𝑙 (𝑡) =𝑃{𝑁(𝑡) =𝑖,𝐼1 (𝑡) =𝑘,𝐼2 (𝑡) =𝑙} 𝐺𝑛 (𝑥) =1−exp {− [𝑟𝜆 1+ (𝛼 𝑎 +𝛽1)𝜂]𝑥}; (6)

𝑝𝑗,𝑘,𝑙 (𝑡, 𝑧) 𝑑𝑧 = 𝑃 {𝑁 (𝑡) =𝑗, (𝑖) 𝑘−1 hence, 𝐺 (𝑡) = 1−𝑃(𝑇𝑛 >𝑡)=1−exp{−[𝑟𝜆+(𝛼1𝑎 +𝛽1)𝜂]}𝑡, 𝑛 𝐼 (𝑡) =𝑘,𝐼 (𝑡) =𝑙,𝑧≤𝑍(𝑡) <𝑧+𝑑𝑧} 𝑖=1,2. 1 2 𝑖 = 0, 1, 2, 3, 4, 5, 𝑗 = 6, 7, 8, 9, 10. Let 𝑁(𝑡) be the system state at time 𝑡;thenwehavethe following. (7)

State 0. At time 𝑡, component 1 is operating, component 2 is Bythenatureofthecoldstandby,andbythefactthatcom- on cold standby, and the repairman is in idle. ponent 1 and component 2 are operating alternately in system, we can obtain that their cycles meet the following relations: State 1. At time 𝑡, component 2 is operating, component 1 is on cold standby, and the repairman is in idle. 𝑘−𝑙=0,1. (8) State 2. At time 𝑡, component 1 is operating; component 2 is being repaired. So the continuous four-dimensional vector Markov process {𝑁(𝑡),1 𝐼 (𝑡),2 𝐼 (𝑡), 𝑍(𝑡), 𝑡≥0} can be converted into a 𝑡 State 3. At time , component 2 is operating; component 1 is continuous three-dimensional vector Markov process {𝑁(𝑡), being repaired. 𝐼1(𝑡), 𝑍(𝑡),,where 𝑡≥0} 𝐼1(𝑡) is the number of cycles of com- 𝑡 𝑡 ponent 1 at time . Then the state probabilities of the system State 4. At time ,component2isbeingrepaired;component1 at time 𝑡 are defined by is waiting for repair.

State 5. At time 𝑡,component1isbeingrepaired;component2 𝑝𝑖,𝑘 (𝑡) =𝑃{𝑁(𝑡) =𝑖,𝐼1 (𝑡) =𝑘} is waiting for repair. 𝑝𝑗,𝑘 (𝑡, 𝑧) 𝑑𝑧 = 𝑃 {𝑁 (𝑡) =𝑗, 𝑡 (9) State 6. At time , component 1 is operating, component 2 is 𝐼 (𝑡) =𝑘,𝑧≤𝑍(𝑡) <𝑧+𝑑𝑧} on cold standby, and the repairman is taking a vacation. 1 𝑖 = 0, 1, 2, 3, 4, 5, 𝑗 = 6, 7, 8, 9, 10. State 7. At time 𝑡, component 2 is operating, component 1 is on cold standby, and the repairman is taking a vacation. Then, we can get each state transition diagram as shown in 𝑡 𝑘−1 𝑘−1 State 8. At time , component 1 is operating, component 2 is Figure 1,where𝑎(𝑘) = 𝜆𝑟1 +(𝛼 𝑎 +𝛽1), 𝑏(𝑘)2 =𝛼 +𝛽2𝑏 . waiting for repair, and the repairman is taking a vacation. By using the probability arguments and limiting transi- tions, we can get the following differential equations for the 𝑡 State 9. At time , component 2 is operating, component 1 is system; for 𝑘≥2,wehave waiting for repair, and the repairman is taking a vacation. 𝑡 State 10. At time , two components are waiting for repair; the 𝑝 (𝑡+Δ𝑡) repairman is taking a vacation. 0,𝑘 𝐸 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Obviously, the state space is = 𝑃 {𝑁 (𝑡 + Δ𝑡1 =0,𝐼 (𝑡+Δ𝑡) =𝑘)} 10},wheretheoperatingstatesetis𝑊 = {0,1,2,3,6,7,8,9} 𝐹={4,5,10} 𝑘−1 and failure state set is . According to the assump- =𝑝0,𝑘 (𝑡) {1 − (𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂)Δ𝑡} tions, {𝑁(𝑡), 𝑡 ≥0} is not a Markov process. Hence, we introduce the following supplementary variables: ∞ + ∫ 𝑝6,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧Δ𝑡 +𝑜 (Δ𝑡) , 0 𝐼𝑖(𝑡): the number of cycles of component 𝑖 at time 𝑡; 𝑝6,𝑘 (𝑡+Δ𝑡,𝑧+Δ𝑡) 𝑍(𝑡) : the elapsed vacation time when the repairman is =𝑃{𝑁 (𝑡+Δ𝑡) =6, taking a vacation at time 𝑡.

𝐼1 (𝑡+Δ𝑡) =𝑘,𝑍(𝑡+Δ𝑡) =𝑧+Δ𝑡} Then {𝑁(𝑡),1 𝐼 (𝑡),2 𝐼 (𝑡), 𝑍(𝑡), 𝑡≥0} is a continuous four- 𝑘−1 dimensional vector Markov process with state space Ω= =𝑝6,𝑘 (𝑡, 𝑧) {1 − [𝛾 (𝑧) +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂]Δ𝑡} {[0, 𝑘, 𝑙], [1,𝑘,𝑙], [2, 𝑘, 𝑙], [3, 𝑘,, 𝑙] [4,𝑘,𝑗], [5,𝑘,𝑙], [6,𝑘,𝑙,𝑧], [7,𝑘,𝑙,𝑧], [8,𝑘,𝑙,𝑧], [9,𝑘,𝑙,𝑧], [10, 𝑘, 𝑙,,where 𝑧]} 𝑘=1,2,...; +𝑜(Δ𝑡) . 𝑙=1,2,.... (10) Mathematical Problems in Engineering 5

