Cost-Effective Updated Sequential Predictive Maintenance Policy for ...

IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 7, NO. 2, APRIL 2010

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Cost-Effective Updated Sequential Predictive Maintenance Policy for Continuously Monitored Degrading Systems Ming-Yi You, Lin Li, Guang Meng, and Jun Ni

Abstract—The importance of maintenance optimization has been recognized over the past decades and is highly emphasized by today’s competitive economy. In this paper, an updated sequential predictive maintenance (USPM) policy is proposed to decide a real-time preventive maintenance (PM) schedule for a continuously monitored degrading system that will minimize maintenance cost rate (MCR) in the long term, by considering the effect of imperfect PM. The USPM model is continuously updated based on the change in the system state to decide an optimal PM schedule. Mathematical analysis of the proposed USPM model demonstrates the existence and uniqueness of an optimal PM schedule under practical conditions. The results validate that: 1) the proposed USPM model yields PM schedules that are consistent with the change in the system states and 2) the USPM model is able to quickly react to drastic degradation of the system and provide an optimal PM schedule in real time. The proposed maintenance policy can provide significant benefits for real-time maintenance decision making. Note to Practitioners—This paper is motivated by the gap that scheduling of commonly applied imperfect preventive maintenance (PM) (e.g., adding lubrication, partial replacement, etc.) scarcely considers a system’s operating condition which is highly correlated with machine health and failures. The updated sequential predictive maintenance (USPM) policy developed in this paper outlines a framework for real-time PM scheduling in a cost-effective way. To implement the proposed method, it is necessary to: 1) monitor a performance variable (e.g., pressure, temperature, etc.) that well indicates the system state; 2) estimate the system lifetime distribution; 3) quantify the PM work orders; and 4) measure the maintenance cost. Although the proposed maintenance policy is based on the objective of minimizing maintenance cost rate (MCR), it can be easily revised according to other practical optimization objectives, i.e., maximizing system availability.

Index Terms—Cost effective, degrading system, imperfect preventive maintenance (PM), updated maintenance scheduling.

Manuscript received November 06, 2008; revised January 19, 2009. First published August 07, 2009; current version published April 07, 2010. This work was recommended for publication by Associate Editor Y. S. Wong and Editor M. Zhou upon evaluation of the reviewers’ comments. This work was supported by the National Natural Science Foundation of China under Grant 10732060 and Grant 10528206 and in part by the U.S. NSF under Grant 0132521. M.-Y. You and G. Meng are with State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]; [email protected]). L. Li and J. Ni are with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TASE.2009.2019964

I. INTRODUCTION HE importance of maintenance has been recognized for over 40 years [1]. In today’s global economy, maintenance activities constitute a large portion of overhead costs for industry [5]. Ineffective maintenance polices will lead to a number of problems such as redundant or insufficient preventive maintenance (PM), unexpected machine failures and poor operation performance. These problems can greatly increase unnecessary maintenance cost and seriously affect productivity and product quality. Therefore, maintenance optimization has continuously been an important and valuable topic for researchers [23]. Most work on maintenance optimization considers equipment underlying lifetime distribution, based on which maintenance schedules are determined [3], [11], [19], [21]. With development of sensor technology and signal processing approaches, the operating condition of specific equipments becomes accessible and can be continuously monitored at a low cost [12]. As most equipment failures are highly correlated with their operating states [16], it is more appropriate to base the maintenance decision on the actual deterioration condition of a system rather than only on its age. Condition-based maintenance (CBM) has been illustrated to improve operational safety and reduce the quantity and severity of in-service system failures [18]. Most CBM models assume that the maintenance action is perfect and restores the equipment to a pristine, “new” condition [6], [9]. This assumption is valid for entire replacement, but is not practical in the case of imperfect maintenance, which does not make the system as good as new [23]. It could be spraying lubricant to a drill bit, replacing a mechanical component from a walking robot, etc. Although imperfect maintenance is common, research addressing the topic of CBM optimization considering imperfect maintenance is limited. Liao et al. [12] proposed a conditionbased availability-limit policy, in which the imperfect effect of PM is quantified by assuming that the system states after PM are randomly distributed. A Gamma process is used for system degradation modeling [26]. Zhou et al. [29] developed a reliability-centered predictive maintenance policy and modeled imperfect corrective maintenance (CM) and PM using the hybrid model that is originally developed in work of PM optimization based on lifetime distribution [13], [20]. The hybrid model has shown to be able to capture the immediate effect of the PM on the system and the lasting effect when the system is put back into service [25]. An optimal reliability threshold is established in [29] to minimize the maintenance cost rate (MCR), assuming that the system hazard rate is a known function of its state.

