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A Dynamic Prognostic Maintenance Policy for Multi-Component Systems
2nd IFAC Workshop on Advanced Maintenance Engineering, Services and Technology Universidad de Sevilla, Sevilla, Spain. November 22-23, 2012
A Dynamic Prognostic Maintenance Policy for Multi-Component Systems Adriaan Van Horenbeek* Liliane Pintelon**
*Department of Mechanical Engineering, KU Leuven, Leuven, Belgium (Tel: +32 16 32 24 97; e-mail: [email protected]). ** Department of Mechanical Engineering, KU Leuven, Leuven, Belgium (e-mail: [email protected]) Abstract: Over the past years, diagnosis and prognosis in maintenance are getting more attention. The use of these techniques for maintenance decision making and optimization in multi-component systems is however a still underexplored area. Consequently, this paper presents a dynamic prognostic maintenance policy for multi-component systems that minimizes the long-term expected maintenance cost per unit time. The proposed maintenance policy is classified as a dynamic method, as each time new information on the current degradation of components becomes available; a prediction of remaining useful life (prognostics) for the considered component is performed and the maintenance schedule is updated accordingly. The performance of the developed dynamic prognostic maintenance policy is compared to several maintenance policies in order to quantify the value of using prognostics in maintenance decision making for multi-component systems. Moreover, the effect of different levels of component dependency on the decision making process is investigated. Keywords: Maintenance, Reliability Analysis, Maintenance optimization, Prognostic maintenance.
setting maintenance policies is illustrated by Camci (2009) and Van Horenbeek and Pintelon (2011).
1. INTRODUCTION The complexity of industrial equipment is ever increasing, which introduces many interdependencies between the components. Neglecting these interdependencies when scheduling maintenance actions leads to inefficient maintenance (e.g. higher costs and downtime). A multicomponent and system approach needs to be taken in maintenance optimization models. Nicolai & Dekker (2007) give an overview of optimal maintenance policies for multicomponent systems based on the dependence between components (stochastic, structural or economic). However, no models that use prognostic information or a prediction of remaining useful life (RUL) are mentioned. This is striking as the use of prognostics in maintenance is increasing over the past years (Jardine et al., 2006, Lee et al., 2006). Therefore, the link between prediction algorithms and decision making based on the resulting remaining useful life distributions should be established.
Although, already some literature exists on multi-component condition-based and prognostic maintenance optimization, the added value of prognostic information in maintenance decision making for multi-component systems considering different levels of dependency between the components is never quantified in the existing literature. In previous studies the dependency is assumed to exist or not, partial dependence has never been considered. The aim of this paper is to develop a prognostic maintenance policy, which builds further on the research performed by Wildeman et al. (1997) and Bouvard et al. (2011), for a multi-component system considering different levels of dependencies between the components (economic, structural and stochastic dependence as defined by Nicolai & Dekker (Nicolai and Dekker, 2007)). The objective is to minimize the long-term expected maintenance cost. The prognostic maintenance policy performance is compared, for the same system but with different dependencies between the components, to a blockbased policy and a condition-based maintenance control limit policy in order to quantify the added value of prognostic information in maintenance optimization for multicomponent systems.
Bouvard et al. (2011) introduce a dynamic condition-based maintenance planning model, where the groups of maintenance operations are rescheduled at each decision moment. A multi-component systems approach for conditionbased maintenance optimization is taken by Tian & Liao (2011) where economic dependence between components exists. An optimal degradation threshold is found by Barata et al. (2002) to minimize the total system cost for on condition maintenance. Yang et al. (2008) schedule maintenance based on the predicted machine degradation by taking into account the complex interaction between components, production process and maintenance operations. The advantage of using prognostic information over threshold 978-3-902823-17-5/12/$20.00 © 2012 IFAC
Section 2 of this paper describes the degradation and maintenance model. The developed prognostic maintenance policy is discussed in Section 3 and Section 4 elaborates on the component dependencies in the considered multicomponent system. A numerical example is given in Section 5. Finally, the conclusions and future work are stated in Section 6.
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2. DEGRADATION AND MAINTENANCE MODEL
this paper by a beta distribution with parameters γ and δ. In this way imperfect maintenance or replacement is included in the developed model.
Consider a system with n non-repairable components, where at failure or preventive maintenance an imperfect replacement is performed. A failure of a component causes the entire system to stop and a failure is noticed immediately without any inspection. Spare parts are assumed to be available whenever they are needed.
3. PROGNOSTIC MAINTENANCE POLICY In order to take the interdependencies between components in a multi-component system into account, grouping of maintenance actions should be considered to find an optimal maintenance policy. Therefore, the presented prognostic maintenance policy is based on a dynamic policy for grouping maintenance activities (Wildeman et al., 1997). This dynamic policy is extended by including prognostic information on the component remaining useful life (Bouvard et al., 2011). One specific preventive or corrective maintenance action can be performed on each component i of the system. A preventive maintenance action has a component-dependent cost ci,p and a system-dependent or setup cost S. A corrective maintenance intervention has a component-dependent cost ci,c and a set-up cost S. The cost S is independent on the performed action and the number of actions at the same time (i.e. economic dependence). The component-dependent cost ci depends on the time-to-failure Ti,F of the considered component:
2.1 Degradation Model The component degradation is characterized by a physical variable Di with i = {1,...,n}, where Di(t), t>0 is a stationary gamma process with shape parameter ν and scale parameter μ and the following properties (van Noortwijk, 2009):
Di(0) = 0 Di(t) has independent increments For t>0 and h>0, Di(t+h) – Di(t) follows a gamma distribution A component i is said to fail when the degradation Di exceeds the failure threshold Di,failure. This deterioration failure threshold Di,failure is, opposed to most of the used degradation models in literature, modelled as a random variable. For each t≥0, the probability of failure in time interval (0,t) can then be written as the convolution integral (Abdel-Hameed, 1975):
PrX t Y
x 0
f X t x PrY xdx x f X t x f Y y dy dx.