𝜕 𝜕 a(k − 1) { + +𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑝 (𝑡, 𝑧) 𝜕𝑡 𝜕𝑧 1 1 8,𝑘

= {𝜆𝑟 + (𝛼 𝑎𝑘−2 +𝛽 )𝜂}𝑝 (𝑡, 𝑧) , 6k(t, y) a(k) 9k(t, y) 7k(t, y) a(k)8k(t, y) a(k) 10k(t, y) 1 1 7,𝑘 𝛾(y) 𝜕 𝜕 𝑘−2 { + +𝛾(𝑦)+𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑝9,𝑘 (𝑡, 𝑧) 𝛾(y) 𝛾(y) 𝛾(y) 𝜕𝑡 𝜕𝑧 = {𝜆𝑟 + (𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑝 (𝑡, 𝑧) , 5k−1 (t) 1k(t) 1 1 6,𝑘 𝛾(y) b(k − 1) a(k − 1) 𝛾(y) 𝜕 𝜕 { + +𝛾(𝑧)}𝑝 (𝑡, 𝑧) 𝜕𝑡 𝜕𝑧 10,𝑘

0k(t) a(k) 3k(t) a(k) 5k(t) 2k(t) a(k) 4k(t) 𝑘−1 = {𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂}𝑝8,𝑘 (𝑡, 𝑧)

3 (t) 𝑘−2 k−1 b(k − 1) + {𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂}𝑝9,𝑘 (𝑡, 𝑧) , b(k − 1) 𝑑 { +𝜆𝑟+(𝛼 𝑎𝑘−1 +𝛽 )𝜂+(𝛼 +𝛽 𝑏𝑘−2)𝜇}𝑝 (𝑡) 𝑑𝑡 1 1 2 2 2,𝑘 7k(t, 0) 6k(t, 0) ∞ b(k − 1) = ∫ 𝑝8,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧 0 Figure 1: The system state transition diagram. 𝑘−2 + (𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂)𝑝1,𝑘 (𝑡)

𝑘−2 +(𝛼2 +𝛽2𝑏 )𝜇𝑝5,𝑘−1 (𝑡) , Let Δ𝑡 tend to zero; we have 𝑑 𝑘−1 𝑘−1 { +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂+(𝛼2 +𝛽2𝑏 )𝜇}𝑝3,𝑘 (𝑡) 𝑑 𝑘−1 𝑑𝑡 { +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑝0,𝑘 (𝑡) 𝑑𝑡 ∞ = ∫ 𝑝9,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧 ∞ 0 = ∫ 𝑝6,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧, 0 𝑘−1 + (𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂)𝑝0,𝑘 (𝑡) 𝜕 𝜕 𝑘−1 { + +𝛾(𝑧) +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑝6,𝑘 (𝑡, 𝑧) =0. 𝑘−2 𝜕𝑡 𝜕𝑧 +(𝛼2 +𝛽2𝑏 )𝜇𝑝4,𝑘 (𝑡) . (11) (12) Their boundary conditions are In the same way, we have 𝑑 [ +𝜂+𝑟𝜆+𝜇]𝑝3,1 (𝑡) =(𝑟𝜆+𝜂)𝑝0,1 (𝑡) ; ∞ 𝑑𝑡 𝑑 𝑘−2 { +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑝1,𝑘 (𝑡) =∫ 𝑝7,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧, 𝑑𝑡 0 𝑑 [ +𝜇]𝑝5,1 (𝑡) =(𝑟𝜆+𝜂)𝑝3,1 (𝑡) 𝑑 𝑑𝑡 { +(𝛼 +𝛽 𝑏𝑘−2)𝜇}𝑝 (𝑡) 𝑑𝑡 2 2 4,𝑘 𝑘−2 𝑝6,𝑘 (𝑡,) 0 =(𝛼2 +𝛽2𝑏 )𝜇𝑝2,𝑘 (𝑡) , ∞ (13) = ∫ 𝑝 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧 𝑘−2 10,𝑘 𝑝7,𝑘 (𝑡,) 0 =(𝛼2 +𝛽2𝑏 )𝜇𝑝3,𝑘−1 (𝑡) , 𝑘≥2. 0 𝑘−1 𝑝 (𝑡,) 0 =0, 𝑗=8,9,10,𝑘≥1; +{(𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂)}𝑝2,𝑘 (𝑡) , 𝑗,𝑘

𝑑 𝑝6,1 (𝑡,) 0 =𝑝7,1 (𝑡,) 0 =0; { +(𝛼 +𝛽 𝑏𝑘−1)𝜇}𝑝 (𝑡) 𝑑𝑡 2 2 5,𝑘 𝑝𝑖,1 (𝑡) =0, 𝑖=1,2,4. ∞ = ∫ 𝑝10,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧 The initial conditions are 0 𝑝0,1 (0) =1; 𝑘−1 +{(𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂)}𝑝3,𝑘 (𝑡) , 𝑝𝑗,1 (0, 𝑧) =0, 𝑗=6,7,8,9,10; (14) 𝜕 𝜕 𝑘−2 { + +𝛾(𝑧) +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑝7,𝑘 (𝑡, 𝑧) =0, 𝜕𝑡 𝜕𝑧 𝑝𝑖,1 (0) =0, 𝑖=1,2,4. 6 Mathematical Problems in Engineering

