Imperfect Maintenance Policy Considering Positive and Negative

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Imperfect Maintenance Policy Considering Positive and Negative Effects for Deteriorating Systems With Variation of Operating Conditions Yunxia Chen, Member, IEEE, Wenjun Gong, Dan Xu, Member, IEEE, and Rui Kang, Member, IEEE

Abstract— This brief develops a degradation-based imperfect maintenance policy considering both positive effect and negative effect for a deteriorating system with variations of operating conditions. The proposed method improves the positive effect on the imperfect maintenance, which further considers the impact of resource on the distribution function of the positive effect after each maintenance process. The positive effect induces the system state into a random interval which is related to maintenance resource applied into each preventive maintenance action and actual maintenance times. Meanwhile, a new negative deteriorating effect model for describing the impacts of imperfect maintenance actions is established. The new model considers that the degradation rate increases after each imperfect maintenance process; meanwhile, it includes the impacts of variations of operating condition bringing to the degradation rate which is described as a distribution subjected to stress as well. Therefore, a condition-based adaptive maintenance policy is applied for a deteriorating system. The optimal maintenance cost allocation and maintenance threshold are determined by maximizing an availability function. Finally, a numerical example is illustrated to demonstrate the application of our maintenance model and policy. Note to Practitioners—This paper was motivated by the problem of computing an optimal commonly agreeable maintenance policy for a deteriorating system to make it running at the longest life span at the limited maintenance cost budget. Consider maintenance effect with maintenance resource, maintenance times, and some other uncontrollable factors to establish a reasonable maintenance schedule for a deteriorating system. The existing approach is to describe the imperfect effect with just assuming that the deterioration level after each preventive maintenance action can be decreased into a fixed variable interval without any distinguished difference. It is thus highly desirable for a deteriorating system with such various operation conditions to have a mechanism that would take into account each maintenance resource arrangement and various stress levels. To come up with a reasonable maintenance model, in this brief, we just assume that maintenance effects are divided into two terms: a positive effect model which induces the system state into a random interval which is related to per maintenance cost which is described as a power function and actual maintenance times. Meanwhile, a negative deteriorating effect model is established for describing the impacts of imperfect maintenance actions on the degradation rate with an increasing trend just like the accelerating process Manuscript received March 31, 2016; revised July 9, 2016 and November 16, 2016; accepted February 23, 2017.This brief was recommended for publication by Associate Editor M. Deng and Editor M. P. Fanti upon evaluation of the reviewers’ comments. This work was supported by the National Science Foundation of China under Grant 51675025 and Grant 61403010. (Corresponding author: Dan Xu.) The authors are with the School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China. (e-mail: gongwenjun@ buaa.edu.cn; [email protected]; [email protected]; kangrui@ buaa.edu.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASE.2017.2675405

after each maintenance action. This maintenance model can solve the problem involved maintenance source arrangement. The users can develop an optimal maintenance schedule in advance. It would be updated in real time based on the change in environmental stress level. Index Terms— Deteriorating system, imperfect maintenance, negative effect, optimization, positive effect.

I. I NTRODUCTION With the development of monitoring and analysis of degradation data, the importance of degradation-based maintenance (DBM) is amplified in many engineering fields, such as fatigue cracks, vibration features, wear, corrosion, voltage, oil concentration, or some other degradation mechanisms in real time [1], [2]. Traditional preventive maintenance (PM) models of DBM are based on the hypotheses that system can be improved as good as new or as bad as old. However, these two extreme cases can only be applicable to structurally simple systems, or highly complex systems [3]. Realistically, equipment after maintenance actions generally stays at a condition between them which we call it imperfect maintenance [4]. Virtual age model and improvement factor model are the two main traditional utilized models to describe PM imperfect effects. The first model assumes that PM reduces system’s physical age; the latter one assumes that PM changes the hazard rate proportionally [5]. For detailed discussions on these various models, Shafiee and Chukova [6] provided a classification scheme for maintenance models based on viewpoints of customer and manufacturer. Wang and Pham [7] made a clear discussion about maintenance policies for different systems. However, these approaches based on the knowledge of the statistical information cannot be suitable for DBM modeling for the reason of limitation of depicting actual system’s condition [8]. The existed literature for describing deterioration process of DBM models involved with PM imperfect effect can be concluded into two aspects: 1) positive effect and 2) negative effect. The positive effect describes per PM action’s positive impact on degradation process to guarantee systems normal operation. Wang and Pham [9] treated positive effect of PM by lifting the degradation threshold proportionally. It is confined to the field of speculation. In [3], [6], and [10]–[13], different degradation models are built to depict deterioration mechanisms, including Gamma process [3], [10], [11], [13] and continuous Markov process [12]. They suppose that deterioration level after PM can restore in a better state which is determined by a random variable with constant distribution functions, such as Beta distribution [10] or truncated normal distribution [3]. Basically, they just assume that each