ci t ci , p , t Ti , F
(1)
ci t ci ,c , t Ti , F The objective is to group maintenance activities to reduce the maintenance cost (total set-up cost). This means when m maintenance operations are performed at their individual optimal times ti*on the considered planning horizon PH, the cost equals:
x 0 y 0
Where X(t) = Di(t) and has a gamma distribution with shape parameter ν and scale parameter μ, and Y = Di,failure has probability density function fY (y) . The random variable Di,failure which models the random failure threshold, is modelled by a Weibull probability distribution with shape parameter α and scale parameter β. Based on the current degradation level Di(t), the failure p0robability function Fi(t) is computed by stochastic simulation of the degradation process over time. Each time new information on Di(t) - e.g. by inspection - becomes available a prognosis of the remaining useful life is made. This prognosis is used in the presented prognostic maintenance policy.
m
C1 ci ti* m S . i 1
(3)
The possibility of grouping maintenance activities in order to reduce the maintenance cost over PH is considered by defining a group of activities as a subset of {1,…,n}. A partition of {1,…,n} is a collection of mutually exclusive groups G1,…,Gj, which cover all activities. Finally, a grouping structure is defined as a partition of {1,…,n} such that all activities within each considered group are jointly
2.2 Imperfect maintenance
* executed at time t G , which is defined as the optimal j
Each maintenance action, corrective or preventive, reduces the degradation level of component i by a factor (1-B), 0≤B≤1, of the total degradation at the time of replacement. B is considered as the improvement factor, when B=1 a minimal replacement is conducted, when B=0 a perfect replacement is performed. The degradation of component i after a replacement, either corrective or preventive, is:
maintenance execution time of group Gj. To determine the maintenance cost when grouping maintenance activities, a grouping structure GSk is defined with u groups of maintenance actions Gj, j {1,...,u} on PH. In this way the grouping maintenance cost is defined as:
Di 0 B Di min Ti ,M ; Ti , F . i1,..., n
u
C 2 ci tG* j u S . j 1iG j
(2)
For each group Gj of n components a cost CGj is saved:
CG j n 1 S ci tG* j ci ti* .
Here Ti,M is the time to preventive replacement and Ti,F is the time-to-failure of component i. The improvement factor has a probability density distribution f (b) which is modelled in
iG j
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(4)
(5)
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Where n 1 S are the savings by grouping n maintenance actions and
dCi * (t i Di (t )) 0. (7) dt where the empty sum in (6) equals zero and the empty product equals one. In (6) ci,p is the component dependent preventive maintenance cost and bi = ci,c – ci,p, with ci,c the component dependent corrective maintenance cost. Both ci,p and ci,c include the cost of downtime. t is the age at which preventive maintenance is performed, ti,p is the downtime due to a preventive maintenance action and ti,c is the downtime caused by a corrective action. pil is the probability that component i survives the next period given that its age equals l at the beginning of the current period and qil = 1-pil. In order to determine the long-term optimal maintenance time ti,l*, with corresponding cost Ci,l*, for each component Di(t) = 0. Short-term optimization to determine the optimal maintenance time of the next maintenance action ti*, with cost Ci*, is based on the prognosis of the failure time distribution which results from the degradation Di(t) at the current time t and the degradation model described in Section 2.1. Based on this short-term prognostic information the long-term maintenance schedule is adapted.
is the additional cost of shifting
ci t*G j
ci t*i
maintenance activity i from the individual optimal time ti* to the optimal group maintenance time t*G j . The prognostic maintenance policy aims at finding the grouping structure that minimizes the maintenance cost C2 on PH. 3.1 Prognostics, estimation of remaining useful life The proposed prognostic maintenance policy is considered as a dynamic maintenance policy, as every time new information on the observed degradation of a component becomes available, the prediction of the remaining useful life of the component is updated. The degradation model described in Section 2.1 is used to predict remaining useful life for each component n, based on the current degradation Di(t). The dynamic or adaptive scheduling of maintenance actions is based on the updated failure probability distribution (i.e. remaining useful life prediction) when new information on the current degradation Di(t) of component i becomes available (e.g. by inspection). In this paper continuous updating of the failure probability distribution is considered, which is possible when component degradation is monitored continuously. First, an optimal maintenance schedule on an infinite horizon is determined. This maintenance schedule is updated on the planning horizon PH, each time a new prediction of the failure probability distribution based on the current degradation Di(t) is made.
Notice that (6) is in fact a discretised version of an agereplacement strategy, but as far as the lifetime distribution of the components is concerned knowledge of the sequence
pli
t 1 l 0
suffices. It is irrelevant whether the lifetimes
themselves have a discrete or continuous probability distribution (van der Duyn Schouten and Vanneste, 1990). 3.3 Penalty functions
3.2 Individual maintenance optimization
When grouping maintenance actions, some components their useful life and failure probability will be increased, while others their useful life will be decreased. In order to define the effect of shifting maintenance actions from their optimal times, penalty functions are constructed. A penalty function hi defines the expected additional cost of shifting the maintenance time from the optimal maintenance time ti* for a component. Penalty functions for both the next optimal maintenance time ti*, based on the short-term information, as for the nth (n > 1) maintenance occurrence, based on the longterm optimal maintenance time ti,l*, are defined. The penalty function, by adopting a long-term shift (Wildeman et al., 1997), for the first maintenance action on component i is defined as (Dekker et al., 1996):
For each individual component an infinite-horizon maintenance optimization model is formulated to find the optimal long-term maintenance time ti,l* for each component. ti,l* represents the maintenance time at which the long-term expected maintenance cost (Ci) on an infinite horizon is minimal. An average use of the components is assumed and the dependencies and interactions between the components are neglected at this stage. In this way, the savings from joint execution of maintenance activities are ignored. This decomposition approach allows the scheduling of many components (Dekker et al., 1996). For an age-based replacement policy the asymptotic cost, where ti,l* is the minimizing argument, is given by van der Duyn Schouten and Vanneste (1990). In this paper this is extended to include non-zero downtimes into the optimization problem as follows:
t 1 ci , p S bi 1 pil l 0 (6) C i t Di t t 1 t j 2 t 1 1 pil pil ti , p 1 pil ti ,c j 2 l 0 l 0 l 0 The optimal maintenance time ti* for a component with degradation Di(t) is deduced from the following equation:
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j 1 ai t 1 j * l qi bi Di t Ci Di t pi l ai j ai if t 0 hi t*i t a 1 . j 1 i Ci* Di t qij bi Di t pil j ai t l ai t if t 0
(8)
According to the long-term shift rule the execution interval of the first maintenance action is changed according to the prognostic short-term information, while all future maintenance intervals remain ti,l*, the long-term optimal
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maintenance time. The penalty function for the nth maintenance action, with n > 1, on component i becomes:
QG j tG* j G j 1 S H G* j .