𝑘−1 𝑘−1 ∗ We introduce the Laplace transform and we denote the {𝑠 + 𝜆𝑟1 +(𝛼 𝑎 +𝛽1)𝜂+(𝛼2 +𝛽2𝑏 )𝜇}𝑝3,𝑘 (𝑠) ∞ 𝑝 (𝑡) 𝑝∗(𝑠) = ∫ 𝑝 (𝑡)𝑒−st𝑑𝑡 𝑖= Laplace transform of 𝑖 by 𝑖 0 𝑖 , ∞ ∞ ∗ ∗ −st = ∫ 𝑝 (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧 0, 1, 2, 3, 4, 5,and𝑝𝑗(𝑡, 𝑧) by 𝑝𝑗 (𝑠, 𝑧) =∫ 𝑝𝑗(𝑡, 𝑧)𝑒 𝑑𝑡, 9,𝑘 0 0 𝑗 = 6, 7, 8, 9,; 10 𝑠>0. The Laplace transforms of the above 𝑘−1 ∗ differential equations are, respectively, given by + (𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂)𝑝0,𝑘 (𝑠) ∞ {𝑠 + 𝜆𝑟 +(𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑝∗ (𝑠) = ∫ 𝑝∗ (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧, 𝑘−2 ∗ 1 1 0,𝑘 6,𝑘 +(𝛼2 +𝛽2𝑏 )𝜇𝑝 (𝑠) . 0 4,𝑘 (15) 𝜕 { +𝑠+𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑝∗ (𝑠,) 𝑧 =0, 𝜕𝑧 1 1 6,𝑘 The boundary conditions are ∞ 𝑘−2 ∗ ∗ {𝑠 + 𝜆𝑟1 +(𝛼 𝑎 +𝛽1)𝜂}𝑝1,𝑘 (𝑠) = ∫ 𝑝7,𝑘 (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧, 0 ∗ ∗ [𝑠+𝜂+𝑟𝜆+𝜇]𝑝3,1 (𝑠) =(𝑟𝜆+𝜂)𝑝0,1 (𝑠) ; 𝑘−2 ∗ ∗ ∗ (16) {𝑠 + (𝛼2 +𝛽2𝑏 )𝜇}𝑝4,𝑘 (𝑠) [𝑠 + 𝜇]5,1 𝑝 (𝑠) =(𝑟𝜆+𝜂)𝑝3,1 (𝑠)

∞ ∗ 𝑘−2 ∗ ∗ 𝑝6,𝑘 (𝑠,) 0 =(𝛼2 +𝛽2𝑏 )𝜇𝑝2,𝑘 (𝑠) , = ∫ 𝑝10,𝑘 (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧 0 (17) 𝑝∗ (𝑠,) 0 =(𝛼 +𝛽 𝑏𝑘−2)𝜇𝑝∗ (𝑠) , 𝑘≥2. 𝑘−1 ∗ 7,𝑘 2 2 3,𝑘−1 + {(𝜆𝑟 1+ (𝛼 𝑎 +𝛽1)𝜂)} 𝑝 (𝑠) , 2,𝑘 ∗ 𝑝𝑗,𝑘 (𝑠,) 0 = 0, 𝑗 = 8, 9, 10, 𝑘 ≥1; 𝑘−1 ∗ {𝑠 + (𝛼2 +𝛽2𝑏 )𝜇}𝑝5,𝑘 (𝑠) ∗ ∗ 𝑝6,1 (𝑠,) 0 =𝑝7,1 (𝑠,) 0 =0; (18) ∞ ∗ 𝑝∗ (𝑠) =0, 𝑖=1,2,4. = ∫ 𝑝10,𝑘 (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧 𝑖,1 0

𝑘−1 ∗ + {(𝜆𝑟 1+ (𝛼 𝑎 +𝛽1)𝜂)}𝑝3,𝑘 (𝑠) , According to (15)–(18), we have

𝜕 { +𝑠+𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎𝑘−2 +𝛽 )𝜂}𝑝∗ (𝑠,) 𝑧 =0, 𝑝∗ (𝑠) =𝑝∗ (𝑠,) 0 𝐻 (𝑠) ; 𝜕𝑧 1 1 7,𝑘 0,𝑘 6,𝑘 𝑘−1 𝑝∗ (𝑠) =𝑝∗ (𝑠,) 0 𝐻 (𝑠) , 𝜕 1,𝑘 7,𝑘 𝑘−2 { +𝑠+𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑝∗ (𝑠,) 𝑧 𝜕𝑧 1 1 8,𝑘 ∗ ∗ ∗ [𝐺 (𝑠) +𝐻𝑘−2 (𝑠)]𝑝7,𝑘 (𝑠,) 0 𝐶𝑘−2 +𝐷𝑘−2𝑝5,𝑘−1(𝑠) 𝑝2,𝑘 (𝑠) = , 𝑘−2 ∗ 𝑠+𝐶𝑘−1 +𝐷𝑘−2 = {𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂}𝑝7,𝑘 (𝑠,) 𝑧 , [𝐺 (𝑠) +𝐻 (𝑠)]𝑝∗ (𝑠,) 0 𝐶 +𝐷 𝑝∗ (𝑠) 𝑝∗ (𝑠) = 𝑘−1 6,𝑘 𝑘−1 𝑘−2 4,𝑘 , 𝜕 𝑘−2 ∗ 3,𝑘 { +𝑠+𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎 +𝛽 )𝜂}𝑝 (𝑠,) 𝑧 𝑠+𝐶𝑘−1 +𝐷𝑘−1 𝜕𝑧 1 1 9,𝑘 1 𝐶 𝑝∗ (𝑠) = [ 𝑘−1 𝑝∗ (𝑠,) 0 +𝐼 (𝑠) 𝑊 (𝑠)], = {𝜆𝑟 + (𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑝∗ (𝑠,) 𝑧 , 4,𝑘 6,𝑘 𝑘 1 1 6,𝑘 𝑠+𝐷𝑘−2 𝐷𝑘−2

𝜕 ∗ ∗ 1 𝐶𝑘−1 ∗ { +𝑠+𝛾(𝑧)}𝑝10,𝑘 (𝑠,) 𝑧 𝑝5,𝑘 (𝑠) = [ 𝑝7,𝑘+1 (𝑠,) 0 +𝐼𝑘 (𝑠) 𝑊 (𝑠)], 𝜕𝑧 𝑠+𝐷𝑘−1 𝐷𝑘−1

𝑘−1 ∗ ∗ −(𝑠+𝐶 )𝑧 ∗ = {𝜆𝑟 + (𝛼 𝑎 +𝛽 )𝜂}𝑝 (𝑠,) 𝑧 𝑘−1 1 1 8,𝑘 𝑝6,𝑘 (𝑠,) 𝑧 =𝑒 𝑉 (𝑧) 𝑝6,𝑘 (𝑠,) 0 ;