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maintenance cost is constant without considering effect of maintenance cost applied into each PM action. Then, Nourelfath et al. [14] presumed the PM cost can be modeled as a linear function of the production’s age and the return product’s age. Aghezzaf et al. [15] and You and Meng [16] built a linear function of PM cost and hazard rate decrement after each imperfect repair. From a practical point of view, Do et al. [3], [12] established a PM cost model for DBM considered as a function of the degradation improvement factor. However, PM impact on degradation process might be affected by human factor, environment fluctuating, or other uncontrollable factors, which determine positive effect to be uncertain. The objective of this paper is to propose a new positive effect model, for depicting degradation-based PM imperfect effect. Moreover, negative effect about PM actions is concerned about adverse influence on degradation process. Some scholars supposed that system could be imperfectly maintained for an infinite number of times. Actually, in some cases [11], [17], [18], systems can be maintained for only some limited number of times due to negative effect. Such as limited availability [19], or the minimal working time of system [10]. In [10] and [27], negative effect is treated by lifting the initial deterioration level after each PM action proportionally. Do et al. [3], [12] considered negative effect of PM on the speed of degradation process with a cumulative function. These methods neglect the variations of operation conditions. Actually, modern systems are requested to be available for a wide range of evolving environment (EE). Naval vessels have to go through multivariate environment conditions (tropical, temperate, or boreal zones; and seas with different salinity). Fighter jets must cover full range of operational requirements. As mentioned in [20], the vibration intensity of the rollers of a guiding system accelerates the roller deterioration due to chipping. Besides, gas turbine is found to exit flouring phenomenon on the surface of the compressor blade which is determined by variation of humidity and salinity [21]–[23]. It can similarly affect deteriorating evolution process of system performance. However, the existing related work [24], [25] neglected the effect of various environmental conditions on deterioration speed or level and actual imperfect state about PM. Thus, the new maintenance model should consider EE’s impact on deterioration rate. Therefore, this paper proposes a degradation-based imperfect maintenance model with both positive effect and negative effect for a continuously monitored degrading system. See Table I for contribution of different authors. The rest of the paper is organized as follows. Section II is devoted to the descriptions of the related assumptions and the characters of the system to be considered. The deterioration model is also described. In Section III, positive effect of PM was described as the random variable of degradation decrement at each PM, considering maintenance cost effect and uncertainty. Then, negative effect was simulated by a degradation acceleration parameter. It can describe the negative impact of PM actions on actual deterioration speed of system. In addition, a random variable determined by a power function is introduced to reflect EE effect during each episode between PM actions.