(13)
If the savings Q are positive, the group G is cost effective,
Gj j , which means it is better to group the maintenance actions j 1 k v ai 1 j * l rather than performing them at their optimal individual times C q b D t 0 p , i ,l i i i i * j ai t t l ai t v k 0 i . The final objective is to find the grouping structure GSk * (9) hi t i t that minimizes the total maintenance cost on the planning 0, k 0 a t 1 j 1 horizon PH. The grouping algorithm developed by Wildeman i q j b D t 0 C * p l , k 1 i ,l i i i i et al. (Wildeman et al., 1997) is used heuristically to find the j ai l ai optimal grouping structure GSk. A constraint is added to this
with v the floor of ((n-1).ti,l*) (Bouvard et al., 2011). When a component i fails, the penalty function hi of the failed component is defined as:
algorithm, as it is not allowed to group two maintenance actions on the same component (Bouvard et al., 2011).
0, t 0 hi ti , F . , t 0
3.5 Maintenance execution and planning update (10)
This means, when a component i fails, preventive maintenance actions on the other components can be performed during the downtime due to the failure of component i. Due to this assumption, opportunistic maintenance is thus included in the model. 3.4 Maintenance activities grouping The aim is to group the maintenance activities on the planning horizon PH in order to minimize the maintenance cost on this planning horizon. The finite planning horizon is defined as:
PH max max ti* ti*,l , i . i1,..., n i
4. COMPONENT DEPENDENCIES
(11)
The parameter εi is defined as the prognostic horizon for component i. The prognostic horizon is the time between two consecutive predictions of remaining useful life of component i, based on newly available component degradation (e.g. by inspection). Each time the prognostic information is updated; this information is used to schedule maintenance actions on the prognostic horizon, as no new information on component degradation becomes available before a time εi. The prognostic horizon εi is assumed to be one time unit in this paper (i.e. continuous updating of component degradation information and corresponding prognosis).
On failure of one of the considered components it is possible that this breakdown influences the deterioration of other components resulting in secondary damage. Three different maintenance or replacement scenarios exist when a corrective maintenance action needs to be performed. In the first maintenance scenario only replacement of the primary failed component is necessary, in the second maintenance scenario replacement of both primary failed component and one secondary damaged component is required, while in the third maintenance scenario replacement of the whole subassembly is necessary due to secondary damage to all components. All corrective maintenance scenarios are initiated by failure of one of the components. In this way stochastic dependence between components is taken into account. The corrective maintenance scenarios are sampled from a multinomial distribution:
maintenance activities on components i Gj are all performed at time tGj instead of their individual optimal times ti*. The optimal maintenance time tGj* of the group is derived by the following equation:
When considering multi-component systems, three major categories of dependencies exist. These are stochastic, structural and economic dependence (Nicolai and Dekker, 2007). These three types of dependencies are also considered in the constructed maintenance model. 4.1 Stochastic dependence
Grouping of maintenance activities on PH can be done by using the defined penalty functions in (8), (9) and (10). Define H Gj t Gj the penalty function of group Gj when
H G j tG* j H G* j min hi t . t iG j
Based on the previous step a maintenance schedule on the planning horizon PH is constructed. Maintenance actions are executed according to the maintenance schedule. A rolling horizon approach is considered as each time the planning horizon is shifted and the maintenance schedule is updated by including newly available information on component degradation and the corresponding remaining useful life. In this way a dynamic and adaptive, since it is based on the currently available predictive information deduced from the component degradation, prognostic maintenance policy is developed.
(12)
x1 xk f ( x; n, p) (n! /( x1!,...,xk !))( p1 ,..., pk ), when ik1 xi n
(14)
where x = (x1,…,xk) gives the number of each of k outcomes in n trials of a process with fixed probabilities p = (p1,…,pk) of individual outcomes in any one trial. The vector p has nonnegative integer components that sum to one. The vector p
The savings QGj by grouping maintenance operations i Gj and executing them at time tGj* can be calculated as follows:
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defines the probabilities of the replacement or failure scenarios for corrective maintenance (pc = (p1 = 0.85, p2 = 0.1, p3 = 0.05)) actions. This means that for corrective maintenance 85% of the actions consist of only replacing the primary failed component, 10% consists of replacing both primary failed component and one secondary damaged component, and in 5% of the cases a replacement of the entire subassembly is necessary.
component. The second policy is a condition-based control limit policy. Preventive maintenance on a component i is performed when the degradation Di(t) exceeds a degradation control limit Di,maint. The degradation Di(t) is continuously monitored over time. The reason to compare the prognostic maintenance policy to these two maintenance policy is the fact that both maintenance policies perform well under certain circumstances. The condition-based maintenance policy performs very well when no dependencies between components exist, while the block-based maintenance policy performs well when high dependencies between components are present. This will also be shown in the results.
4.2 Economic and structural dependence In order to be able to determine the performance of the proposed prognostic maintenance policy when considering different levels of dependence between the components, a dependence parameter α is introduced. This parameter α reflects the advantage of performing maintenance on multiple components at once compared to maintenance on a single component, in other words it affects the set-up cost S by adapting the savings QGj (see (13)) when grouping maintenance as follows:
QG j tG* j G j 1 S H G* j .
Table 1. Component degradation parameters Component n
νi
μi
αi
βi
γi
δi
1
2,00
1
100
20
0,2
3
2
0,40
5
100
3
0,2
3
3
0,32
5
100
3
0,2
3
Table 2. Cost and time parameters
(15)
The dependence parameter α is assumed to incorporate the effect of both economic and structural dependence between the components in the considered system. α ranges from 0 to 1, where α = 0 means no economic and structural dependence, while α = 1 means maximal economic and/or structural dependence between the components.
ci,p
ci,c
twait(h)
treplace(h)
tinst(h)
tsecD(h)
€
€
μ
σ
μ
σ
μ
σ
μ
σ
605
5805
10
1
3
0,5
3
0,5
10
1
665
5865
10
1
3
0,5
3
0,5
10
1
475
5675
10
1
3
0,5
3
0,5
10
1
5.3 Results
5. NUMERICAL EXAMPLE A numerical example is presented in order to validate the developed prognostic maintenance policy and to compare its performance to a block-based maintenance policy and a condition-based control limit maintenance policy. This is performed for the same system but with different dependencies between the components (Section 4.2).