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It can be observed that the existing CBM model considering imperfect maintenance is built either on a predetermined stochastic model [12], or on a known deteriorating path for a specific unit [29]. The maintenance models remain unchanged regardless of a system’s real-time degradation states over its life cycle. Therefore, the current CBM models lack the functionality for real-time maintenance decision making, and leave a gap in research. To assess the changing state of a system, it is necessary to continuously update the PM schedule based on the most recent system state. In this paper, we propose a cost-effective updated sequential predictive maintenance (USPM) policy to decide real-time optimal PM schedules for continuously monitored degrading systems. The USPM policy is based on the objective of minimizing maintenance cost per unit operational time (MCR) in the long term, by considering the effect of imperfect PM. Mathematical analysis of the proposed USPM model validates the existence and uniqueness of an optimal PM schedule under practical conditions. In addition, a real-time maintenance decision-making scheme is recommended based on the output of the USPM model. A numerical study based on industrial data illustrates that 1) the proposed USPM model yields optimal PM schedules that are consistent with the system state change and 2) the proposed USPM model quickly reacts to the system’s drastic degradation and provides a cost-effective PM schedule in real time. The rest of this paper is organized as follows. Section II develops the cost effective USPM model and proposes a scheme for real-time maintenance decision making. Section III presents a numerical study based on industrial data. Finally, Section IV gives conclusions and potential research directions.

Fig. 1. Degradation process.

In (1), the prediction of conditional system reliability is based is on the forecasted system performance variable. defined as condition-based system reliability function at time , given system performance from to , where ; is defined as probability density function of at time given from to . In (1), is discrete since values of a performance variable are usually discrete. Lu et al. [15] further remarked that approaches the continuous case and acts similarly as traditional when . Consequently, we have reliability function

(2) is condition-based system hazard rate funcwhere given system performance from to . When tion at is large, we could approximate by treating as a constant in period , and then (2) becomes

II. USPM POLICY

(3)

A. Condition-Based System Reliability In CBM, performance variables that affect system health are usually monitored [7]. The variables usually include current, vibration, acoustics, temperature, pressure, etc. A well selected performance variable should ideally exhibit a monotonic trend as the system gradually deteriorates and finally exceeds a critical level (failure threshold) when the system is considered failed [10], [28]. This critical level is used to define system failure and can be determined based on industrial standards or operation experience [8]. Note that value of a performance variable may exhibit an increasing/decreasing trend depending on the type of the performance variable and the degradation. The way in which a performance variable evolves (increasing case) due to system’s deterioration from initialization to failure is schematically shown in Fig. 1. Assume a system has one performance variable at time given , which also has one failure mode defined in terms of the by . Then, the conditional system reliability over critical level estimated at time could be assessed by [15], an interval [16] (1)

A simple derivation of (3) yields (4) Note that

(5) Several useful forecasting techniques may facilitate the implementation of the condition-based system reliability. For real-time modeling, a self-generating and self-updating forecasting method is generally preferred. Lu et al. [15] pointed out that exponential smoothing is a simple and effective approach for real-time performance data modeling and short-term forecasting, which is able to capture system dynamics without lags. Lu et al. [16] used an integrated random walk (IRW) model, coupled with Kalman filtering, to recursively model and predict system performance. When the forecasting step length is relatively long and enough time is given for offline analysis, autoregressive moving average (ARMA), and related techniques

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YOU et al.: COST-EFFECTIVE UPDATED SEQUENTIAL PREDICTIVE MAINTENANCE POLICY

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provide effective forecasting methods [22]. Recently, the match matrix method has been proposed [14] for accurate long-term prediction at expense of computation effort. In addition to the above forecasting methods, there are many other data-driven methods for performance forecasting, for example, recurrent neural network (RNN), discussed in [4] and multiresolution wavelet models, discussed in [24]. B. USPM Modeling Fig. 2. Timing in the USPM model.

