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A DECISION TOOL FOR OPPORTUNISTIC MAINTENANCE STRATEGY TO PRESERVE COMPONENT PERFORMANCES
A DECISION TOOL FOR OPPORTUNISTIC MAINTENANCE STRATEGY TO PRESERVE COMPONENT PERFORMANCES E. Levrat, E. Thomas, B. Iung Nancy University - CRAN – UMR 7039 CNRS UHP INPL – Faculty of Sciences - BP239 54506 Vandoeuvre-lès-Nancy (France)
Abstract: Today, a new role for maintenance exists to enhance the eco-efficiency of the product life cycle. The concept of “life cycle maintenance” emerged to stress this role leading to push, at the manufacturing stage, an innovative culture wherein maintenance activities become of equal importance to actual production activities. This equivalence requires mainly considering the integration of the maintenance and production strategy planning for developing opportunistic maintenance task keeping conjointly the product – production – equipment performances. In this paper, a novel approach is proposed for integrating maintenance and production planning. The approach uses the “odds algorithm” and is based upon the theory of optimal stopping. The objective is to select, among all the production stops already planned, those which will be optimal to develop maintenance tasks keeping the expected product conditions. It uses criteria such as component performance and component reliability. An approach evaluation is shown on a practical example. Copyright © 2007 IFAC Keywords: Opportunist Maintenance, Production, Decision-Making, Odds algorithm.
1. INTRODUCTION Even if maintenance is a necessity, maintenance has a negative image and suffers from a deficiency of understanding and respect. It is usually recognised as a cost, a necessary evil, not as a contributor. Moreover traditionally the scope of maintenance activities has been limited to the production vs. operation phase. But as the paradigm of manufacturing shift towards realizing a sustainable society, the role of maintenance has to change to take into account a life-cycle management oriented approach (Takata, et al., 2004). Indeed, limits on resources and energy consumption imply a sharp change in the objectives of manufacturing, shifting from the need to produce more efficiently to the need to produce new assets as late as possible while still ensuring customers’ satisfaction and profits. From a global perspective of life cycle management, the role of e-maintenance is now to enhance the ecoefficiency (Desimone and Popoff, 1997) of the product life cycle while preserving the product “characteristics” (Cunha and Caldeira Duarte, 2004). In that way, maintenance has to be considered not only in production vs. operation phase but also in product design, product disassembly, and product recycling (Van Houten, et al., 1998). The concept of “life cycle maintenance” (Takata, et al., 2004) emerge to stress this new role leading to push, at the operation stage where the product characteristics can have a huge influence on manufacturing system performance, an innovative culture wherein maintenance activities become of equal importance to actual production activities.
However as a unified maintenance/production framework does not yet exist, this equivalence requires mainly to consider the integration of the maintenance and production strategy planning for developing opportunistic maintenance task (synchronised with production) keeping simultaneously the production and the equipment performances to preserve product conditions. This notion of opportunistic maintenance, which is not yet defined by a standard, has changed during the last decades. At first, an opportunistic inspection policy implies the inspection of a non-monitored part conditioned by the state (good or failed) of a monitored part (McCall, 1963). Then an opportunistic maintenance is any action that combines a preventive maintenance action and a corrective maintenance action (Dekker, et al., 1997), or the grouping of maintenance actions on different components for economical reasons (Berk and Moinzadeh, 2000). For (Jhang and Sheu, 1999), an opportunity appears as soon as an element is idle, consequently allowing to perform simultaneous maintenance actions. This idleness can be due to failures of some other elements in the system. More generally, (Budai, et al., 2006) define an opportunity as being a moment (1) at which the units to be maintained are less needed for their function than normally, (2) that occurs occasionally and (3) that is difficult to predict in advance. These opportunities appear not only in the case of failure of other elements, but also at an interruption of production. This view of an opportunity will be considered in the sequel. Opportunistic maintenance is often criticised for not being plannable long in advance and therefore no
work preparation is possible. But opportunistic maintenance can help save set-up costs and guarantee the expected performances for the system. Developing opportunistic maintenance tasks (synchronised with production) means to move from conventional maintenance strategies towards new just-in-time condition-based or predictive ones performed only when a certain level of equipment deterioration (impacting product conditions) occurs rather than after a specified period of time. It leads to have at one's disposal not only statistical and historical information about the system operation but also just-in-time information processed by means of prognosis process (Muller, et al., 2006) in order to assess new product/system situation and to anticipate product deteriorations and system failures. More precisely, in relation to ISO13381, the prognosis process aims at foreseeing how a component will evolve, until its failure and then until the system’s breakdown (i.e. residual lifetime). From this process result, the main opportunistic maintenance issue has to be formulated as follows: Taking into account the residual life time of a component, is it possible to select one of the production stops already planned in order to carry out an opportunistic maintenance action guaranteeing a compromise between costs, safety, impact on production conditions (i.e. functional performances) and on component availability… If it is the case, how it is possible to classify the selected production stops in decreasing order related to relevance criteria. In relation to this issue, some academic work has been already achieved by proposing (a) common maintenance-production scheduling (Kianfar, 2005), (b) production stop selection based on expert judgement (Rosqvist, 2002) or (c) decreasing of the maintenance operation time. The originality of the approach developed in this paper and based on “odds algorithm” is, firstly, to keep the initial production scheduling without to modify it for operating maintenance action, secondly, to ensure an optimality of the maintenance decision compared to a criterion (i.e. functional performance), and finally, to use current system information delivered by prognosis process . In that way, section 2 aims at formalising the “production stop” problem in terms of mathematical equations consistent with “odds algorithm”, then section 3 shows the implementation of the odds algorithm and section 4 its feasibility and interest for a specific application case. Finally conclusions and prospective are developed in section 5. 2
FORMALISATION OF THE “PRODUCTION STOP” PROBLEM
To solve the problem proposed, a mathematical result, issued from Bruss’ work (Bruss, 2000) and based on the theory of optimal stopping (Chow, et al., 1991), is used. This method calculates the optimal behaviour in some situations where future is uncertain.