The long-term average cost per unit time of the prognostic maintenance policy is compared to an optimal block-based and condition-based control limit maintenance policy for different values of the dependence parameter α. The results can be seen in Figure 1.
5.1 Input parameters As an example a three component system is considered. The component degradation parameters, as described in detail in Section 2.1, are given in Table 1. The corresponding cost and time parameters for all components are shown in Table 2. twait stands for the waiting time, treplace for the actual replacement time, tinst for the installation time and the start-up time of the system and finally tsecD stands for the time to repair secondary damage. All these parameters determine the downtime due to preventive maintenance ti,p(treplace, tinst) and corrective maintenance ti,c(twait, treplace, tinst, tsecD). The cost of working (70€/h), cost of transportation (120€) and downtime cost (200€/h) are also considered in the numerical example. 5.2 Maintenance policy comparison Fig. 1. Long-term average maintenance cost per unit time in relation to the dependence α for the considered maintenance policies.
The prognostic maintenance policy is compared to two other maintenance policies. The first one is a block-based maintenance policy, where a component i is replaced every ti time units, independent of the failure history of the 119
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As foreseen the condition-based maintenance policy performs very well when no or low dependencies are present between components. This is the case because the real degradation of each component is monitored separately, and when this degradation reaches the control limit, maintenance is performed regardless the state of the other components. This is also the reason why the cost of this policy stays the same regardless the dependencies between components. On the other hand, the block-based maintenance policy outperforms the condition-based maintenance when considering high interdependencies. This is because the optimal block-based maintenance policy reduces to a group-based maintenance policy, which takes advantage of grouping maintenance actions, from the moment α reaches 0.5.
REFERENCES Abdel-Hameed, M. (1975). A Gamma Wear Process. IEEE Transactions on Reliability, volume (24), 152-153. Barata, J., Soares, C. G., Marseguerra, M. and Zio, E. (2002). Simulation modelling of repairable multi-component deteriorating systems for 'on condition' maintenance optimisation. Reliability Engineering & System Safety, volume (76), 255-264. Bouvard, K., Artus, S., Bérenguer, C. and Cocquempot, V. (2011). Condition-based dynamic maintenance operations planning & grouping. Application to commercial heavy vehicles. Reliability Engineering & System Safety, volume (96), 601-610. Camci, F. 2009. System maintenance scheduling with prognostics information using genetic algorithm. IEEE Transactions on Reliability, volume (58), 539-552. Dekker, R., Wildeman, R. E. and Van Egmond, R. (1996). Joint replacement in an operational planning phase. European Journal of Operational Research, volume (91), 74-88. Jardine, A. K. S., Lin, D. and Banjevic, D. (2006). A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mechanical Systems and Signal Processing, volume (20), 1483-1510. Lee, J., Ni, J., Djurdjanovic, D., Qiu, H. and Liao, H. (2006). Intelligent prognostics tools and e-maintenance. Computers in Industry, volume (57), 476-489. Nicolai, R. P. and Dekker, R. (2007). Optimal Maintenance of Multi-component Systems: A Review. Complex System Maintenance Handbook. Springer London. Tian, Z. and Liao, H. (2011). Condition based maintenance optimization for multi-component systems using proportional hazards model. Reliability Engineering & System Safety, volume (96), 581-589. Van Der Duyn Schouten, F. A. and Vanneste, S. G. (1990). Analysis and computation of (n, N)-strategies for maintenance of a two-component system. European Journal of Operational Research, volume (48), 260-274. Van Horenbeek, A. and Pintelon, L. (2011). Optimal prognostic maintenance planning for multi-component systems. In: Proceedings of the European Safety and Reliability Conference: ESREL 2011, 910-917. CRC Press/Balkema, Troyes, France. Van Noortwijk, J. M. (2009). A survey of the application of gamma processes in maintenance. Reliability Engineering & System Safety, volume (94), 2-21. Wildeman, R. E., Dekker, R. and Smit, A. C. J. M. (1997). A dynamic policy for grouping maintenance activities. European Journal of Operational Research, volume (99), 530-551. Yang, Z. M., Djurdjanovic, D. and Ni, J. (2008). Maintenance scheduling in manufacturing systems based on predicted machine degradation. Journal of intelligent manufacturing, volume (19), 87-98.
The prognostic maintenance policy performs at least as good as or better than the other considered maintenance policies. In fact, the prognostic policy combines the advantages of both other maintenance policies. On the one hand, it considers the real component degradation by continuously updating the remaining useful life of the components, while on the other hand it takes into account the possible advantage of grouping maintenance activities due to the dependencies in the system. The prognostic maintenance policy is able to react to different degradation patterns and dependencies between components while always guaranteeing an optimal policy. 6. CONCLUSIONS A prognostic maintenance policy for multi-component systems considering different levels of component dependence is presented in this paper. It is a dynamic and adaptive policy as the maintenance schedule is always updated on a finite horizon based on the current component degradation and deduced prognostic information. The prognostic maintenance policy is able to handle complex maintenance problems, as the problem defined includes components with different degradation patterns, random failure thresholds, imperfect maintenance, non-zero downtimes and all types of dependencies (i.e. stochastic, structural and economic dependence). The performance (i.e. long-term average cost per unit time) of the prognostic maintenance policy is compared to two other maintenance policies (i.e. block-based preventive and condition-based control limit maintenance policy). The results show that the prognostic maintenance policy combines the advantages of both maintenance policies, which causes the prognostic maintenance policy to outperform the other maintenance policies. The prognostic maintenance policy guarantees optimal maintenance for different degradation patterns and dependencies between components in a multicomponent system. Future work will be on the analysis of the effect of several parameters of the model (e.g. imperfect maintenance) on the added value of the use of prognostic information in maintenance optimization and decision making. Moreover, the inclusion of an inventory policy and production schedule into the prognostic maintenance policy are interesting research subjects that will be covered. 120
A Dynamic Prognostic Maintenance Policy for Multi-Component Systems Adriaan Van Horenbeek* Liliane Pintelon**
*Department of Mechanical Engineering, KU Leuven, Leuven, Belgium (Tel: +32 16 32 24 97; e-mail: [email protected]). ** Department of Mechanical Engineering, KU Leuven, Leuven, Belgium (e-mail: [email protected]) Abstract: Over the past years, diagnosis and prognosis in maintenance are getting more attention. The use of these techniques for maintenance decision making and optimization in multi-component systems is however a still underexplored area. Consequently, this paper presents a dynamic prognostic maintenance policy for multi-component systems that minimizes the long-term expected maintenance cost per unit time. The proposed maintenance policy is classified as a dynamic method, as each time new information on the current degradation of components becomes available; a prediction of remaining useful life (prognostics) for the considered component is performed and the maintenance schedule is updated accordingly. The performance of the developed dynamic prognostic maintenance policy is compared to several maintenance policies in order to quantify the value of using prognostics in maintenance decision making for multi-component systems. Moreover, the effect of different levels of component dependency on the decision making process is investigated. Keywords: Maintenance, Reliability Analysis, Maintenance optimization, Prognostic maintenance.