𝑘−2 ∗ ∗ −(𝑠+𝐶𝑘−2)𝑧 ∗ + {𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂}𝑝9,𝑘 (𝑠,) 𝑧 , 𝑝7,𝑘 (𝑠,) 𝑧 =𝑒 𝑉 (𝑧) 𝑝7,𝑘 (𝑠,) 0 ,

∗ ∗ −(𝑠+𝐶 )𝑧 −(𝑠+𝐶 )𝑧 𝑝 (𝑠,) 𝑧 =𝑝 (𝑠,) 0 𝑉 (𝑧) (𝑒 𝑘−2 −𝑒 𝑘−1 ) 𝑘−1 𝑘−2 ∗ 8,𝑘 7,𝑘 {𝑠 + 𝜆𝑟1 +(𝛼 𝑎 +𝛽1)𝜂+(𝛼2 +𝛽2𝑏 )𝜇}𝑝2,𝑘 (𝑠) 𝐶𝑘−2 ∞ × , ∗ 𝑘−2 𝛼1𝜂𝑎 (𝑎−1) = ∫ 𝑝8,𝑘 (𝑠, 𝑦) 𝛾 (𝑧) 𝑑𝑧 0 ∗ ∗ −(𝑠+𝐶 )𝑧 −(𝑠+𝐶 )𝑧 𝑝 (𝑠,) 𝑧 =𝑝 (𝑠,) 0 𝑉 (𝑧) (𝑒 𝑘−2 −𝑒 𝑘−1 ) 𝑘−2 ∗ 9,𝑘 6,𝑘 +(𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂)𝑝1,𝑘 (𝑠) 𝐶 × 𝑘−1 , +(𝛼 +𝛽 𝑏𝑘−2)𝜇𝑝∗ (𝑠) , 𝑘−2 2 2 5,𝑘−1 𝛼1𝜂𝑎 (𝑎−1) Mathematical Problems in Engineering 7

∗ −𝑠𝑧 𝐷 𝑝10,𝑘 (𝑠,) 𝑧 =𝑒 𝑉 (𝑧) 𝐼𝑘 (𝑠) ∗ 𝑘−2 𝑝7,𝑘 (𝑠,) 0 = 𝑠+𝐶𝑘−2𝐷𝑘−2 −𝐶 −𝐶 {𝐶 (1 − 𝑒 𝑘−2 )−𝐶 (1 − 𝑒 𝑘−1 )} × 𝑘−1 𝑘−2 , 𝑘−2 ×{[𝐻 (𝑠) +𝐺(𝑠)]𝑝 (𝑠,) 0 𝐶 𝛼1𝜂𝑎 (𝑎−1) 𝑘−2 6,𝑘−1 𝑘−2 1 ∗ 𝐶 𝐷 𝑝0,1 (𝑠) = ; 𝑘−2 𝑘−3 𝑠+𝑟𝜆+𝜂 + 𝑝6,𝑘−1 (𝑠,) 0 𝐷𝑘−3 (𝑠 +𝑘−3 𝐷 )

∗ 𝑟𝜆 +𝜂 𝑝 (𝑠) = , 𝐷𝑘−3 3,1 (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂) + 𝑊 (𝑠) 𝐼𝑘−1 (𝑠)}. 𝑠+𝐷𝑘−3 2 (21) ∗ (𝑟𝜆 + 𝜂) 𝑝5,1 (𝑠) = , ∗ ∗ (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂)(𝑠+𝜇) As the formulae 𝑝6,2(𝑠, 0), 𝑝7,2(𝑠, 0) have been calculated, ∗ when 𝑘=3,by(17), we can obtain 𝑝7,3(𝑠, 0);combining (𝑟𝜆 +2 𝜂)𝜇 [(𝜇/ (𝑠 + 𝜇))+(𝑟𝜆 +𝜂)𝐻 (𝑠) +𝐺(𝑠)] 𝑝∗ (𝑠, 0) 𝑝∗ (𝑠, 0) 𝑝∗ (𝑠,) 0 = 0 , 7,3 with (17), then we can get 6,3 .Bytherecur- 6,2 𝑝∗ (𝑠, 0) 𝑝∗ (𝑠, 0) (𝑠 + 𝑟𝜆 + 𝜂 +𝜇)(𝑠 +𝑟𝜆1 +𝜂)(𝑠+𝑟𝜆(𝛼 𝑎+𝛽1)𝜂) rence formula (17), we can obtain 6,𝑘 , 7,𝑘 when ∗ ∗ 𝑘≥3.Sowhen𝑘≥2, 𝑝𝑗,𝑘(𝑠, 𝑧), 𝑗 = 6, 7, 8, 9,,and 10 𝑝𝑖,𝑘(𝑠), 𝜇(𝑟𝜆+𝜂) 𝑖 = 0, 1, 2, 3, 4, 5 𝑝∗ (𝑠,) 0 = , ,canbeobtained. 7,2 (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂) (19) 4. Reliability Indices 4.1.SystemAvailability. By the definition of the system avail- where ability 𝐴(𝑡),wehave