2.1 Problem statement The product life cycle phase concerned by this problem is the production phase. In this phase, an observation horizon being given, the production vs. maintenance expert can access to the calendar of all the production stops already planned in order to carry out a maintenance action taking into account the residual life time of the components of the manufacturing system. Indeed let S be a system. The prognosis process provides a remaining lifetime of T time units for S, which defines a bounded uncertain observation horizon. The expert has at his disposal, before time T, the ‘production stops’ which are defined by means of the ‘beginning instants’ (a sequence (ai)1≤i≤n where n≥1 and ai ∈ [0; T ], 1 ≤ i ≤ n of respective ‘durations’ n
(di) 1≤i≤n, with min1≤ i ≤ n di > 0 and
∑d < T . i
i =1
Among these production stops, some of them should be appropriated for developing “just-in-time” maintenance action which was not scheduled by the maintenance manager but required due to the remaining lifetime. Thus an appropriated stop means that (a) the system is survival at instant a and maintainable during the stop, (b) the deterioration of the product/production performances are acceptable at instant a but also (c) all the logistic resources (spare parts, qualified manpower…) are available for developing maintenance action. These selected stops will be called ‘successes’ in the following and the main issue can be formulated with this way: Determine the last success at which a maintenance action can be performed in order to restore the system or one of its components into a nominal state for preserving the expected product/production performances. 2.2 Thomas Bruss’ results A way to solve the aforementioned problem is to use Bruss Theorem which can be described as follow. Let (I i )1≤i≤ n be n indicators of random and
independent events ( Ai )1≤i≤ n which are defined upon
the same probability space . (Ω; G; P ) It is possible to observe I 1 , I 2 , L sequentially and stop at any of these observations but may not recall on the previous ones. A ‘success’ is determined as being an observation equals 1. Let Bk denote the sigma-field generated by (I i )1≤i≤ k and ξ the class of all rules t such that the event {t = k } is Bk –measurable, for 1 ≤ k ≤ n . The scope is to find a stop rule τ n ∈ ξ that will maximise P I t = 1 ; I j = 0 ; t + 1 ≤ j ≤ n over every t ∈ ξ and
(
)
its value (i.e. probability to stop the observations precisely at the last success of the sequence). For developing the rule, the following quantities are used:
p j := E ( I j ) = P ( A j ) , q j := 1 − p j , rj := p j / q j ,
(1) 1≤ j ≤ n
Then it is necessary to add the numbers rk from left
The quantities rj are traditionally called the ‘odds’ which are related to the odds-theorem, (Bruss, 2003) defined as follow: an optimal rule τ n for finding the last success exists and is to stop on the first index (if any) k with I k = 1 and k ≥ s , where : ⎛ ⎧ s := sup ⎜⎜1; sup ⎨1 ≤ k ≤ n ⎩ ⎝
∑
n j=k
⎫⎞ rj ≥ 1⎬ ⎟⎟ (2) ⎭⎠
with the convention sup(∅)= −∞ . The optimal reward is given by : n n ν s = ⎛⎜ q ⎞⎟ . ⎛⎜ r ⎞⎟ (3) j=s j ⎠ ⎝ j=s j ⎠ ⎝
∏
∑
2.3 Thomas Bruss results adapted to production stop selection Based on the previous formalisation, the odds algorithm is a decision-making tool that is relevant to answer the question of the best choice, among the production stops, for performing optimal “just-intime” maintenance action. In that way, the probability space has to be particularised by (Ω; G; P ) = ([0; T ] ; ℘([0; T ]) ; P ) . In the following is made the assumption that the production stops are independent random variables leading to consider the beginning of the stops and their respective duration as independent. The n independent random variables Ai are the interval repartitions materialising the production stops. To apply the theorem, it is necessary to assess the probability that these variables are successes. P is the probability that a production stop (a;d) allows to perform a maintenance action. It is defined by the product (i) of the probability that the system is survival at instant a (reliability point of view) or the probability that the expected production performance is sufficient to guarantee the product quality at instant a (performance point of view) by (ii) the probability that the system is maintainable during the interval [a; a + d ] . The reliability or performance distributions and the maintainability distribution, can be provide in practice by means of a prognosis process (Muller et al., 2006). 3
e.g. pn , pn−1 , pn− 2 ,L p1 , knowing that for each index 1 ≤ k ≤ n, pk ∈ ] 0;1 [, pk + qk = 1, rk = pk / qk .