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TABLE I C ONTRIBUTION OF D IFFERENT AUTHORS

Section IV presents an optimal PM policy that is to obtain maximal availability of the repairable system at the limited affordable maintenance cost and minimal requests for each episode of working time between each PM action and total duration time. Then, a numerical example has been introduced in Section V. Finally, Section VI presents the conclusion drawn from this paper. II. S YSTEM D ESCRIPTION AND A SSUMPTIONS A. Assumptions and Notations Without any repair or replacement, the evolution of the system deterioration is assumed to be strictly increasing. It should satisfy the following assumptions. 1) The state of the system X t degrading to failure can be stimulated by a stochastic process (X t , t), and the initial state X 0 = 0 indicates that the system is in a new state. 2) The system is monitored continuously and perfectly, which can reveal the true state of system instantaneously. 3) The failure threshold of the system can be seen as the deterioration level L. 4) The PM maintenance actions are assumed to be imperfect. They cannot bring the system to the good-as-new state, except corrective replacement. 5) After i th PM, the system should be guaranteed to stay at the working state for at least a duration time Tmin before next PM action. B. Deterioration Modeling Assume that the degradation process of systems is stochastic Gamma process with continuous time, which is a stochastic process with independent nonnegative increments having a gamma distribution with identical scale parameter u (u > 0) and a varying shape parameter v > 0 [3]. Considering the PM maintenance actions, the probability density function of the deterioration process X t of the system between the i th and the (i + 1)th maintenance actions can be given by f v i (t −s),u (x) = Ga(x|v i · (t − s), u).

(1)


Fig. 1.

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Fig. 2. Illustration of cost–benefit function with different value of parameter a.

Illustration about choosing different random variables.

And the expectation and variance of the stochastic process X (t) can be obtained by vi · t μ X (t ) = E[X (t)] = u vi · t 2 σ X (t ) = Var[X (t)] = 2 . (2) u Generally, the imperfect PM maintenance action can affect the evolution of system’s deterioration process. In this paper, we assume that wi = v i /u represents the mean deterioration speed of the system between the i th and the (i + 1)th maintenance actions. After each imperfect PM, the value of wi is changing to reflect the effect of imperfect PM. And it will be described in the next section. However, after a corrective replacement action, the system becomes as good as new. It means that the degradation level is reset to zero, and the deterioration behavior with time return to the initial speed w0 = v 0 /u. III. I MPERFECT M AINTENANCE M ODEL D ESCRIPTION A. Positive Effect About Maintenance Actions Given that degradation state before i th PM is X (PM− i ), a new state of system after i th PM is X (PM+ ). It will at least i return a better state than the maintenance threshold γs . That is, X (PM+ i ) falls randomly in the interval, [0, γs ] as shown in Fig. 1. From [1], a truncated normal distribution is given as x i −μi 1 σ σi · I[0,γs ] (x) gμi ,σi ,γs (x i ) = (3) 0−μi i − γs −μ σi σi where I[0,γs ] (x) = 1 if 0 ≤ x i ≤ γs , and I[m i ,ni ] (x) = 0 otherwise. √ (ξ ) = 1/ 2πexp(−(1/2)ξ 2 ) is the probability density function of the standard normal distribution x i = X (PM+ i ), μi = γs · [1 − exp(−i μ/ci )] and σi = σ. Note that, μ > 0 and σ > 0 are constants, referred to as the rectification average and variance respectively, which are assumed to be independent of γs · i represents PM times, ci (ci ≥ 1) is the cost factor that mainly reflects the impact of maintenance cost on i th PM. ci is defined as ci = (CPMi /Cth )a + εCPMi

(4)

where a(a > 0) is constant determined by actual engineering data. CPMi represents the i th PM cost. Cth is the imperfect PM cost threshold which means that each actual PM cost should overpass the value of Cth . ε RCPM denote the impact i of other uncontrollable factors. We suppose that ε RCPM obeys i to the normal distribution that can be described as ε RCPM ∼ i N(o, σε ). From (5), we can see that the value of a has a great influence on cost function. And different kinds of maintenance functions can be found depending on the value of a. As shown in Fig. 2, it can be divided into these three cases. 1) When 0 < a < 1, the cost function is a concave function, it means that it will cost more to counterbalance maintenance action times. 2) When a = 1, the value of CPMi /Cth is proportional to ci . 3) When a > 1, the cost function is a convex function: maintenance cost increases less than counterbalance maintenance action times. B. Negative Effect About Maintenance Actions Based on [3] and [23], the negative effect and environment evolving effect has been considered in our model. Assume the shape parameter of i th PM cycle is v i , the shape parameter of (i − 1)th PM cycle is v i−1 . Considering the actual observation point is a group of time series data, the linear mixed effect model are chosen to describe the cumulative effect about imperfect maintenance actions which can be obtained as wi = ρ · wi−1 + d(¯si , s¯i−1 )