𝑘−1 𝐴 (𝑡) =𝑃(𝑁 (𝑡) ∈𝑊) 𝐶𝑘−1 =𝑟𝜆+(𝛼1𝑎 +𝛽1)𝜂; ∞ 3 ∞ 9 (22) 𝑘−1 = ∑ ∑𝑝𝑖,𝑘 (𝑡) + ∑ ∑𝑝𝑗,𝑘 (𝑡, 𝑧) . 𝐷𝑘−1 =(𝛼2 +𝛽2𝑏 )𝜇, 𝑘=1 𝑖=0 𝑘=1 𝑗=6 V∗ (𝑠 + 𝐶 )−V∗ (𝑠 + 𝐶 ) 𝐴(𝑡) 𝐺 (𝑠) = 𝑘−2 𝑘−1 ; The Laplace transform of is given by 𝛼 𝑎𝑘−2 (𝑎−1) 1 ∞ 3 ∞ 9 𝐴∗ (𝑠) = ∑ ∑𝑝∗ (𝑠) + ∑ ∑𝑝∗ (𝑠,) 𝑧 V∗ (𝑠 + 𝐶 ) 𝑖,𝑘 𝑗,𝑘 𝑘−1 𝑘=1 𝑖=0 𝑘=1 𝑗=6 𝐻𝑘−1 (𝑠) = , 𝑠+𝐶𝑘−1 3 9 ∗ ∗ 𝑊 (𝑠) = ∑𝑝𝑖,1 (𝑠) + ∑𝑝𝑗,1 (𝑠,) 𝑧 𝑖=0 𝑗=6 ∗ ∗ ∗ ∗ 𝐶𝑘−1 {V (𝑠)−V (𝑠+𝐶𝑘−2)} −𝐶𝑘−2 {V (𝑠)−V (𝑠 +𝑘−1 𝐶 )} = ; ∞ 3 ∞ 9 𝑘−2 ∗ ∗ 𝛼1𝜂𝑎 (𝑎−1) + ∑ ∑𝑝𝑖,𝑘 (𝑠) + ∑ ∑𝑝𝑗,𝑘 (𝑠,) 𝑧 (23) 𝑘=2 𝑖=0 𝑘=2 𝑗=6 ∗ ∗ 𝐼𝑘 (𝑠) =𝑝7,𝑘 (𝑠,) 0 +𝑝6,𝑘 (𝑠,) 0 . 𝑠 + 2𝑟𝜆 + 2𝜂+𝜇 (20) = (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂)

𝑘≥3 ∞ 3 ∞ 9 When ,wehave ∗ ∗ + ∑ ∑𝑝𝑖,𝑘 (𝑠) + ∑ ∑𝑝𝑗,𝑘 (𝑠,) 𝑧 , 𝑘=2 𝑖=0 𝑘=2 𝑗=6 ∗ 𝐷𝑘−2 𝑝6,𝑘 (𝑠,) 0 = 𝑠+𝐶𝑘−1𝐷𝑘−2 where

∗ 1 𝑝0,1 (𝑠) = ; ×{[𝐻𝑘−2 (𝑠) +𝐺(𝑠)]𝑝7,𝑘 (𝑠,) 0 𝐶𝑘−2 𝑠+𝑟𝜆+𝜂

∗ 𝑟𝜆 +𝜂 𝐶𝑘−2𝐷𝑘−2 𝑝3,1 (𝑠) = , + 𝑝7,𝑘 (𝑠,) 0 (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂) (24) 𝐷𝑘−2 (𝑠 +𝑘−2 𝐷 ) ∗ ∗ 𝑝1,1 (𝑠) =𝑝2,1 (𝑠) =0; 𝐷𝑘−2 + 𝑊 (𝑠) 𝐼 (𝑠)}, ∗ 𝑘−1 𝑝 (𝑠, 𝑦) = 0, 𝑗 = 6, 7,8,9. 𝑠+𝐷𝑘−2 𝑗,1 8 Mathematical Problems in Engineering

Then, according to the Tauberian theorem, the limiting avail- when the repairman vacation is over; one component is oper- ability of the system is given by ating, while the other is on cold standby. Thus, the idle prob- 𝑡 ∗ ability of the repairman at time is given by 𝐴= lim 𝐴 (𝑡) = lim𝑠𝐴 (𝑠) =0. (25) 𝑡→∞ 𝑠→0 𝐷 (𝑡) =𝑃{𝑁 (𝑡) =0} +𝑃{𝑁 (𝑡) =1}

This is consistent with our intuition. Since neither of ∞ (30) the two components is as good as new after repair, system = ∑ [𝑝0,𝑘 (𝑡) +𝑝1,𝑘 (𝑡)]. availability will tend to 0 with 𝑡→+∞. 𝑘=1

The Laplace transform of 𝐼(𝑡) is 4.2.SystemROCOF. Let 𝑀𝑓(𝑡) be the expected number of the system failures in the (0, 𝑡]. Its derivative 𝑚𝑓(𝑡) is called ∞ ∗ ∗ ∗ the rate of occurrence of the system failure (ROCOF) at time 𝐷 (𝑠) = ∑ [𝑝0,𝑘 (𝑠) +𝑝1,𝑘 (𝑠)] 𝑡.Then, 𝑘=1 𝑡 ∞ 𝑀 (𝑡) = ∫ 𝑚 (𝑥) 𝑑𝑥. ∗ ∗ ∗ ∗ 𝑓 𝑓 (26) =𝑝0,1 (𝑠) +𝑝1,2 (𝑠) + ∑ [𝑝0,𝑘 (𝑠) +𝑝1,𝑘 (𝑠)] (31) 0 𝑘=2 With the result [18] and in view of the system model analysis, 1 ∞ = + ∑ [𝑝∗ (𝑠) +𝑝∗ (𝑠)]. we have 𝑠+𝑟𝜆+𝜂 0,𝑘 1,𝑘 𝑘=2 ∞ 𝑚 (𝑡) = ∑ {(𝑟𝜆+(𝛼 𝑎𝑘−1 +𝛽 )𝜂) 𝑓 1 1 Then, according to the Tauberian theorem, the idle probabil- 𝑘=1 ity of the repairman in the steady state is given by ∞ ∗ ×[𝑝2,𝑘 (𝑡) + ∫ 𝑝8,𝑘 (𝑡, 𝑧) 𝑑𝑧]} 𝐷= lim 𝐷 (𝑡) = lim𝑠𝐷 (𝑠) 0 𝑡→∞ 𝑠→0 (27) ∞ ∞ 𝑘−2 1 ∗ ∗ + ∑ {(𝑟𝜆+(𝛼1𝑎 +𝛽1)𝜂) = lim𝑠{ + ∑ [𝑝 (𝑠) +𝑝 (𝑠)]} 𝑠→0 𝑠+𝑟𝜆+𝜂 0,𝑘 1,𝑘 𝑘=2 𝑘=2 (32) ∞ ∞ ×[𝑝 (𝑡) + ∫ 𝑝 (𝑡, 𝑧) 𝑑𝑧]} . 𝑠 ∗ ∗ 3,𝑘 9,𝑘 = lim + lim𝑠∑ [𝑝 (𝑠) +𝑝 (𝑠)] 0 𝑠→0𝑠+𝑟𝜆+𝜂 𝑠→0 0,𝑘 1,𝑘 𝑘=2 𝑚 (𝑡) The Laplace transform of 𝑓 is =0.