IMPLEMENTATION OF THE ALGORITHM
To be relevant for industrial applications, the objective is now to translate the previous mathematical formalisation, into practice. To support this action, the quantities pk , qk , rk have to be successively written with beginning from the last time
to right, until the sum is equal to or greater than 1. It means to add, for decreasing values of n, Rs := rn + rn−1 + L + rs , until the value of Rs exceeds 1. The value of s, for which this sum exceeds 1 at the first time, is the stop index we are looking for (i.e. success time for performing maintenance action). We then choose the next success after the stop index i.e. the stop (after index s) which has the greatest odds value. But if the value 1 is not reached once all the terms have been added, let s be 1 (formula provided by the theorem). Finally the product Qs = qn qn−1 L qs has to be compute for providing the win probability of the optimal strategy as equal to vs = Qs Rs . The probability function is defined as the product of a function X (X can represent reliability or performance function) by the maintainability function. Every event Ai of the sequence is therefore characterised by a couple (ai;di) and : pi = X (ai ). M ( di ) ,
(4)
The odds can thus be defined by: ri = X (ai ). M ( d i ) /(1 − X (ai ). M ( di ))
(5)
The odds is calculated for every single production stop and sum up the odds from the last one until reaching (or going beyond) the value of 1 (at index s). The probability vs that this particular production stop be optimal is given by: ⎛ vs = ⎜ ⎜ ⎝
⎞⎛ (1 − X (ai ). M ( di )) ⎟.⎜ ⎟⎜ j=s ⎠⎝ n
∏
4
n
X (ai ). M ( di ) ⎞
∑ 1 − X (a ). M (d ) ⎟⎟ (6) i
j=s
i
⎠
ALGORITHM APPLICATION
The feasibility and interest of the algorithm are considering in relation to an application case extracted from TELMA platform already explained in (Iung, et al., 2005). Bobbin Changing
Strip Accumulation
Stamping/ Cutting
Advance
Next Bobbins ¼ turn
Stubbing position
Current Bobbin Strip Position Sensor
Bobbin of raw material
Continuous Strip
End Product
Strip Recycling
Fig. 1. Architecture of the e-maintenance platform TELMA
TELMA allows to unwind metal bobbins and to stamp or to cut the metal strip. One subset of TELMA is dedicated to lock/unlock the axis supporting the metal bobbin, in the changing bobbin part (Figure 1). This process is fulfilled by operating a mandrel unit action and is broken up into four sub-processes in relation to 4 supports: a PLC, a distributor, a cylinder, and a mandrel (in contact with the axis). The example is only focusing on the sub-process named “To actuate the shaft of the cylinder” which is supported by the component “cylinder”. It enables to transform the “regulated pneumatic energy” flow (input flow) into a “translation movement” flow (i.e. Shaft movement). The input flow is characterised by two static variables: input pressure P and airflow rate Fr. The output flow is characterised by two static variables: final static force F and average speed S of translation. Figure 2 presents a part of TELMA process analysis, according to 3 functionnal levels from highest level (‘To transform metal bobbin in end products’ process) to component/cylinder level (‘To actuate the shaft of the cylinder’).
results given by the prognosis model are reused here as main inputs of the maintenance decision making process. For example, after an operation period of 200h for the cylinder, it is observed vibrations on the axis attesting that the gripping force (F) is decreasing. Thus it is necessary to carry out a “just-in-time” maintenance action which was not initially planned. This action aims at controlling the cylinder degradation and its direct impact on the process performance (shaft movement) and indirectly on the “product” quality (tightening in right position of the metal bobbin).
Cylinder states : OK : Normal state D1 : degraded 1 D2 : degraded 2 OK : Failure state
Fig.3. DBN of ”To actuate the shaft of the cylinder”
Legend Process
Component
Finality flow
Secondary flow
Fig. 2. Process analysis of TELMA The causal relationships between (deviated) inputoutput flows and (degraded) component allow to define oriented arcs linking the potential cause variables (P, Fr and TL (k+1)) to effect variables (F and S), where TL (k+1) is the (nominal or degraded) state of the component supporting the process (Figure 3). This functional vs. dysfunctional knowledge is used in (Iung, et al., 2005) for formalising a prognosis model of this sub-process thanks to Dynamic Bayesian Network (DBN). The numerical values and
This action has to be operated between the current instant (current time) and the next scheduled maintenance action (planned at T=450h) but while seeking not to stop the system apart from the stops already planned by the production. These stops are known in advance as listed in Table 1 and materialise times required for tool changes, series changes ... on TELMA. The stop durations are assessed by maintenance experts in dialogue with production expert. If some new stops occur (operator unavailability, lack of raw material …) it is possible progressively to integrate them in the stop list and to make new iteration for finding the best stop. According to selected values of the parameters (λ1, λ2 …) (Iung, et al., 2005), the process supported by the cylinder is considered as running well with preserving “product” performances, when the static Force (F) developed by the cylinder is higher or equal to 600 Newton. Thus this criterion will be used for assessing, with odds algorithm, the optimality of the stop required for maintenance action and it represents a first step for taking into account not only component conditions (i.e. reliability) but also “finality” conditions (global performance) in this optimality (Thomas, et al., 2006). Therefore the approach and algorithm proposed are enough generic to be implemented at higher abstraction levels until processes directly in charge with the product (i.e. metal bobbin) transformed. At the “cylinder” level, the finality performance is the probability that the static force is greater than 600 N (X(t)=Prob(F(t)>600 N)) whereas the reliability is the probability that the cylinder is in state OK (X(t)=R(t)). By means of the prognosis model developed in (Iung, et al., 2005), the evolution of R(t) or Prob(F(t)>600 N) can be known all over the time. In that way, Figure 2 shows the reliability and performance (Prob(F(t)>600 N)) curves between T=0 to T=1000 h. The maintainability function of the
system is supposed to be exponential with µ parameter equals to 0.3 (MTTR=3.3h). 1 Prob(F(t)>600 N)
R(t)
0,9 0,8
Probability
0,7 0,6
Moreover for improving the decision making process, the expert can have at his disposal the classification of the production stops as shown in Table 2 by decreasing order of relevance. The other solutions are found by deleting, in the index alternative list, the solution coming from higher row and by applying the algorithm again.