In this section, we develop the USPM policy that utilizes condition-based system reliability information. In this maintenance policy, the system’s operating condition (performance variable ) and lifetime distribution are simultaneously considered to provide a cost-effective updated PM schedule within one replacement cycle in real time. The detailed descriptions of the proposed USPM policy, as well as its underlying assumptions are given as follows. 1) The system is continuously monitored and the monitoring does not affect the system’s performance. 2) The objective is to optimally schedule PM work orders for a continuously monitored degrading system in real time in order to achieve minimum overall MCR, which is a commonly used objective [23], [27]. , when PM actions have 3) At the current time been completed at time , the PM. The remaining system has operated for after the PM work orders in the replacement cycle are scheduled at time , where is the expected number of PM work orders after time in is system operating time one replacement cycle, PM and the uncompleted PM between the (estimated at ), the last PM is the replacement and ends the cycle. For clarity, the timing in the USPM policy is shown in Fig. 2. 4) The system reliability is predicted based on the value of the in the present operation cycle performance variable . The operaduring the time tion cycle is defined as the system operating time between the PM, where . the PM), the lifetime 5) After the upcoming PM (the distribution of the system is used to estimate the system reliability. Because it is difficult to accurately predict system performance variable after a PM, this technique provides a general and reliable estimate of the system performance after a PM. 6) The effect of imperfect PM is evaluated and quantified by the hybrid model [13], [25], that is, the hazard rate of the PM system based on its life-time distribution after the is given by

age reduction factor can be estimated from the historical data and are not a function of the current time , where . Note that , and . Note that perfect maintenance is a special case when and . 7) The system undergoes minimal repair at failures between replacements. Minimal repair implies that the system is set to be operational without changing its hazard rate. The practicality of this assumption has been recognized since 1960s [2]. 8) After replacement, the system is assumed to be in a “new” condition, and planning time (also shown in Fig. 2) is set back to zero. , if In the proposed maintenance policy, at time , the expected maintenance cost of a system for the remaining time in the replacement cycle consists of 1) replacement cost: ; , 2) imperfect PM cost: where is imperfect PM cost per repair; 3) minimal repair cost (minimal CM cost) is composed of contributions from two main sources. • Expected CM cost based on predicted system operating condition for the remaining time in the present operation cycle

(7)

where is minimal repair cost per repair. We support the interpretation of by analogy with that of [29]. • Expected CM cost based on lifetime distribution and hyPM brid imperfect PM model after the

(8) (6) , and is the system’s effective age where PM, which we will discuss in immediately prior to the more detail later. The hazard rate increase factor and the

where , . Then, the overall CM cost is given by


(9)

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where

(10) are the actual and predicted effective ages of the system. Based on the above three kinds of cost, the expected overall maintenance cost for the remaining system operating time in the is replacement cycle estimated at time

system remaining operating time. Based on (11)–(15), we have (16) shown at the bottom of the page. , Note that in the expression of is estimated based on the forecasted system performance as shown in (1). For comparison, the maintenance variable models based only on lifetime distribution can be referred to [13], [20], and [25]. As the system state is continuously changing, the USPM model defined in (16) needs to be continuously updated based on the newest known system state and thereby determine an optimal PM schedule. , the USPM model in (16) is updated When as

(11) Note that when

, (11) becomes (12)

When , the expected time period from until the replacement estimated at consists of two parts. • The remaining system operating time in the present opera. tion cycle: PM • The remaining operation cycles after the

Based on the above periods, the expected remaining system operating time in the replacement cycle is (13) Then, using (13) with (10), we obtain

(17) Based on (16) and (17), the USPM model and the updating scheme are formulated. C. Model Optimization In the USPM model described in (16), the effective system and the estimated number of PM work orage: are treated as decision variders after ables. Therefore, the objective is to find the optimal values of and that minimize the MCR. For clarity in the rest of paper, we write , , unless otherwise noted. Additionally, we disin its differentiable case. cuss to be a constant and differFor our method, we first set entiate with respect to , then loop over . This method is common in maintenance optimization based on lifetime distribution as in [13] and [25]. can be directly found from The optimal solution when (12) and (15), that is, we set

(14) Note that when

, (14) reduces to

(18) By the quotient rule, we obtain

(15) Thus, the MCR for the remaining system operating time in the replacement cycle estimated at can be computed as the ratio between the expected maintenance cost and the expected

(19)

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Since (19) is an implicit function of , where both terms on the left hand side (LHS) are predicted based on system performance variable and the right hand side (RHS) is a constant, that satisfies (19) can be found. Then, the optimal value is the MCR when

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is in fact the unique soludetermined by (23). Therefore, . tion that is of practical interest and is denoted as Finally, we discuss the practicality of the condition given in (24). The condition in (24) implies

(26)

(20) Thus, if

s.t.