(𝑟𝜆 2+ 𝜂) 𝑚∗ (𝑠) = This conclusion is also consistent with our intuition. In 𝑓 (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂) fact, after repair, the components are not as good as new; their consecutive operating time is stochastically decreasing and ∞ 𝑘−1 their consecutive repair time is stochastically increasing. + ∑ {(𝑟𝜆+(𝛼1𝑎 +𝛽1)𝜂) 𝑘=2 Finally, they will be irreparable. Thus, the repairman has to repair them frequently, forever. This implies that the idle ∞ ∗ ∗ probability of the repairman will be zero, as 𝑡→∞. ×[𝑝2,𝑘 (𝑠) + ∫ 𝑝8,𝑘 (𝑠,) 𝑧 𝑑𝑧]} (28) 0 𝑉(𝑡) ∞ 4.4. The Vacation Probability of the Repairman. Let be 𝑘−2 + ∑ {(𝑟𝜆+(𝛼1𝑎 +𝛽1)𝜂) the repairman vacation probability, by the system analysis; 𝑘=2 whenthesystemisat6,7,8,9,and10states,therepairmanis ∞ taking vacation. So the vacation probability of repairman at ∗ ∗ time 𝑡 is ×[𝑝3,𝑘 (𝑠) + ∫ 𝑝9,𝑘 (𝑠,) 𝑧 𝑑𝑧]} , 0 ∞ ∞ 10 𝑉 (𝑡) = ∑ ∫ ∑𝑝 (𝑡, 𝑧) 𝑑𝑧. where 𝑖,𝑘 (33) 𝑘=1 0 𝑖=6 ∗ ∗ 𝑟𝜆 +𝜂 𝑝 (𝑠) =0; 𝑝 (𝑠) = ; 𝑉(𝑡) 2,1 3,1 (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂) The Laplace transform of is

∗ ∞ ∞ 10 𝑝𝑗,1 (𝑠, 𝑦) = 0, 𝑗 =8,9. ∗ ∗ 𝑉 (𝑠) = ∑ ∫ ∑𝑝𝑖,𝑘 (𝑠,) 𝑧 𝑑𝑧 (29) 𝑘=1 0 𝑖=6 (34) ∞ ∞ 10 4.3.TheIdleProbabilityoftheRepairman. By the system anal- ∗ = ∑ ∫ ∑𝑝𝑖,𝑘 (𝑠,) 𝑧 𝑑𝑧. ysis, we can obtain that the repairman is idle when and only 𝑘=2 0 𝑖=6 Mathematical Problems in Engineering 9

∗ ∗ 4.5. System Reliability. In order to obtain the reliability of the Let 𝑄𝑖 (𝑠), 𝑄𝑗 (𝑠, 𝑧) be the Laplace transform of 𝑄𝑖(𝑡), 𝑄𝑗(𝑡, 𝑦), ∞ system, we let the above three failure states 4, 5, and 10 be the 𝑄∗(𝑠) = ∫ 𝑄 (𝑡)𝑒−st𝑑𝑡 𝑖 = 0, 1, 2, 3 respectively, so we have 𝑖 0 𝑖 , , ∞ absorbing states, and we can obtain another vector Markov ∗ −st ̃ ̃ and 𝑄𝑗 (𝑠, 𝑧) =∫ 𝑄𝑗(𝑡, 𝑧)𝑒 𝑑𝑡, 𝑗 = 6, 7, 8, 9; 𝑠>0.The process {𝑁(𝑡), 𝐼(𝑡), 𝑍(𝑡),.Let 𝑡≥0} 𝑄𝑗,𝑘(𝑡, 𝑧)𝑑𝑧𝑁(𝑡) =𝑃{ = 0 Laplacetransformsoftheabovedifferentialequationsare, 𝑗,1 𝐼 (𝑡) = 𝑘, 𝑧 ≤ 𝑍(𝑡), <𝑧+𝑑𝑧} 𝑗 = 6, 7, 8,.Let 9 𝑄𝑖,𝑘(𝑡) = ̃ respectively, given by 𝑃{𝑁(𝑡) = 𝑖,1 𝐼 (𝑡) = ,𝑘} 𝑖 = 0, 1, 2, 3.Denotethestateprob- abilities of the system at time 𝑡.Usingthemethodsimilarto ∞ Section 3, we have the following differential equations: 𝑘−1 ∗ ∗ {𝑠 + 𝜆𝑟1 +(𝛼 𝑎 +𝛽1)𝜂}𝑄0,𝑘 (𝑠) = ∫ 𝑄6,𝑘 (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧, 0 𝑑 𝑘−1 { +𝜆𝑟+(𝛼 𝑎 +𝛽 )𝜂}𝑄 (𝑡) ∞ 𝑑𝑡 1 1 0,𝑘 𝑘−2 ∗ ∗ {𝑠 + 𝜆𝑟1 +(𝛼 𝑎 +𝛽1)𝜂}𝑄1,𝑘 (𝑠) = ∫ 𝑄7,𝑘 (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧, ∞ 0 = ∫ 𝑄6,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧, 𝑘−1 𝑘−2 ∗ 0 {𝑠 + 𝜆𝑟1 +(𝛼 𝑎 +𝛽1)𝜂+(𝛼2 +𝛽2𝑏 )𝜇}𝑄2,𝑘 (𝑠) 𝑑 ∞ { +𝜆𝑟+(𝛼 𝑎𝑘−2 +𝛽 )𝜂}𝑄 (𝑡) ∗ 1 1 1,𝑘 = ∫ 𝑄8,𝑘 (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧 𝑑𝑡 0 ∞ +(𝜆𝑟+(𝛼 𝑎𝑘−2 +𝛽 )𝜂)𝑄∗ (𝑠) , = ∫ 𝑄7,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧, 1 1 1,𝑘 0 {𝑠 + 𝜆𝑟 +(𝛼 𝑎𝑘−1 +𝛽 )𝜂+(𝛼 +𝛽 𝑏𝑘−1)𝜇}𝑄∗ (𝑠) 𝑑 𝑘−1 𝑘−2 1 1 2 2 3,𝑘 { +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂+(𝛼2 +𝛽2𝑏 )𝜇}𝑄2,𝑘 (𝑡) 𝑑𝑡 ∞ ∗ = ∫ 𝑄9,𝑘 (𝑠,) 𝑧 𝛾 (𝑧) 𝑑𝑧 ∞ 0 = ∫ 𝑄8,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧 0 𝑘−1 ∗ +(𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂)𝑄0,𝑘 (𝑠) , 𝑘−2 + (𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂)𝑄1,𝑘 (𝑡) , 𝜕 𝑘−1 ∗ { +𝑠+𝛾(𝑧) +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑄6,𝑘 (𝑠,) 𝑧 =0, 𝑑 𝑘−1 𝑘−1 𝜕𝑧 { +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂+(𝛼2 +𝛽2𝑏 )𝜇}𝑄3,𝑘 (𝑡) 𝑑𝑡 𝜕 { +𝑠+𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎𝑘−2 +𝛽 )𝜂}𝑄∗ (𝑠,) 𝑧 =0, ∞ 𝜕𝑧 1 1 7,𝑘 = ∫ 𝑄9,𝑘 (𝑡, 𝑧) 𝛾 (𝑧) 𝑑𝑧 0 𝜕 { +𝑠+𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑄∗ (𝑠,) 𝑧 𝑘−1 𝜕𝑧 1 1 8,𝑘 + (𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂)𝑄0,𝑘 (𝑡) , 𝑘−2 ∗ 𝜕 𝜕 = {𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂}𝑄7,𝑘 (𝑠,) 𝑧 , { + +𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑄 (𝑡, 𝑧) =0, 𝜕𝑡 𝜕𝑧 1 1 6,𝑘 𝜕 𝑘−2 ∗ { +𝑠+𝛾(𝑧) +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑄9,𝑘 (𝑠,) 𝑧 𝜕 𝜕 𝑘−2 𝜕𝑧 { + +𝛾(𝑧) +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑄7,𝑘 (𝑡, 𝑧) =0, 𝜕𝑡 𝜕𝑧 𝑘−1 ∗ = {𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂}𝑄6,𝑘 (𝑠,) 𝑧 . 𝜕 𝜕 { + +𝛾(𝑧) +𝜆𝑟+(𝛼 𝑎𝑘−1 +𝛽 )𝜂}𝑄 (𝑡, 𝑧) (37) 𝜕𝑡 𝜕𝑧 1 1 8,𝑘