0,5
Table 2 Complete results for both approaches
0,4 0,3 0,2 0,1 0 1
201
401
601
801
1001
Time unit
Fig. 2. Reliability R(t) and Prob(F(t)>600 N) curves During this observation time (1000h), 13 production stops are considered and distributed in time as indicated in Table 1. It is noticed that certain durations of production stops do not allow, in average time, carrying out the intervention (i.e. stop 2 and 4 have a duration less than 3.3h). On the basis of values given in Table 1, the odds algorithm is applied as formulated in section 3. Table 1 Details of the 13 production stoppages Index 1 2 3 4 5 6 7 8 9 10 11 12 13
Beginning (h) 200 210 230 235 250 256 310 320 400 420 425 430 450
Duration 3 2 4 2 1 4 4 2 1 1 3 2 5
4.1 The decision-making The results of the odds algorithm execution are given in Table 2. The case A corresponds to the reliability point of view whereas the case B corresponds to the finality performance point of view (Prob(F(t)>600 N). It should be noted that in practice the index s has not to be calculated for the two probability functions. Indeed the two cases are developed here, to highlight the interest to take into account, in the optimality, the “finality” conditions (case B) instead of component conditions (case A). In relation to case A, the optimal solution is obtained for index s equals to 7 with a win probability equals to 40.2%. This percentage may seem low, but in fact it is excellent, as far as a general decision making algorithm is concerned. In relation to case B, the optimal solution is obtained for index s equals to 13 with a win probability equals to 46.5%.
1 2 3 4 5 6 7
Stop Number s A B 7 13 6 11 13 8 4 7 3 6 2 4 1 3
Win Probability A B 0.4020 0.4650 0,4028 0.4349 0,3997 0,4285 0,4012 0,4564 0,4098 0,4640 0,4024 0,4124 0,4081 0,4129
Decision Optimal Degraded Degraded Degraded Degraded Degraded Degraded
Only the first solution (s=7 in case A; s=13 in case B) can be defined as ‘optimal’ because calculated by the odds algorithm according to the initial problem statement. For the other stops, the decision is necessary a non optimal decision (degraded decision). Some discussions can be done according to the Table 2 results. In relation to case A, stop 7 (s=7) is the last stop enough long (duration = 4h) before the stop concerned by the scheduled maintenance action (stop 13). Stop 1, 2, 3 and 6 have a win probability higher than stop 7 but they cannot be chosen as optimal because the odds algorithm is looking for the stop which must be the last success. Nevertheless stop 2 belongs to the list even if the stop duration is lower than MTTR (in average) because its probability of survival (Figure 2) is high. Stop 13 is not selected too because its win probability is lower than those affiliated to stop 7 but it appears on the list because its duration is higher than those required. The probability of survival of stop 11 is too weak for selecting this stop in the list (even if its duration is adapted). In relation to case B, a more global view is adopted by focusing on the process success according to its finality (its goal). On the Figure 2, the curve Prob(F(t) 600N) is always above the reliability curve R(t) leading to consider a optimistic vision compared to case A. This fact is also materialised in Table 2 with the win probabilities which are higher in the case B. As the stop durations do not change, the two visions (A & B) must be in agreement but with a temporal interval. Following the stops 13, 11, and 8, the stops proposed in the case B are similar to those proposed in the case A. In both cases, the algorithm is making a compromise between the two points of views in regards with Maintainability. Therefore the role of the expert is to use all these proposals for decision making but by integrating the uncertainties and constraints related to the industrial context. One advantage concerns the possibility for dynamically taking into account each new production stop in the future. If new information occurs, it is
integrated in the opportunity list and the ‘odds algorithm’ is applied again. For example, at time T=249h an information from the production staff indicates that a new production stop will be done at T=380h for 4h (named stop 8b). An emergent question is for maintenance point of view: is this stop an opportunity to optimize decision or should we keep our previous decisions? Table 3 underlies that if the reliability of the component cylinder is only considered, it is possible to move the maintenance task forecasted at Tp=310h to Tn=380h. Table 3 Results for both approaches
1
Stop Number s A B 8b 13
Win Probability A B 0.4041 0.4650
Decision Optimal
The win probability is almost the same. If economical and logistic criteria are better than for Tn then the new decision is selected. If we focus at the finality performance view, this opportunity doesn’t replace the previous decision. 5
CONCLUSIONS
In this paper, an odds algorithm-based approach is proposed to select a production stop in order to perform a just-in-time maintenance action preserving finality performances and not only component conditions. This approach is proved to be optimal and in face with system evolution by taking into account the results of a prognosis process. An application on a specific process (one component) of TELMA platform has been carried out to show the feasibility of this approach. Based on these results, further developments should be implemented to demonstrate the real efficiency and added value of this approach. It should consist mainly in: - Developing the approach to a global system (a lot of components) at different abstraction levels highlighting the impact on the real product delivered by the system. In this case the European IP DYNAMITE project can provide an experimentation site, for example, in FIAT or VOLVO plants. - Integrating the logistic and costs aspects in the algorithm, - Combining the local optimal decisions (for each component) to deliver a global optimal decision (for the system). REFERENCES Berk, E. and K. Moinzadeh (2000). Analysis of maintenance policies for M machines with deteriorating performance. IIE Transactions, 32, pp. 433–444. Bruss, F.T. (2000). Sum the odds to one and stop. Annals of Probability, 28, pp. 1384–1391 Bruss, F.T. (2003). A note on bounds for the oddstheorem of optimal stopping. Annals of Probability, 31, pp. 1859–1861.