(5)

where ρ(ρ > 1) is the cumulative effect factor, it represents the impact of imperfect actions on the evolution deterioration of the system. d(¯si , s¯i−1 ) represents the error term considering the effect caused by the difference between the environment or working stress level before PM action and after PM action. s¯i−1 is the synthetic stress level after (i − 1)th PM action, s¯i is the synthetic stress level after i th PM action. Suppose that d(¯si , s¯i−1 ) obeys the normal distribution described as d(¯si , s¯i−1 ) obeys to the normal distribution N(μd , σd ). The impact of difference between the two working stresses on the system degradation level can be described by



a separate power function which is defined as ⎧ s¯i ϕ ⎪ ⎪ −1 (¯si ≥ s¯i−1 ) ⎪ ⎪ ⎨ s¯i−1 μd = ⎪ s¯i −ϕ ⎪ ⎪ ⎪ (¯si ≥ s¯i−1 ) 1 − ⎩ s¯i−1

(6)

where s¯i /¯si−1 is the stress ratio factor, ϕ, ϕ (ϕ, ϕ > 1) are nonnegative real numbers. In this equation, it is assumed that the synthesis stress level is measurable. Generally, the value of s¯i , s¯i−1 will be easy to be obtained if there is only one working stress parameter. However, the environment or working stresses that the system has to face with are very complex. Mostly, we need to consider a parameter vector to indicate real working space. And several methods can be used to evaluate the stress level at which system works after each PM action. Traditional methods in this regard include analytic hierarchy process, fuzzy evaluation, or expert scoring method [29]. These methods in some cases are quite practical, while sometimes it is not quite ideal owing to the lack of rigorous mathematical derivation process, or excessive doping subjective factors.

From [10], the expectation of Mi which represents the time required to perform the i th maintenance can be described by E(Mi ) = αγs exp(iβγs )

(7)

where α > 0, β ≥ 0 are constants which can be estimated by fitting exponential distribution to maintenance times. And we can concluded that E(Mi ) ≥ E(M j ) for any i > j ≥ 1. IV. O PTIMAL M AINTENANCE P OLICY

max Availability of system ⎧ ⎪ 0 < γs < L s ⎪ ⎪ N+1 ⎪ ⎪

⎪ ⎪ ⎪ TPMi ≥ T0 ⎪ ⎨Ttotal = E i=1 subject to ⎪ E(TPMi ) ≥ Tmin ; i = 0, 1, 2, . . . , N − 1 ⎪ ⎪ ⎪ N ⎪

⎪ ⎪ ⎪ CPMi ≤ C S . ⎪ ⎩

expected total uptime . expected total uptime + downtime (8)

Suppose that the expected uptime Ttotal and downtime Mtotal in a cycle including N PM actions are Ttotal = E[TPM1 + TPM2 + · · · + TPM N + TPM N+1 ]

The second constraint ensures that the system will not be removed from service nor be replaced before a period of time, Ttotal, has elapsed. The third constraint is an index to tell the consumers that we need to make a replacement if the duration time TN+1 after the Nth PM action cannot reach the minimal request for operating duration time, Tmin . Note that, under the Gamma process model, the unconditional expectation of intermaintenance time, Ti can be expressed as ⎞ ⎛

γs ∞ ut, γs v−x ⎠ · f + (x)dtd x. ⎝1 − E[Ti ] = X PMi−1 (ut) 0 0 B. Optimization Algorithm

The proposed imperfect maintenance policy is to maximize the system availability as described in [10]. It is defined as

E(Mi )

(9)

where TPMi represents the duration time before i th PM, TPM N+1 is the last duration time before replacement; the value of Mtotal can be calculated by (8). Tr represents the time needed for a replacement. Thus, the achieved availability is As =

The optimal maintenance policy can be obtained by solving the following optimization problem:

(11)

A. Formulation of the Optimization Problem

Mtotal = E[M1 + M2 + · · · + M N ] =

Illustration of degradation behavior and the proposed maintenance

i=1

C. Maintenance Time Model

Availability of system =

Fig. 3. policy.