𝑘−2 = {𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂}𝑄7,𝑘 (𝑡, 𝑧) , The solutions of above equations can be written as

𝑄∗ (𝑠) =𝑄∗ (𝑠,) 0 𝐻 (𝑠) ; 𝜕 𝜕 𝑘−2 0,𝑘 6,𝑘 𝑘−1 { + +𝛾(𝑧) +𝜆𝑟+(𝛼1𝑎 +𝛽1)𝜂}𝑄9,𝑘 (𝑡, 𝑧) 𝜕𝑡 𝜕𝑧 ∗ ∗ 𝑄1,𝑘 (𝑠) =𝑄7,𝑘 (𝑠,) 0 𝐻𝑘−2 (𝑠) , 𝑘−1 = {𝜆𝑟 +1 (𝛼 𝑎 +𝛽1)𝜂}𝑄6,𝑘 (𝑡, 𝑧) . ∗ ∗ [𝐺 (𝑠) +𝐻𝑘−2 (𝑠)]𝑄7,𝑘 (𝑠,) 0 𝐶𝑘−2 𝑄2,𝑘 (𝑠) = , (35) 𝑠+𝐶𝑘−1 +𝐷𝑘−2 ∗ The initial conditions are ∗ [𝐺 (𝑠) +𝐻𝑘−1 (𝑠)]𝑄6,𝑘 (𝑠,) 0 𝐶𝑘−1 𝑄3,𝑘 (𝑠) = , 𝑠+𝐶𝑘−1 +𝐷𝑘−1 𝑄0,1 (0) =1; ∗ −(𝑠+𝐶 )𝑧 ∗ 𝑄 (𝑠,) 𝑧 =𝑒 𝑘−1 𝑉 (𝑧) 𝑄 (𝑠,) 0 , 𝑄𝑗,1 (0, 𝑧) =0, 𝑗=6,7,8,9; (36) 6,𝑘 6,𝑘

∗ −(𝑠+𝐶𝑘−2)𝑦 ∗ 𝑄1,1 (0) =𝑄2,1 (0) =0. 𝑄7,𝑘 (𝑠,) 𝑧 =𝑒 𝑉 (𝑧) 𝑄7,𝑘 (𝑠,) 0 , 10 Mathematical Problems in Engineering

Table 1: The result of the approximation 𝑄0,1(𝑡), 𝑄3,1(𝑡) at time 𝑡. 𝑡=10 𝑡=20 𝑡=30 𝑡=40 𝑡=50 𝑡=60 𝑡=70𝑡=80