Budai, G., Dekker R. and Nicolai R. P. (2006). A review of planning models for maintenance and production. Economic Institute report, EI 200644. Chow, Y.S., Robbins H. and Siegmund, D., (1991). The theory of optimal stopping. Dover, New York. Cunha, P.F. and Caldeira Duarte, J.A., (2004). Development of a productive service module based on a life cycle perspective of maintenance issues, Annals of the CIRP, 53/1, pp. 13–16. Dekker, R., Van der Duyn Schouten F. A. and Wildeman R. E. (1997). A review of multicomponent models with economic dependence. Mathematical methods of operational research, 45 (3), pp. 411–435. DeSimone, L. D., Popoff, F. with The WBCSD, (1997). Eco-Efficiency, MIT Press. Iung, B., Veron, M., Suhner, M.C. and Muller, A., (2005). Integration of maintenance strategies into prognosis process to decision-making aid on system operation, Annals of the CIRP 54/1, 1, pp. 5–8. Jhang, J. P. and Sheu S. H. (1999). Opportunity-based age replacement policy with minimal repair. Reliability engineering and system safety, 64 (3), pp. 339–344. Kianfar, F., (2005). A numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system. Applied Mathematics and Computation, 170, pp. 924– 940. McCall, J. J. (1963). Operating characteristics of opportunistic replacement and inspection policies. Management Science, 10 (1), pp. 85–97. Muller, A., Suhner, M-C., Iung B., (2006). Proactive Maintenance for industrial system operation based on a formalised prognosis process. Reliability Engineering and System Safety (RESS) – Publication on line – doi:10.1016/j.ress.2006.12.004 Rosqvist, T., (2002). Stopping time optimisation in condition monitoring. Reliability Engineering and maintenance system, 11th IFAC Symposium INCOM04, Salvador, Brasil, April 5-7. Scarf, P. A. (1997). On the application of mathematical models in maintenance. European Journal of Operational Research, 99 (3), pp. 493–506. Takata, S., Kimura, F., Van Houten, F.J.A.M., Westkämper, E., Shpitalni, M., Ceglarek, D., Lee, J., (2004). Maintenance: Changing Role in Life Cycle Management, Annals of the CIRP, 53/2, pp. 643–656. Thomas, E., Levrat, E., Iung, B. and Monnin, M., (2006). ‘Odds algorithm’-based opportunitytriggered preventive maintenance with production policy, 6th IFAC Symposium Safeprocess'06, China, pp. 835–840. Van Houten, F.J.A.M., Tomiyama, T. and Salomons, O.W., (1998). Product modelling for model-based maintenance, Annals of the CIRP, 47/1, pp. 123–129.
Abstract: Today, a new role for maintenance exists to enhance the eco-efficiency of the product life cycle. The concept of “life cycle maintenance” emerged to stress this role leading to push, at the manufacturing stage, an innovative culture wherein maintenance activities become of equal importance to actual production activities. This equivalence requires mainly considering the integration of the maintenance and production strategy planning for developing opportunistic maintenance task keeping conjointly the product – production – equipment performances. In this paper, a novel approach is proposed for integrating maintenance and production planning. The approach uses the “odds algorithm” and is based upon the theory of optimal stopping. The objective is to select, among all the production stops already planned, those which will be optimal to develop maintenance tasks keeping the expected product conditions. It uses criteria such as component performance and component reliability. An approach evaluation is shown on a practical example. Copyright © 2007 IFAC Keywords: Opportunist Maintenance, Production, Decision-Making, Odds algorithm.
1. INTRODUCTION Even if maintenance is a necessity, maintenance has a negative image and suffers from a deficiency of understanding and respect. It is usually recognised as a cost, a necessary evil, not as a contributor. Moreover traditionally the scope of maintenance activities has been limited to the production vs. operation phase. But as the paradigm of manufacturing shift towards realizing a sustainable society, the role of maintenance has to change to take into account a life-cycle management oriented approach (Takata, et al., 2004). Indeed, limits on resources and energy consumption imply a sharp change in the objectives of manufacturing, shifting from the need to produce more efficiently to the need to produce new assets as late as possible while still ensuring customers’ satisfaction and profits. From a global perspective of life cycle management, the role of e-maintenance is now to enhance the ecoefficiency (Desimone and Popoff, 1997) of the product life cycle while preserving the product “characteristics” (Cunha and Caldeira Duarte, 2004). In that way, maintenance has to be considered not only in production vs. operation phase but also in product design, product disassembly, and product recycling (Van Houten, et al., 1998). The concept of “life cycle maintenance” (Takata, et al., 2004) emerge to stress this new role leading to push, at the operation stage where the product characteristics can have a huge influence on manufacturing system performance, an innovative culture wherein maintenance activities become of equal importance to actual production activities.