, we have the following results. In the case that , set the derivative of (16) with respect to For the case equal zero. This implies

(21) For the case , of (16) with respect to

, set the derivative equal zero. This implies

(22) , set the derivative of (16) with respect For the case to equal zero. This implies (23) Three theorems are derived to find the optimal solutions for with a fixed . Additionally, we discuss the physical interpretation and the practicality of the conditions. exists for a Theorem 1: The condition under which given For a given , if satisfying

(27) Then, the condition in (24) is satisfied. In a general sense, the LHS of (27) reflects the system health based on performance , the RHS of (27) reflects the system health based variable on lifetime distribution. A physical interpretation of a situation in which (27) is satisfied could be that the system is degrading much faster than another imaginary system whose condition-based reliability is ideally the same as that based only on the lifetime distribution. This condition is possible in real applications especially near the failure of a specific system when the system is much degraded and rapidly approaching failure. Theorem 2: The condition under which , exists and is unique for a given For a given , the solution for which satisfies (22) exists if and is strictly increasing. In addition, the solution is unique if is and is strictly increasing. differentiable, , when , we have Proof: Existence: If (28) If

is strictly increasing, we have (29)

(24) then the optimal solution exists. , the conProof: For the effective age ditional reliability of the system is 1 because the system is still operating. Therefore, since it is impossible for the system to fail . This implies at that moment, we have from (21)

When

, we have

(30) Therefore, by the intermediate value theorem, satisfying (22). is differentiable, and is Uniqueness: When strictly increasing, taking the derivative of the LHS of (22) with respect to , we obtain

(25) Based on the results shown in (24) and (25), the intermediate value theorem implies that with s.t. (21) is satisfied. To address uniqueness, assume the solution to (21) is not unique, that is, which satisfies (21). However, from a practical point of view, a PM work order should be scheduled at , which will prevent potential failure most effectively without increasing the MCR

(31) That is, the LHS of (22) is an strictly increasing function of . Therefore, the solution of (22) is unique. Finally, we discuss the practicality of the conditions. The conmeans that the hazard rate increase factor dition should be smaller than the reciprocal of the age reduction factor . The practicability of the condition lies in the fact that PM tends to improve the system’s working condition and the


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effect of PM is reflected by and . The smaller the values and , the better the improvement. In the following, it is of . assumed that PM is good enough so that Theorem 3: The condition under which exists and is unique is differentiable and strictly increasing, the solution of If exists and is unique. (23) with respect to Proof: Substituting (16) into (23) and expending, we have

(32) When

, we have (33)

Taking the derivative of LHS of (32) with respect to we obtain

,

(34) Fig. 3. Optimization algorithm.

Note must hold for a PM schedule to have physical meaning (see Fig. 2). We see that LHS of (32) and smaller than the is a strictly increasing function of RHS of (32) when . Therefore, the solution exists and is unique. of (23) with respect to Based on (21)–(23) and theorems 1–3, an overall optimization algorithm is proposed to find the optimal PM schedule at . In the optimization algorithm, we take the initial time when , value to find the optimal solution then we gradually increase . The termination criterion in computation is set as a small value so that the error in the optimal PM schedule is negligible. After finding the optimal solutions , for a given , the optimal number of remaining PM work oris determined to minimize the ders in the replacement cycle MCR based on (23). More details about the algorithm is provided in Fig. 3. Note that the optimization algorithm is still valid , provided that the USPM model is updated when accordingly. Following the optimization procedures in Fig. 3, the optimal that minimize the MCR solutions estimated at time

can be obtained. When the next recommended PM is far in , the system is allowed to the future, i.e. continue operating without performing a PM. Then, the USPM for a small interval model is updated at time as shown in (17). In addition, the optimization procedures to find the subsequent opare repeated at time PM will be performed when timal PM schedule. The , where denotes the number of small time intervals elapsed until a PM is recommended. Replacement is necessary in two cases. 1) . 2) The recommended remaining PM work orfor ders are too frequent, that is . After the PM, the USPM model will continued to be updated as shown in (16) and (17) to schedule the remaining PM work orders. The scheme for real-time PM decision making is summarized in Fig. 4. From the scheme for real-time maintenance decision making proposed in the last section, the next scheduled PM time is . From (21), we see chosen in real time based on , is determined by that for a given comparing two functions and .