E[T P M1 + T P M2 + · · · + T P M N + T P M N+1 ] . (10) E[M1 + M2 + · · · + M N ] + Tr

We now describe the optimization algorithm that obtains the optimum PM cost allocation plan, over the range (0, Cs ] that maximizes availability during the system life span. And the optimization algorithm can be described as follows (see Fig. 3). 1) Start with a small value of γs within the range (0, L s ]. 2) Choose the value of CPMi . 3) Calculate every inter-maintenance time, E(TPMi ) and E(Mi ). 4) Search and set N as the number of maintenance actions at which the average duration time constraint, Tmin , is first violated. N+1 TPMi ) if 5) Calculate the total service time, E( i=1 N+1 E( i=1 TPMi ) < T0 go to step 2, otherwise go to next. 6) Adjust each CPMi by changing maintenance action methN ods with a small increment unless i=1 CPMi ≥ Cs and repeat steps 2–5. 7) Choose that maximizes N+1 the feasible CPMi E( i=1 TPMi ).


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TABLE II D ATA OF C OSTS AND I MPACT OF I MPERFECT A CTIONS

Fig. 4.

Structure of the actuator for illustration.

V. N UMERICAL E XAMPLE OF M AINTENANCE M ODEL In this section, an example of actuator will be introduced to show how the proposed maintenance policy can be used in maintenance optimization considering imperfect effect. The actuator consists of the comprehensive controller, feedback lever and actuating cylinder (see Fig. 4). The actuator is a repairable component that adjusts aircraft aileron angular deviation into an affordable range. Because of continuous working condition, the failure modes of actuator can be detected directly, and the key failure mechanism we know is the wear evolution of the spool valve. From the observation of output performance level, we can find that the actuator develops an evident deteriorating process which is related to the working stress level. Based on the operation and maintenance field data, the degradation process of the actuator can be governed by a gamma process with the parameters u = 4 and v = 4. Maintenance time can be described by exponential distribution with parameters α = 0.02 and β = 0.05. The replacement action is implemented when the degradation measurement crosses the threshold, L s = 20. Besides, the actuator needs to be maintained when it is upon maintenance threshold γs = 16, and be replaced if the short-run time after PM, Tmin = 3.5 cannot be sustained based on actual operation data. In addition, the rectification average and variance of the maintenance action are assumed to be μ = 0.5 and σ 2 = 0.05, respectively. The coefficient a of the PM cost function [see (5)] satisfies a = 1 from the knowledge of service man. Then, the option of each PM cost can be classified in three choices: 1) low level that each PM cost equals 10; 2) medium level that each PM cost equals 20; and 3) high level that each PM cost equals 30. It means that minimal PM cost expenditure Cth = 10, depending on maintenance site conditions. The total cost of PM actions should not exceed the cost limit that C S = 60 in this case. Besides, the synthetic stress levels after each PM action in this case are assumed to be equal which is Sa , it means that μd = 0. And the value of σd is assumed to be 0.05. The aggregation of parameters of maintenance model is presented in Table II. If we set the initial deterioration level is zero, then, the optimal decision about maintenance actions can be calculated through comparison of different maintenance policies. Table II illustrates the system availability with different maintenance policy. The calculation process of first maintenance policy (10, 10) for illustration can be described as follows. The initial duration time T1 before first PM is 5.1 (5.1 > Tmin ), After first PM action with maintenance cost 10, the duration time T2 of actuator becomes 4.1 (4.1 > Tmin ). Then, after

Fig. 5.