However as a unified maintenance/production framework does not yet exist, this equivalence requires mainly to consider the integration of the maintenance and production strategy planning for developing opportunistic maintenance task (synchronised with production) keeping simultaneously the production and the equipment performances to preserve product conditions. This notion of opportunistic maintenance, which is not yet defined by a standard, has changed during the last decades. At first, an opportunistic inspection policy implies the inspection of a non-monitored part conditioned by the state (good or failed) of a monitored part (McCall, 1963). Then an opportunistic maintenance is any action that combines a preventive maintenance action and a corrective maintenance action (Dekker, et al., 1997), or the grouping of maintenance actions on different components for economical reasons (Berk and Moinzadeh, 2000). For (Jhang and Sheu, 1999), an opportunity appears as soon as an element is idle, consequently allowing to perform simultaneous maintenance actions. This idleness can be due to failures of some other elements in the system. More generally, (Budai, et al., 2006) define an opportunity as being a moment (1) at which the units to be maintained are less needed for their function than normally, (2) that occurs occasionally and (3) that is difficult to predict in advance. These opportunities appear not only in the case of failure of other elements, but also at an interruption of production. This view of an opportunity will be considered in the sequel. Opportunistic maintenance is often criticised for not being plannable long in advance and therefore no
work preparation is possible. But opportunistic maintenance can help save set-up costs and guarantee the expected performances for the system. Developing opportunistic maintenance tasks (synchronised with production) means to move from conventional maintenance strategies towards new just-in-time condition-based or predictive ones performed only when a certain level of equipment deterioration (impacting product conditions) occurs rather than after a specified period of time. It leads to have at one's disposal not only statistical and historical information about the system operation but also just-in-time information processed by means of prognosis process (Muller, et al., 2006) in order to assess new product/system situation and to anticipate product deteriorations and system failures. More precisely, in relation to ISO13381, the prognosis process aims at foreseeing how a component will evolve, until its failure and then until the system’s breakdown (i.e. residual lifetime). From this process result, the main opportunistic maintenance issue has to be formulated as follows: Taking into account the residual life time of a component, is it possible to select one of the production stops already planned in order to carry out an opportunistic maintenance action guaranteeing a compromise between costs, safety, impact on production conditions (i.e. functional performances) and on component availability… If it is the case, how it is possible to classify the selected production stops in decreasing order related to relevance criteria. In relation to this issue, some academic work has been already achieved by proposing (a) common maintenance-production scheduling (Kianfar, 2005), (b) production stop selection based on expert judgement (Rosqvist, 2002) or (c) decreasing of the maintenance operation time. The originality of the approach developed in this paper and based on “odds algorithm” is, firstly, to keep the initial production scheduling without to modify it for operating maintenance action, secondly, to ensure an optimality of the maintenance decision compared to a criterion (i.e. functional performance), and finally, to use current system information delivered by prognosis process . In that way, section 2 aims at formalising the “production stop” problem in terms of mathematical equations consistent with “odds algorithm”, then section 3 shows the implementation of the odds algorithm and section 4 its feasibility and interest for a specific application case. Finally conclusions and prospective are developed in section 5. 2
FORMALISATION OF THE “PRODUCTION STOP” PROBLEM
To solve the problem proposed, a mathematical result, issued from Bruss’ work (Bruss, 2000) and based on the theory of optimal stopping (Chow, et al., 1991), is used. This method calculates the optimal behaviour in some situations where future is uncertain.