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Fig. 5. Condition-based cdf of system failure. (Solid line: conditional cdf of system failure surviving to hole 37; dashed line: conditional cdf of system failure surviving to hole 38). TABLE I NECESSARY INFORMATION FOR IMPLEMENTING USPM POLICY

Fig. 4. Scheme for real-time PM decision making.

The condition-based hazard rate function of the system which reflects the degradation state of a specific system: . The hazard rate function based on the system’s lifetime distribution and the effect of imperfect PM, which reflects the expected mean state of the whole population: . Therefore, the factor that determines the upcoming PM schedule is the deviation of the system’s actual degradation state from the expected degradation state. When the system exhibits very high degradation rates, that is, increases much faster is so small that than , and a PM work order should be performed.

III. NUMERICAL CASE STUDY To demonstrate the condition-based system reliability in predictive maintenance scheduling and real-time maintenance decision making, the thrust force of an in-operation drill bit subject to an excessive wear failure mode is considered as performance variable. The data was obtained from [15]. Exponential smoothing is used as an online data modeling and forecasting approach. The condition-based system reliability of a drill bit, given survival to hole 37 and hole 38, respectively, has been estimated in [15]. The condition-based cumulative distribution function (cdf) of system failure is shown in Fig. 5. For this study, we assume that no prior PM work orders have been performed. Therefore, we study the condition-based system reliability in the first operation cycle. For the remaining operation cycles before replacement, the hazard rate function

is chosen based on a Weibull lifetime distribution with shape parameter and scale parameter as (35) Given the condition-based of system failure and the Weibull parameters shown in Fig. 5 and (35), the cost parameand ) and the improvement factors ( and ) ters ( , should be known in order to compute the optimal PM schedule. and For the cost parameters, only the cost ratios need to be known, as shown in (32). The improvement factors can be estimated from historical data. The cost ratios and the improvement factors are selected as suggested by [25] to illustrate the model and results presented earlier in this paper. and are chosen using typical values. For Furthermore, clarity, in Table I, we summarize all the necessary information for implementing USPM policy and the corresponding illustrative equations in the paper. All the parameters used in this case study are shown in Table II. Using the condition-based of system failure and given parameters, the expected MCR given that the drill bit survives until hole 37 and hole 38 have been drilled is minimized using the proposed optimization algorithm (where ). Tables III and IV show the optimal number of PM work orders


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TABLE II PARAMETERS IN USPM MODEL

TABLE IV UPDATED OPTIMAL PM SCHEDULE AT HOLE 38

TABLE III OPTIMAL PM SCHEDULE ESTIMATED AT HOLE 37

and the corresponding PM schedule estimated at hole 37 and then updated at hole 38 with respect to different ratios. Note that the unit of in Tables III and IV is the number of drilled holes. The results in Tables III and IV show that the optimal number of PM work orders in one replacement cycle increases as increases. Therefore, it is beneficial to conduct more imperfect PM work orders when replacement is more expensive. In addi, which tion, we always have coincides with the remarks made in [13]. As mentioned in [20], it would be reasonable to do frequent PM with age, but it would be better to do the last PM as late as possible because the system should be replaced at the last PM. Furthermore, the comparison between the results in Tables III and IV shows that there is no significant difference between the recommended optimal PM schedules (especially time to the next PM work order). This is because the condition of the system does not change significantly between model updates, as illustrated by the two condiof system failure with similar shapes shown in tion-based Fig. 5. It illustrates that the proposed USPM model yields optimal PM schedules that are consistent with the change in the system state. All of the first PM schedules in Tables III and IV allow the system continue operating without performing a PM (for example, for hole 37 with , we have ). Therefore, the proposed USPM model should be examined to determine whether it is able to deal with rapid degradation of the system and suggest a PM action. To test this, a condition-based of system failure that increases much faster than the cdfs in Fig. 5 is simulated and shown in Fig. 6, by assuming that the drill bit survives to hole 41. The corresponding first PM schedule with respect to ratios is listed in Table V. different

Fig. 6. Simulated condition-based cdf of system failure.