setting maintenance policies is illustrated by Camci (2009) and Van Horenbeek and Pintelon (2011).
1. INTRODUCTION The complexity of industrial equipment is ever increasing, which introduces many interdependencies between the components. Neglecting these interdependencies when scheduling maintenance actions leads to inefficient maintenance (e.g. higher costs and downtime). A multicomponent and system approach needs to be taken in maintenance optimization models. Nicolai & Dekker (2007) give an overview of optimal maintenance policies for multicomponent systems based on the dependence between components (stochastic, structural or economic). However, no models that use prognostic information or a prediction of remaining useful life (RUL) are mentioned. This is striking as the use of prognostics in maintenance is increasing over the past years (Jardine et al., 2006, Lee et al., 2006). Therefore, the link between prediction algorithms and decision making based on the resulting remaining useful life distributions should be established.
Although, already some literature exists on multi-component condition-based and prognostic maintenance optimization, the added value of prognostic information in maintenance decision making for multi-component systems considering different levels of dependency between the components is never quantified in the existing literature. In previous studies the dependency is assumed to exist or not, partial dependence has never been considered. The aim of this paper is to develop a prognostic maintenance policy, which builds further on the research performed by Wildeman et al. (1997) and Bouvard et al. (2011), for a multi-component system considering different levels of dependencies between the components (economic, structural and stochastic dependence as defined by Nicolai & Dekker (Nicolai and Dekker, 2007)). The objective is to minimize the long-term expected maintenance cost. The prognostic maintenance policy performance is compared, for the same system but with different dependencies between the components, to a blockbased policy and a condition-based maintenance control limit policy in order to quantify the added value of prognostic information in maintenance optimization for multicomponent systems.
Bouvard et al. (2011) introduce a dynamic condition-based maintenance planning model, where the groups of maintenance operations are rescheduled at each decision moment. A multi-component systems approach for conditionbased maintenance optimization is taken by Tian & Liao (2011) where economic dependence between components exists. An optimal degradation threshold is found by Barata et al. (2002) to minimize the total system cost for on condition maintenance. Yang et al. (2008) schedule maintenance based on the predicted machine degradation by taking into account the complex interaction between components, production process and maintenance operations. The advantage of using prognostic information over threshold 978-3-902823-17-5/12/$20.00 © 2012 IFAC
Section 2 of this paper describes the degradation and maintenance model. The developed prognostic maintenance policy is discussed in Section 3 and Section 4 elaborates on the component dependencies in the considered multicomponent system. A numerical example is given in Section 5. Finally, the conclusions and future work are stated in Section 6.
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this paper by a beta distribution with parameters γ and δ. In this way imperfect maintenance or replacement is included in the developed model.
Consider a system with n non-repairable components, where at failure or preventive maintenance an imperfect replacement is performed. A failure of a component causes the entire system to stop and a failure is noticed immediately without any inspection. Spare parts are assumed to be available whenever they are needed.
3. PROGNOSTIC MAINTENANCE POLICY In order to take the interdependencies between components in a multi-component system into account, grouping of maintenance actions should be considered to find an optimal maintenance policy. Therefore, the presented prognostic maintenance policy is based on a dynamic policy for grouping maintenance activities (Wildeman et al., 1997). This dynamic policy is extended by including prognostic information on the component remaining useful life (Bouvard et al., 2011). One specific preventive or corrective maintenance action can be performed on each component i of the system. A preventive maintenance action has a component-dependent cost ci,p and a system-dependent or setup cost S. A corrective maintenance intervention has a component-dependent cost ci,c and a set-up cost S. The cost S is independent on the performed action and the number of actions at the same time (i.e. economic dependence). The component-dependent cost ci depends on the time-to-failure Ti,F of the considered component:
2.1 Degradation Model The component degradation is characterized by a physical variable Di with i = {1,...,n}, where Di(t), t>0 is a stationary gamma process with shape parameter ν and scale parameter μ and the following properties (van Noortwijk, 2009):
Di(0) = 0 Di(t) has independent increments For t>0 and h>0, Di(t+h) – Di(t) follows a gamma distribution A component i is said to fail when the degradation Di exceeds the failure threshold Di,failure. This deterioration failure threshold Di,failure is, opposed to most of the used degradation models in literature, modelled as a random variable. For each t≥0, the probability of failure in time interval (0,t) can then be written as the convolution integral (Abdel-Hameed, 1975):
PrX t Y
x 0
f X t x PrY xdx x f X t x f Y y dy dx.
ci t ci , p , t Ti , F
(1)
ci t ci ,c , t Ti , F The objective is to group maintenance activities to reduce the maintenance cost (total set-up cost). This means when m maintenance operations are performed at their individual optimal times ti*on the considered planning horizon PH, the cost equals:
x 0 y 0
Where X(t) = Di(t) and has a gamma distribution with shape parameter ν and scale parameter μ, and Y = Di,failure has probability density function fY (y) . The random variable Di,failure which models the random failure threshold, is modelled by a Weibull probability distribution with shape parameter α and scale parameter β. Based on the current degradation level Di(t), the failure p0robability function Fi(t) is computed by stochastic simulation of the degradation process over time. Each time new information on Di(t) - e.g. by inspection - becomes available a prognosis of the remaining useful life is made. This prognosis is used in the presented prognostic maintenance policy.