2.1 Problem statement The product life cycle phase concerned by this problem is the production phase. In this phase, an observation horizon being given, the production vs. maintenance expert can access to the calendar of all the production stops already planned in order to carry out a maintenance action taking into account the residual life time of the components of the manufacturing system. Indeed let S be a system. The prognosis process provides a remaining lifetime of T time units for S, which defines a bounded uncertain observation horizon. The expert has at his disposal, before time T, the ‘production stops’ which are defined by means of the ‘beginning instants’ (a sequence (ai)1≤i≤n where n≥1 and ai ∈ [0; T ], 1 ≤ i ≤ n of respective ‘durations’ n
(di) 1≤i≤n, with min1≤ i ≤ n di > 0 and
∑d < T . i
i =1
Among these production stops, some of them should be appropriated for developing “just-in-time” maintenance action which was not scheduled by the maintenance manager but required due to the remaining lifetime. Thus an appropriated stop means that (a) the system is survival at instant a and maintainable during the stop, (b) the deterioration of the product/production performances are acceptable at instant a but also (c) all the logistic resources (spare parts, qualified manpower…) are available for developing maintenance action. These selected stops will be called ‘successes’ in the following and the main issue can be formulated with this way: Determine the last success at which a maintenance action can be performed in order to restore the system or one of its components into a nominal state for preserving the expected product/production performances. 2.2 Thomas Bruss’ results A way to solve the aforementioned problem is to use Bruss Theorem which can be described as follow. Let (I i )1≤i≤ n be n indicators of random and
independent events ( Ai )1≤i≤ n which are defined upon
the same probability space . (Ω; G; P ) It is possible to observe I 1 , I 2 , L sequentially and stop at any of these observations but may not recall on the previous ones. A ‘success’ is determined as being an observation equals 1. Let Bk denote the sigma-field generated by (I i )1≤i≤ k and ξ the class of all rules t such that the event {t = k } is Bk –measurable, for 1 ≤ k ≤ n . The scope is to find a stop rule τ n ∈ ξ that will maximise P I t = 1 ; I j = 0 ; t + 1 ≤ j ≤ n over every t ∈ ξ and
(
)
its value (i.e. probability to stop the observations precisely at the last success of the sequence). For developing the rule, the following quantities are used:
p j := E ( I j ) = P ( A j ) , q j := 1 − p j , rj := p j / q j ,
(1) 1≤ j ≤ n
Then it is necessary to add the numbers rk from left
The quantities rj are traditionally called the ‘odds’ which are related to the odds-theorem, (Bruss, 2003) defined as follow: an optimal rule τ n for finding the last success exists and is to stop on the first index (if any) k with I k = 1 and k ≥ s , where : ⎛ ⎧ s := sup ⎜⎜1; sup ⎨1 ≤ k ≤ n ⎩ ⎝
∑
n j=k
⎫⎞ rj ≥ 1⎬ ⎟⎟ (2) ⎭⎠
with the convention sup(∅)= −∞ . The optimal reward is given by : n n ν s = ⎛⎜ q ⎞⎟ . ⎛⎜ r ⎞⎟ (3) j=s j ⎠ ⎝ j=s j ⎠ ⎝
∏
∑
2.3 Thomas Bruss results adapted to production stop selection Based on the previous formalisation, the odds algorithm is a decision-making tool that is relevant to answer the question of the best choice, among the production stops, for performing optimal “just-intime” maintenance action. In that way, the probability space has to be particularised by (Ω; G; P ) = ([0; T ] ; ℘([0; T ]) ; P ) . In the following is made the assumption that the production stops are independent random variables leading to consider the beginning of the stops and their respective duration as independent. The n independent random variables Ai are the interval repartitions materialising the production stops. To apply the theorem, it is necessary to assess the probability that these variables are successes. P is the probability that a production stop (a;d) allows to perform a maintenance action. It is defined by the product (i) of the probability that the system is survival at instant a (reliability point of view) or the probability that the expected production performance is sufficient to guarantee the product quality at instant a (performance point of view) by (ii) the probability that the system is maintainable during the interval [a; a + d ] . The reliability or performance distributions and the maintainability distribution, can be provide in practice by means of a prognosis process (Muller et al., 2006). 3
e.g. pn , pn−1 , pn− 2 ,L p1 , knowing that for each index 1 ≤ k ≤ n, pk ∈ ] 0;1 [, pk + qk = 1, rk = pk / qk .
IMPLEMENTATION OF THE ALGORITHM
To be relevant for industrial applications, the objective is now to translate the previous mathematical formalisation, into practice. To support this action, the quantities pk , qk , rk have to be successively written with beginning from the last time
to right, until the sum is equal to or greater than 1. It means to add, for decreasing values of n, Rs := rn + rn−1 + L + rs , until the value of Rs exceeds 1. The value of s, for which this sum exceeds 1 at the first time, is the stop index we are looking for (i.e. success time for performing maintenance action). We then choose the next success after the stop index i.e. the stop (after index s) which has the greatest odds value. But if the value 1 is not reached once all the terms have been added, let s be 1 (formula provided by the theorem). Finally the product Qs = qn qn−1 L qs has to be compute for providing the win probability of the optimal strategy as equal to vs = Qs Rs . The probability function is defined as the product of a function X (X can represent reliability or performance function) by the maintainability function. Every event Ai of the sequence is therefore characterised by a couple (ai;di) and : pi = X (ai ). M ( di ) ,
(4)
The odds can thus be defined by: ri = X (ai ). M ( d i ) /(1 − X (ai ). M ( di ))
(5)
The odds is calculated for every single production stop and sum up the odds from the last one until reaching (or going beyond) the value of 1 (at index s). The probability vs that this particular production stop be optimal is given by: ⎛ vs = ⎜ ⎜ ⎝
⎞⎛ (1 − X (ai ). M ( di )) ⎟.⎜ ⎟⎜ j=s ⎠⎝ n
∏
4
n
X (ai ). M ( di ) ⎞
∑ 1 − X (a ). M (d ) ⎟⎟ (6) i
j=s
i
⎠
ALGORITHM APPLICATION
The feasibility and interest of the algorithm are considering in relation to an application case extracted from TELMA platform already explained in (Iung, et al., 2005). Bobbin Changing
Strip Accumulation
Stamping/ Cutting
Advance
Next Bobbins ¼ turn
Stubbing position
Current Bobbin Strip Position Sensor
Bobbin of raw material
Continuous Strip
End Product
Strip Recycling
Fig. 1. Architecture of the e-maintenance platform TELMA
TELMA allows to unwind metal bobbins and to stamp or to cut the metal strip. One subset of TELMA is dedicated to lock/unlock the axis supporting the metal bobbin, in the changing bobbin part (Figure 1). This process is fulfilled by operating a mandrel unit action and is broken up into four sub-processes in relation to 4 supports: a PLC, a distributor, a cylinder, and a mandrel (in contact with the axis). The example is only focusing on the sub-process named “To actuate the shaft of the cylinder” which is supported by the component “cylinder”. It enables to transform the “regulated pneumatic energy” flow (input flow) into a “translation movement” flow (i.e. Shaft movement). The input flow is characterised by two static variables: input pressure P and airflow rate Fr. The output flow is characterised by two static variables: final static force F and average speed S of translation. Figure 2 presents a part of TELMA process analysis, according to 3 functionnal levels from highest level (‘To transform metal bobbin in end products’ process) to component/cylinder level (‘To actuate the shaft of the cylinder’).