𝑄0,1(𝑡) 0.4966 0.2466 0.1225 0.0608 0.0302 0.0150 0.0074 0.0037

𝑄3,1(𝑡) 0.1101 0.0574 0.0286 0.0142 0.0070 0.0035 0.0017 0.0008

∗ ∗ −(𝑠+𝐶𝑘−2)𝑧 −(𝑠+𝐶𝑘−1)𝑧 𝑄8,𝑘 (𝑠,) 𝑧 =𝑄7,𝑘 (𝑠,) 0 𝑉 (𝑧) (𝑒 −𝑒 ) 5. Numerical Examples 𝐶 × 𝑘−2 , Basedontheresultsabove,wecanfindthatisdifficultto 𝑘−2 𝛼1𝜂𝑎 (𝑎−1) obtain the transient results of the reliability indices for the system proposed in this paper. We can only obtain the steady- ∗ ∗ −(𝑠+𝐶𝑘−2)𝑧 −(𝑠+𝐶𝑘−1)𝑧 𝑄9,𝑘 (𝑠,) 𝑧 =𝑄6,𝑘 (𝑠,) 0 𝑉 (𝑧) (𝑒 −𝑒 ) state results of some reliability of the system. Thus, in engineering applications, a numerical method based on Runge- 𝐶 × 𝑘−1 , Kutta method is often adopted. Here, an approximative 𝑘−2 𝑅(𝑡) 𝛼1𝜂𝑎 (𝑎−1) solution of is given to illustrate the numerical method. To validate the above derivation, we conduct the following 1 𝑄∗ (𝑠) = ; numerical experiments. Here, we assume 0,1 𝑠+𝑟𝜆+𝜂 1−𝑒−𝜃𝑥,𝑥>0; 𝑄𝑗,1 (𝑠,) 0 =0, 𝑗=6,7,8,9, 𝐾 (𝑥) ={ 0, else, 𝑟𝜆 +𝜂 𝑄∗ (𝑠) = . (43) 3,1 (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂) 1−𝑒−(𝜃/4)𝑥, 𝑥>0; Φ (𝑥) ={ (38) 0, else. With the definition of the system reliability 𝑅(𝑡),wehave Thenwehave ∞ 3 ∞ ∞ 9 𝑅 (𝑡) = ∑ ∑𝑄 (𝑡) + ∑ ∫ ∑𝑄 (𝑡, 𝑧) 𝑑𝑧. ∞ 𝑖,𝑘 𝑗,𝑘 (39) ̂ ̂ ̂ 𝑘=1 𝑖=0 𝑘=1 0 𝑗=6 𝑟=𝑃{𝑋𝑛 >𝜏𝑛}=∫ 𝑃{𝜏𝑛 <𝑥|𝑋𝑛 =𝑥}𝑑𝑃{𝑋𝑛 ≤𝑥} 0 With the above explicit expressions, we obtain that the ∞ −(𝜃/4)𝑥 −𝜃𝑥 1 Laplace transform of 𝑅(𝑡) is = ∫ (1 − 𝑒 )𝑑(1−𝑒 )= . 0 5 ∞ 3 ∞ ∞ 9 ∗ ∗ ∗ (44) 𝑅 (𝑠) = ∑ ∑𝑄𝑖,𝑘 (𝑠) + ∑ ∫ ∑𝑄𝑗,𝑘 (𝑠,) 𝑧 𝑑𝑧 𝑘=1 𝑖=0 𝑘=1 0 𝑗=6 We make inverse transformation for (40); let 𝜆 = 0.3, 𝜂= 𝑠 + 2𝑟𝜆 + 2𝜂+𝜇 0.01,and𝜇 = 0.3.When𝑘=1, 𝑡 = 10, 20, 30, 40, 50, 60, = (40) (𝑠+𝑟𝜆+𝜂+𝜇)(𝑠+𝑟𝜆+𝜂) 70, 80, by MATLAB, we can obtain these approximative solutions of 𝑄0,1(𝑡), 𝑄3,1(𝑡) (see Table 1). ∞ 3 ∞ ∞ 9 + ∑ ∑𝑄∗ (𝑠) + ∑ ∫ ∑𝑄∗ (𝑠,) 𝑧 𝑑𝑧. Thus, we can further obtain these approximative solutions 𝑖,𝑘 𝑗,𝑘 𝑅(𝑡) 𝑅(𝑡) ≈𝑄 (𝑡) + 𝑄 (𝑡) 𝑡 = 10, 20, 30, 40, 50, 𝑘=2 𝑖=0 𝑘=2 0 𝑗=6 of , 0,1 3,1 .When 60, 70, 80,theresultsof𝑅(𝑡) are 𝑅(10) = 0.6067, 𝑅(20) = 0.3040 𝑅(30) = 0.1510 𝑅(40) = 0.0750 𝑅(50) = 0.0372 𝑇 , , , , 4.6. System MTTFF. Let be the lifetime of the system before 𝑅(60) = 0.0185, 𝑅(70) = 0.0092,and𝑅(80) = 0.0046. Clearly, being failed. According to the definition of the mean time to we can see that the value of the system reliability 𝑅(𝑡) is the first failure (MTTFF), we have decreasing when 𝑡 increases (see Figure 2). For a deteriorating ∞ ∞ system, the observation is consistent with our intuition. MTTFF =𝐸(𝑇) = ∫ 𝑡𝑑𝐹(𝑡) = ∫ 𝑅 (𝑡) 𝑑𝑡 𝜆=0.3 𝜂= 0 0 We make inverse transformation for (23); let , 0.01 𝜇 = 0.3 𝑘=1𝑡 = 10, 20, 30, 40, 50, 60, 70, ∞ (41) ,and .When , − ∗ ∗ 80 = lim ∫ 𝑒 st𝑅 (𝑡) 𝑑𝑡 = lim𝑅 (𝑠) =𝑅 (0) . , by MATLAB, we can obtain the approximative solutions 𝑠→0 0 𝑠→0 of 𝑝0,1(𝑡) just as 𝑄0,1(𝑡) in Table 1; we express it in Figure 3. Clearly, we can see that the repairman idle probability 𝐷(𝑡) is With (40), then we can get decreasing when 𝑡 increases. Since two components are not as 2𝑟𝜆 + 2𝜂 +𝜇 ∞ 3 good as new after repair, their consecutive operating time is = + ∑ ∑𝑄∗ (0) MTTFF (𝑟𝜆+𝜂+𝜇)(𝑟𝜆+𝜂) 𝑖,𝑘 stochastically decreasing and their consecutive repair time is 𝑘=2 𝑖=0 stochastically increasing. Finally, they will be irreparable. (42) ∞ ∞ 9 Thus, the repairman has to repair them frequently, forever; ∗ + ∑ ∫ ∑𝑄𝑗,𝑘 (0, 𝑧) 𝑑𝑧. this is consistent with practical experience. With the similar 𝑘=2 0 𝑗=6 approach, other results can be simulated. Mathematical Problems in Engineering 11

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