TABLE V OPTIMAL FIRST PM SCHEDULE AT HOLE 41

Each case in Table V recommends immediate PM action and validates that the USPM model quickly reacts to drastic deterioration of the system state and provides a cost-effective optimal PM schedule. IV. CONCLUSION AND FUTURE WORK Compared with traditional CBM decision-making methodologies that only consider the equipment lifetime distribution without state update, in this paper, a cost-effective updated sequential predictive maintenance (USPM) policy is proposed to decide a real-time PM schedule for continuously monitored degrading systems that will minimize MCR in the long term, considering the effect of imperfect PM. In the USPM policy, the maintenance model is continuously updated based on the change of the system state to decide upon a cost-effective PM schedule in real time. Mathematical analysis of the USPM model demonstrates the existence and uniqueness of an optimal PM schedule under practical conditions. The results from a numerical study based on industrial data validate that 1) the



USPM model yields optimal PM schedules that are consistent with the change in the system state and 2) the USPM model quickly reacts to drastic degradation and provide a PM schedule in real time. For future work, it is necessary to benchmark the effectiveness of the proposed policy with other maintenance polices based only on lifetime distributions as a systematic experiment. In addition, we may investigate incorporating the effect of uncertain critical levels for the performance variable. Uncertain critical levels could be used to define system failure in case that a clear understanding of the failure boundary is unavailable. REFERENCES [1] R. E. Barlow and F. Prochan, Mathematical Theory of Reliability. New York: Wiley, 1965. [2] R. E. Barlow and L. C. Hunter, “Optimum preventive maintenance policies,” Oper. Res., vol. 8, no. 1, pp. 90–100, Feb. 1960. [3] J.-K. Chan and L. Shaw, “Modeling repairable systems with failure rates that depend on age and maintenance,” IEEE Trans. Reliab., vol. 42, no. 4, pp. 566–570, Dec. 1993. [4] J. T. Connor, R. D. Martin, and L. E. Atlas, “Recurrent neural networks and robust time series prediction,” IEEE Trans. Neural Networks, vol. 5, no. 2, Mar. 1994. [5] R. Dekker, “Applications of maintenance optimization models: A review and analysis,” Reliab. Eng. Syst. Saf., vol. 51, no. 3, pp. 229–240, Mar. 1996. [6] L. Dieulle, C. Berenquer, A. Grall, and M. Roussignol, “Sequential condition-based maintenance scheduling for a deteriorating system,” Eur. J. Oper. Res., vol. 150, no. 2, pp. 451–461, Oct. 2003. [7] N. Gebraeel, M. A. Lawley, R. Li, and J. K. Ryan, “Residual-life distributions from component degradation signals: A Bayesian approach,” IIE Trans., vol. 37, no. 6, pp. 543–557, Jun. 2005. [8] N. Gebraeel, “Sensory-updated residual life distributions for components with exponential degradation patterns,” IEEE Trans. Autom. Sci. Eng., vol. 3, no. 4, pp. 382–393, Oct. 2006. [9] A. Grall, L. Dieulle, C. Berenquer, and M. Roussignol, “Continuoustime predictive maintenance scheduling for a deteriorating system,” IEEE Trans. Reliab., vol. 51, no. 2, pp. 141–150, Jun. 2002. [10] R. Huang, L. Xi, X. Li, L. C. Liu, H. Qiu, and J. Lee, “Residual life predictions for ball bearings based on self-organizing map and back propagation neural network methods,” Mech. Syst. Signal Process., vol. 21, no. 1, pp. 193–207, Jan. 2007. [11] V. Jayabalan and D. Chaudhuri, “Cost optimization of maintenance scheduling for a system with assured reliability,” IEEE Trans. Reliab., vol. 41, no. 1, pp. 21–25, Mar. 1992. [12] H. Liao, E. A. Elsayed, and L.-Y. Chan, “Maintenance of continuously monitored degrading systems,” Eur. J. Oper. Res., vol. 175, no. 2, pp. 821–835, Dec. 2006. [13] D. Lin, M. J. Zuo, and R. C. M. Yam, “Sequential imperfect preventive maintenance models with two categories of failure modes,” Nav. Res. Logist., vol. 48, no. 2, pp. 172–183, Mar. 2001. [14] J. Liu, D. Djurdjanovic, J. Ni, N. Casoettlo, and J. Lee, “Similarity based method for manufacturing process performance prediction and diagnosis,” Comput. Ind., vol. 58, no. 6, pp. 558–566, Aug. 2007. [15] H. Lu, W. J. Kolarik, and S. Lu, “Real-time performance reliability prediction,” IEEE Trans. Reliab., vol. 50, no. 4, pp. 353–357, Dec. 2001. [16] S. Lu, Y.-C. Tu, and H. Lu, “Predictive condition-based maintenance for continuously deteriorating systems,” Qual. Reliab. Eng. Int., vol. 23, no. 1, pp. 71–81, Feb. 2007. [17] W. Q. Meeker and L. A. Escobar, Statistical Methods for Reliability Data. New York: Wiley, 1998. [18] R. K. Mobley, An Introduction to Predictive Maintenance. New York: Butterworth-Heinemann, 1989. [19] T. Nakagawa, “Periodic and sequential preventive maintenance policies,” J. Appl. Probab., vol. 23, no. 2, pp. 536–542, Jun. 1986. [20] T. Nakagawa, “Sequential imperfect preventive maintenance policies,” IEEE Trans. Reliab., vol. 37, no. 3, pp. 295–298, Aug. 1988. [21] D. G. Nguyen and D. N. P. Murthy, “Optimal preventive maintenance policies for repairable systems,” Oper. Res., vol. 29, no. 6, pp. 1181–1194, Dec. 1981. [22] S. M. Pandit and S.-M. Wu, Time Series and System Analysis With Applications. New York: Wiley, 1983. [23] H. Pham and H. Wang, “Imperfect maintenance,” Eur. J. Oper. Res., vol. 94, no. 3, pp. 425–438, Nov. 1996.