m
C1 ci ti* m S . i 1
(3)
The possibility of grouping maintenance activities in order to reduce the maintenance cost over PH is considered by defining a group of activities as a subset of {1,…,n}. A partition of {1,…,n} is a collection of mutually exclusive groups G1,…,Gj, which cover all activities. Finally, a grouping structure is defined as a partition of {1,…,n} such that all activities within each considered group are jointly
2.2 Imperfect maintenance
* executed at time t G , which is defined as the optimal j
Each maintenance action, corrective or preventive, reduces the degradation level of component i by a factor (1-B), 0≤B≤1, of the total degradation at the time of replacement. B is considered as the improvement factor, when B=1 a minimal replacement is conducted, when B=0 a perfect replacement is performed. The degradation of component i after a replacement, either corrective or preventive, is:
maintenance execution time of group Gj. To determine the maintenance cost when grouping maintenance activities, a grouping structure GSk is defined with u groups of maintenance actions Gj, j {1,...,u} on PH. In this way the grouping maintenance cost is defined as:
Di 0 B Di min Ti ,M ; Ti , F . i1,..., n
u
C 2 ci tG* j u S . j 1iG j
(2)
For each group Gj of n components a cost CGj is saved:
CG j n 1 S ci tG* j ci ti* .
Here Ti,M is the time to preventive replacement and Ti,F is the time-to-failure of component i. The improvement factor has a probability density distribution f (b) which is modelled in
iG j
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(4)
(5)
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Where n 1 S are the savings by grouping n maintenance actions and
dCi * (t i Di (t )) 0. (7) dt where the empty sum in (6) equals zero and the empty product equals one. In (6) ci,p is the component dependent preventive maintenance cost and bi = ci,c – ci,p, with ci,c the component dependent corrective maintenance cost. Both ci,p and ci,c include the cost of downtime. t is the age at which preventive maintenance is performed, ti,p is the downtime due to a preventive maintenance action and ti,c is the downtime caused by a corrective action. pil is the probability that component i survives the next period given that its age equals l at the beginning of the current period and qil = 1-pil. In order to determine the long-term optimal maintenance time ti,l*, with corresponding cost Ci,l*, for each component Di(t) = 0. Short-term optimization to determine the optimal maintenance time of the next maintenance action ti*, with cost Ci*, is based on the prognosis of the failure time distribution which results from the degradation Di(t) at the current time t and the degradation model described in Section 2.1. Based on this short-term prognostic information the long-term maintenance schedule is adapted.
is the additional cost of shifting
ci t*G j
ci t*i
maintenance activity i from the individual optimal time ti* to the optimal group maintenance time t*G j . The prognostic maintenance policy aims at finding the grouping structure that minimizes the maintenance cost C2 on PH. 3.1 Prognostics, estimation of remaining useful life The proposed prognostic maintenance policy is considered as a dynamic maintenance policy, as every time new information on the observed degradation of a component becomes available, the prediction of the remaining useful life of the component is updated. The degradation model described in Section 2.1 is used to predict remaining useful life for each component n, based on the current degradation Di(t). The dynamic or adaptive scheduling of maintenance actions is based on the updated failure probability distribution (i.e. remaining useful life prediction) when new information on the current degradation Di(t) of component i becomes available (e.g. by inspection). In this paper continuous updating of the failure probability distribution is considered, which is possible when component degradation is monitored continuously. First, an optimal maintenance schedule on an infinite horizon is determined. This maintenance schedule is updated on the planning horizon PH, each time a new prediction of the failure probability distribution based on the current degradation Di(t) is made.
Notice that (6) is in fact a discretised version of an agereplacement strategy, but as far as the lifetime distribution of the components is concerned knowledge of the sequence
pli
t 1 l 0
suffices. It is irrelevant whether the lifetimes
themselves have a discrete or continuous probability distribution (van der Duyn Schouten and Vanneste, 1990). 3.3 Penalty functions
3.2 Individual maintenance optimization
When grouping maintenance actions, some components their useful life and failure probability will be increased, while others their useful life will be decreased. In order to define the effect of shifting maintenance actions from their optimal times, penalty functions are constructed. A penalty function hi defines the expected additional cost of shifting the maintenance time from the optimal maintenance time ti* for a component. Penalty functions for both the next optimal maintenance time ti*, based on the short-term information, as for the nth (n > 1) maintenance occurrence, based on the longterm optimal maintenance time ti,l*, are defined. The penalty function, by adopting a long-term shift (Wildeman et al., 1997), for the first maintenance action on component i is defined as (Dekker et al., 1996):
For each individual component an infinite-horizon maintenance optimization model is formulated to find the optimal long-term maintenance time ti,l* for each component. ti,l* represents the maintenance time at which the long-term expected maintenance cost (Ci) on an infinite horizon is minimal. An average use of the components is assumed and the dependencies and interactions between the components are neglected at this stage. In this way, the savings from joint execution of maintenance activities are ignored. This decomposition approach allows the scheduling of many components (Dekker et al., 1996). For an age-based replacement policy the asymptotic cost, where ti,l* is the minimizing argument, is given by van der Duyn Schouten and Vanneste (1990). In this paper this is extended to include non-zero downtimes into the optimization problem as follows:
t 1 ci , p S bi 1 pil l 0 (6) C i t Di t t 1 t j 2 t 1 1 pil pil ti , p 1 pil ti ,c j 2 l 0 l 0 l 0 The optimal maintenance time ti* for a component with degradation Di(t) is deduced from the following equation:
117
j 1 ai t 1 j * l qi bi Di t Ci Di t pi l ai j ai if t 0 hi t*i t a 1 . j 1 i Ci* Di t qij bi Di t pil j ai t l ai t if t 0
(8)
According to the long-term shift rule the execution interval of the first maintenance action is changed according to the prognostic short-term information, while all future maintenance intervals remain ti,l*, the long-term optimal
2nd IFAC A-MEST Sevilla, Spain. November 22-23, 2012
maintenance time. The penalty function for the nth maintenance action, with n > 1, on component i becomes:
QG j tG* j G j 1 S H G* j .