System deterioration process with optimal PM actions (30, 30). TABLE III

S YSTEM AVAILABILITY W ITH D IFFERENT M AINTENANCE P OLICIES

choosing the second PM policy that maintenance cost is 10, the duration time T3 changes to be 3.2 (3.2 < Tmin ). Thus, the actuator is stopped with a replacement action. By analog, other maintenance policies can be calculated as in Fig. 3. The optimization algorithm is an enumeration method, considering the amount of calculation is not great. With options of each PM cost increasing, the calculation shows the geometric progression. Then, other optimal optimization algorithms should be considered to deal with different combinations of objectives and constraints. From the result of Table III, the optimal maintenance policy is (30, 30). It means that cost of each PM action is 30; there are only two maintenance actions are taken during the life span of system. Actual deterioration process with the optimal PM actions can be simulated as shown in Fig. 5. Comparing with other maintenance policies, the high level maintenance policy taken at the first PM point is wiser to acquire higher availability that can bring actual improvement of the initial duration time of system.



TABLE IV

TABLE V

O PTIMAL M AINTENANCE P OLICY W ITH A G IVEN M AINTENANCE T HRESHOLD

O PTIMAL M AINTENANCE P OLICY W ITH D IFFERENT C OMBINATIONS OF S TRESS L EVELS

are excluded because the total service time less than the required 12 units in time. B. Sensitivity Analysis to the Environmental Stress Variety

Fig. 6. Relationship between maintenance threshold and total duration time.

Maintenance cost allocation scheme is necessary to be considered to assist policy makers to provide a reasonable maintenance policy. The method in [10] about availability-objective maintenance policy with a short-run availability constraint provides a general scheme for maintenance action without consideration of PM cost effect. However, with requirements increasing, affordability is essential for consumers and users. A reasonable PM cost allocation scheme can be provided based on our method in this section, which supplies more information for policy makers preparing the actual repair plans. In addition, a sensitivity analysis of the proposed maintenance policy with respect to the threshold of imperfect maintenance within a life cycle is studied in the following section. A. Sensitivity Analysis to the Imperfect Maintenance Threshold The proposed method in this paper strongly depends on their related maintenance threshold, which is characterized by γs . Table IV reports the optimum maintenance policy and the optimum availability of system for different values of γs . Fig. 6 shows the evolution trend of the system availabilities as maintenance threshold varies. The results shows optimum maintenance policy is the same as (30, 30). The optimum policy for our case study is obtained as γs = 13, and As = 0.79. And we can see that maintenance threshold has an obvious effect on availability of system. The reason is that maintenance threshold both influence maintenance time and duration time after maintenance. It induces to a balance situation in this case. It also gives the total service time for various maintenance thresholds, and the cases when γs < 12

From the result of Table III, we can see that optimal maintenance policy about cost arrangement is (30, 30), and As equals 0.774. This entire outcome is based on the assumption that environmental or working stress level is identical during the life span of system. In this section, we suppose that system will go through three different stress levels which are represented by Sa , Sb , Sc . They satisfy equations that Sa = 1.1Sb , Sb = 1.1Sc . Other parameters are the same as those shown in Table II. Different combinations of these stress levels will have an impact on availability of system. Table V just presents the results about sensitivity analysis of the environmental stress. It can be seen that availability of system is sensitive to the deviation of stress level during the duration time between each maintenance actions. The best arrangement for operating conditions is (Sc , Sa , Sb ). It means that the stress level of duration time before the first PM action is Sc ; then the second is Sa ; and the third is Sb . Besides, the data from Table V also tell us that initial working condition should be the least severe working environment to increase availability of system. Compared with the method in [3], our method takes into account the environment stress effect to acquire more practicable information about availability or life cycle of equipment, which can help to make a reasonable usage allocation scheme. VI. C ONCLUSION In this paper, a degradation-based imperfect maintenance strategy for a deteriorating system under multiple operating conditions is proposed. Full consideration of the impact of the four aspects about imperfect PM modeling has been included as shown in Table I. The new positive effect model considering PM cost effect contributes to PM cost allocation scheme. The new negative effect model on the degradation speed of the system includes the item that describes impact of operating condition which is presented by a power function. Compared with previous processing modes of imperfect maintenance actions, the method in this paper is useful for engineers and developers to implement degradation-based PM. The optimal maintenance policy in this paper with the purpose of system availability is a more utility performance measure in comparison of the cost objective function which was usually proposed


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