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[24] O. Renaud, J.-L. Starck, and F. Murtagh, “Wavelet-based combined signal filtering and prediction,” IEEE Trans. Syst., Man, Cybern. B, vol. 35, no. 6, pp. 1241–1251, Dec. 2005. [25] S. EI-Ferik and M. Ben-Daya, “Age-based hybrid model for imperfect preventive maintenance,” IIE Trans., vol. 38, no. 4, pp. 365–375, Apr. 2006. [26] N. J. M. Van, M. R. Cooke, and M. Kok, “A Bayesian failure model based on isotropic deterioration,” Eur. J. Oper. Res., vol. 82, no. 2, pp. 270–282, Apr. 1995. [27] H. Wang, “A survey of maintenance policies of deteriorating systems,” Eur. J. Oper. Res., vol. 139, no. 3, pp. 469–489, Jun. 2002. [28] W. Wang, “A model to determine the critical level and the monitoring intervals in condition-based maintenance,” Int. J. Prod. Res., vol. 38, no. 6, pp. 1425–1436, Apr. 2000. [29] X. Zhou, L. Xi, and J. Lee, “Reliability-centered predictive maintenance scheduling for a continuously monitored system subject to degradation,” Reliab. Eng. Syst. Saf., vol. 92, no. 4, pp. 530–534, Apr. 2007. Ming-Yi You received the B.S. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, where is currently pursuing the Ph.D. degree in the State Key Laboratory of Mechanical System and Vibration. He was a visiting student of S. M. Wu Manufacturing Research Center, University of Michigan, Ann Arbor, from 2007 to 2008. His research interests include reliability modeling and testing, fault diagnosis and prognosis, and maintenance scheduling.

Lin Li received the B.S. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2001, and the M.S.E. degree in mechanical engineering, the M.S.E. degree in industrial and operations engineering, and the Ph.D. degree in mechanical engineering from the University of Michigan, Ann Arbor, in 2003, 2005, and 2007, respectively. Currently, he is an Assistant Research Scientist in the Department of Mechanical Engineering, University of Michigan. His research interests are in manufacturing system modeling, performance analysis and prediction, and mass customization; reliability engineering; quality control methods; intelligent maintenance systems; condition monitoring and fault diagnosis and prognosis.

Guang Meng received the Ph.D. degree in vibration engineering from the Northwestern Polytechnical University, Xi’an, Shaanxi, China, in 1988. He is currently a “Cheung Kong” Chair Professor and the Dean of the School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China. His research interests include vibration analysis and control, nonlinear vibration, MEMS dynamics, rotor dynamics, and fault diagnosis.

Jun Ni received the B.S. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 1982, and the M.S. and Ph.D. degrees in mechanical engineering from the University of Wisconsin, Madison, in 1984 and 1987, respectively. Currently, he is Shien-Ming (Sam) Wu Collegiate Professor of Manufacturing Science, and a Professor in the Department of Mechanical Engineering, University of Michigan, Ann Arbor. His research interests are in manufacturing process modeling, analysis and prediction; precision engineering and metrology; cutting tool development, quality control methods, intelligent maintenance systems, monitoring, and fault diagnosis.