(13)
If the savings Q are positive, the group G is cost effective,
Gj j , which means it is better to group the maintenance actions j 1 k v ai 1 j * l rather than performing them at their optimal individual times C q b D t 0 p , i ,l i i i i * j ai t t l ai t v k 0 i . The final objective is to find the grouping structure GSk * (9) hi t i t that minimizes the total maintenance cost on the planning 0, k 0 a t 1 j 1 horizon PH. The grouping algorithm developed by Wildeman i q j b D t 0 C * p l , k 1 i ,l i i i i et al. (Wildeman et al., 1997) is used heuristically to find the j ai l ai optimal grouping structure GSk. A constraint is added to this
with v the floor of ((n-1).ti,l*) (Bouvard et al., 2011). When a component i fails, the penalty function hi of the failed component is defined as:
algorithm, as it is not allowed to group two maintenance actions on the same component (Bouvard et al., 2011).
0, t 0 hi ti , F . , t 0
3.5 Maintenance execution and planning update (10)
This means, when a component i fails, preventive maintenance actions on the other components can be performed during the downtime due to the failure of component i. Due to this assumption, opportunistic maintenance is thus included in the model. 3.4 Maintenance activities grouping The aim is to group the maintenance activities on the planning horizon PH in order to minimize the maintenance cost on this planning horizon. The finite planning horizon is defined as:
PH max max ti* ti*,l , i . i1,..., n i
4. COMPONENT DEPENDENCIES
(11)
The parameter εi is defined as the prognostic horizon for component i. The prognostic horizon is the time between two consecutive predictions of remaining useful life of component i, based on newly available component degradation (e.g. by inspection). Each time the prognostic information is updated; this information is used to schedule maintenance actions on the prognostic horizon, as no new information on component degradation becomes available before a time εi. The prognostic horizon εi is assumed to be one time unit in this paper (i.e. continuous updating of component degradation information and corresponding prognosis).
On failure of one of the considered components it is possible that this breakdown influences the deterioration of other components resulting in secondary damage. Three different maintenance or replacement scenarios exist when a corrective maintenance action needs to be performed. In the first maintenance scenario only replacement of the primary failed component is necessary, in the second maintenance scenario replacement of both primary failed component and one secondary damaged component is required, while in the third maintenance scenario replacement of the whole subassembly is necessary due to secondary damage to all components. All corrective maintenance scenarios are initiated by failure of one of the components. In this way stochastic dependence between components is taken into account. The corrective maintenance scenarios are sampled from a multinomial distribution:
maintenance activities on components i Gj are all performed at time tGj instead of their individual optimal times ti*. The optimal maintenance time tGj* of the group is derived by the following equation:
When considering multi-component systems, three major categories of dependencies exist. These are stochastic, structural and economic dependence (Nicolai and Dekker, 2007). These three types of dependencies are also considered in the constructed maintenance model. 4.1 Stochastic dependence
Grouping of maintenance activities on PH can be done by using the defined penalty functions in (8), (9) and (10). Define H Gj t Gj the penalty function of group Gj when
H G j tG* j H G* j min hi t . t iG j
Based on the previous step a maintenance schedule on the planning horizon PH is constructed. Maintenance actions are executed according to the maintenance schedule. A rolling horizon approach is considered as each time the planning horizon is shifted and the maintenance schedule is updated by including newly available information on component degradation and the corresponding remaining useful life. In this way a dynamic and adaptive, since it is based on the currently available predictive information deduced from the component degradation, prognostic maintenance policy is developed.
(12)
x1 xk f ( x; n, p) (n! /( x1!,...,xk !))( p1 ,..., pk ), when ik1 xi n
(14)
where x = (x1,…,xk) gives the number of each of k outcomes in n trials of a process with fixed probabilities p = (p1,…,pk) of individual outcomes in any one trial. The vector p has nonnegative integer components that sum to one. The vector p
The savings QGj by grouping maintenance operations i Gj and executing them at time tGj* can be calculated as follows:
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defines the probabilities of the replacement or failure scenarios for corrective maintenance (pc = (p1 = 0.85, p2 = 0.1, p3 = 0.05)) actions. This means that for corrective maintenance 85% of the actions consist of only replacing the primary failed component, 10% consists of replacing both primary failed component and one secondary damaged component, and in 5% of the cases a replacement of the entire subassembly is necessary.
component. The second policy is a condition-based control limit policy. Preventive maintenance on a component i is performed when the degradation Di(t) exceeds a degradation control limit Di,maint. The degradation Di(t) is continuously monitored over time. The reason to compare the prognostic maintenance policy to these two maintenance policy is the fact that both maintenance policies perform well under certain circumstances. The condition-based maintenance policy performs very well when no dependencies between components exist, while the block-based maintenance policy performs well when high dependencies between components are present. This will also be shown in the results.
4.2 Economic and structural dependence In order to be able to determine the performance of the proposed prognostic maintenance policy when considering different levels of dependence between the components, a dependence parameter α is introduced. This parameter α reflects the advantage of performing maintenance on multiple components at once compared to maintenance on a single component, in other words it affects the set-up cost S by adapting the savings QGj (see (13)) when grouping maintenance as follows:
QG j tG* j G j 1 S H G* j .
Table 1. Component degradation parameters Component n
νi
μi
αi
βi
γi
δi
1
2,00
1
100
20
0,2
3
2
0,40
5
100
3
0,2
3
3
0,32
5
100
3
0,2
3
Table 2. Cost and time parameters
(15)
The dependence parameter α is assumed to incorporate the effect of both economic and structural dependence between the components in the considered system. α ranges from 0 to 1, where α = 0 means no economic and structural dependence, while α = 1 means maximal economic and/or structural dependence between the components.
ci,p
ci,c
twait(h)
treplace(h)
tinst(h)
tsecD(h)
€
€
μ
σ
μ
σ
μ
σ
μ
σ
605
5805
10
1
3
0,5
3
0,5
10
1
665
5865
10
1
3
0,5
3
0,5
10
1
475
5675
10
1
3
0,5
3
0,5
10
1
5.3 Results
5. NUMERICAL EXAMPLE A numerical example is presented in order to validate the developed prognostic maintenance policy and to compare its performance to a block-based maintenance policy and a condition-based control limit maintenance policy. This is performed for the same system but with different dependencies between the components (Section 4.2).
The long-term average cost per unit time of the prognostic maintenance policy is compared to an optimal block-based and condition-based control limit maintenance policy for different values of the dependence parameter α. The results can be seen in Figure 1.
5.1 Input parameters As an example a three component system is considered. The component degradation parameters, as described in detail in Section 2.1, are given in Table 1. The corresponding cost and time parameters for all components are shown in Table 2. twait stands for the waiting time, treplace for the actual replacement time, tinst for the installation time and the start-up time of the system and finally tsecD stands for the time to repair secondary damage. All these parameters determine the downtime due to preventive maintenance ti,p(treplace, tinst) and corrective maintenance ti,c(twait, treplace, tinst, tsecD). The cost of working (70€/h), cost of transportation (120€) and downtime cost (200€/h) are also considered in the numerical example. 5.2 Maintenance policy comparison Fig. 1. Long-term average maintenance cost per unit time in relation to the dependence α for the considered maintenance policies.
The prognostic maintenance policy is compared to two other maintenance policies. The first one is a block-based maintenance policy, where a component i is replaced every ti time units, independent of the failure history of the 119
2nd IFAC A-MEST Sevilla, Spain. November 22-23, 2012
As foreseen the condition-based maintenance policy performs very well when no or low dependencies are present between components. This is the case because the real degradation of each component is monitored separately, and when this degradation reaches the control limit, maintenance is performed regardless the state of the other components. This is also the reason why the cost of this policy stays the same regardless the dependencies between components. On the other hand, the block-based maintenance policy outperforms the condition-based maintenance when considering high interdependencies. This is because the optimal block-based maintenance policy reduces to a group-based maintenance policy, which takes advantage of grouping maintenance actions, from the moment α reaches 0.5.
REFERENCES Abdel-Hameed, M. (1975). A Gamma Wear Process. IEEE Transactions on Reliability, volume (24), 152-153. Barata, J., Soares, C. G., Marseguerra, M. and Zio, E. (2002). Simulation modelling of repairable multi-component deteriorating systems for 'on condition' maintenance optimisation. Reliability Engineering & System Safety, volume (76), 255-264. Bouvard, K., Artus, S., Bérenguer, C. and Cocquempot, V. (2011). Condition-based dynamic maintenance operations planning & grouping. Application to commercial heavy vehicles. Reliability Engineering & System Safety, volume (96), 601-610. Camci, F. 2009. System maintenance scheduling with prognostics information using genetic algorithm. IEEE Transactions on Reliability, volume (58), 539-552. Dekker, R., Wildeman, R. E. and Van Egmond, R. (1996). Joint replacement in an operational planning phase. European Journal of Operational Research, volume (91), 74-88. Jardine, A. K. S., Lin, D. and Banjevic, D. (2006). A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mechanical Systems and Signal Processing, volume (20), 1483-1510. Lee, J., Ni, J., Djurdjanovic, D., Qiu, H. and Liao, H. (2006). Intelligent prognostics tools and e-maintenance. Computers in Industry, volume (57), 476-489. Nicolai, R. P. and Dekker, R. (2007). Optimal Maintenance of Multi-component Systems: A Review. Complex System Maintenance Handbook. Springer London. Tian, Z. and Liao, H. (2011). Condition based maintenance optimization for multi-component systems using proportional hazards model. Reliability Engineering & System Safety, volume (96), 581-589. Van Der Duyn Schouten, F. A. and Vanneste, S. G. (1990). Analysis and computation of (n, N)-strategies for maintenance of a two-component system. European Journal of Operational Research, volume (48), 260-274. Van Horenbeek, A. and Pintelon, L. (2011). Optimal prognostic maintenance planning for multi-component systems. In: Proceedings of the European Safety and Reliability Conference: ESREL 2011, 910-917. CRC Press/Balkema, Troyes, France. Van Noortwijk, J. M. (2009). A survey of the application of gamma processes in maintenance. Reliability Engineering & System Safety, volume (94), 2-21. Wildeman, R. E., Dekker, R. and Smit, A. C. J. M. (1997). A dynamic policy for grouping maintenance activities. European Journal of Operational Research, volume (99), 530-551. Yang, Z. M., Djurdjanovic, D. and Ni, J. (2008). Maintenance scheduling in manufacturing systems based on predicted machine degradation. Journal of intelligent manufacturing, volume (19), 87-98.
The prognostic maintenance policy performs at least as good as or better than the other considered maintenance policies. In fact, the prognostic policy combines the advantages of both other maintenance policies. On the one hand, it considers the real component degradation by continuously updating the remaining useful life of the components, while on the other hand it takes into account the possible advantage of grouping maintenance activities due to the dependencies in the system. The prognostic maintenance policy is able to react to different degradation patterns and dependencies between components while always guaranteeing an optimal policy. 6. CONCLUSIONS A prognostic maintenance policy for multi-component systems considering different levels of component dependence is presented in this paper. It is a dynamic and adaptive policy as the maintenance schedule is always updated on a finite horizon based on the current component degradation and deduced prognostic information. The prognostic maintenance policy is able to handle complex maintenance problems, as the problem defined includes components with different degradation patterns, random failure thresholds, imperfect maintenance, non-zero downtimes and all types of dependencies (i.e. stochastic, structural and economic dependence). The performance (i.e. long-term average cost per unit time) of the prognostic maintenance policy is compared to two other maintenance policies (i.e. block-based preventive and condition-based control limit maintenance policy). The results show that the prognostic maintenance policy combines the advantages of both maintenance policies, which causes the prognostic maintenance policy to outperform the other maintenance policies. The prognostic maintenance policy guarantees optimal maintenance for different degradation patterns and dependencies between components in a multicomponent system. Future work will be on the analysis of the effect of several parameters of the model (e.g. imperfect maintenance) on the added value of the use of prognostic information in maintenance optimization and decision making. Moreover, the inclusion of an inventory policy and production schedule into the prognostic maintenance policy are interesting research subjects that will be covered. 120