results given by the prognosis model are reused here as main inputs of the maintenance decision making process. For example, after an operation period of 200h for the cylinder, it is observed vibrations on the axis attesting that the gripping force (F) is decreasing. Thus it is necessary to carry out a “just-in-time” maintenance action which was not initially planned. This action aims at controlling the cylinder degradation and its direct impact on the process performance (shaft movement) and indirectly on the “product” quality (tightening in right position of the metal bobbin).
Cylinder states : OK : Normal state D1 : degraded 1 D2 : degraded 2 OK : Failure state
Fig.3. DBN of ”To actuate the shaft of the cylinder”
Legend Process
Component
Finality flow
Secondary flow
Fig. 2. Process analysis of TELMA The causal relationships between (deviated) inputoutput flows and (degraded) component allow to define oriented arcs linking the potential cause variables (P, Fr and TL (k+1)) to effect variables (F and S), where TL (k+1) is the (nominal or degraded) state of the component supporting the process (Figure 3). This functional vs. dysfunctional knowledge is used in (Iung, et al., 2005) for formalising a prognosis model of this sub-process thanks to Dynamic Bayesian Network (DBN). The numerical values and
This action has to be operated between the current instant (current time) and the next scheduled maintenance action (planned at T=450h) but while seeking not to stop the system apart from the stops already planned by the production. These stops are known in advance as listed in Table 1 and materialise times required for tool changes, series changes ... on TELMA. The stop durations are assessed by maintenance experts in dialogue with production expert. If some new stops occur (operator unavailability, lack of raw material …) it is possible progressively to integrate them in the stop list and to make new iteration for finding the best stop. According to selected values of the parameters (λ1, λ2 …) (Iung, et al., 2005), the process supported by the cylinder is considered as running well with preserving “product” performances, when the static Force (F) developed by the cylinder is higher or equal to 600 Newton. Thus this criterion will be used for assessing, with odds algorithm, the optimality of the stop required for maintenance action and it represents a first step for taking into account not only component conditions (i.e. reliability) but also “finality” conditions (global performance) in this optimality (Thomas, et al., 2006). Therefore the approach and algorithm proposed are enough generic to be implemented at higher abstraction levels until processes directly in charge with the product (i.e. metal bobbin) transformed. At the “cylinder” level, the finality performance is the probability that the static force is greater than 600 N (X(t)=Prob(F(t)>600 N)) whereas the reliability is the probability that the cylinder is in state OK (X(t)=R(t)). By means of the prognosis model developed in (Iung, et al., 2005), the evolution of R(t) or Prob(F(t)>600 N) can be known all over the time. In that way, Figure 2 shows the reliability and performance (Prob(F(t)>600 N)) curves between T=0 to T=1000 h. The maintainability function of the
system is supposed to be exponential with µ parameter equals to 0.3 (MTTR=3.3h). 1 Prob(F(t)>600 N)
R(t)
0,9 0,8
Probability
0,7 0,6
Moreover for improving the decision making process, the expert can have at his disposal the classification of the production stops as shown in Table 2 by decreasing order of relevance. The other solutions are found by deleting, in the index alternative list, the solution coming from higher row and by applying the algorithm again.
0,5
Table 2 Complete results for both approaches
0,4 0,3 0,2 0,1 0 1
201
401
601
801
1001
Time unit
Fig. 2. Reliability R(t) and Prob(F(t)>600 N) curves During this observation time (1000h), 13 production stops are considered and distributed in time as indicated in Table 1. It is noticed that certain durations of production stops do not allow, in average time, carrying out the intervention (i.e. stop 2 and 4 have a duration less than 3.3h). On the basis of values given in Table 1, the odds algorithm is applied as formulated in section 3. Table 1 Details of the 13 production stoppages Index 1 2 3 4 5 6 7 8 9 10 11 12 13
Beginning (h) 200 210 230 235 250 256 310 320 400 420 425 430 450
Duration 3 2 4 2 1 4 4 2 1 1 3 2 5
4.1 The decision-making The results of the odds algorithm execution are given in Table 2. The case A corresponds to the reliability point of view whereas the case B corresponds to the finality performance point of view (Prob(F(t)>600 N). It should be noted that in practice the index s has not to be calculated for the two probability functions. Indeed the two cases are developed here, to highlight the interest to take into account, in the optimality, the “finality” conditions (case B) instead of component conditions (case A). In relation to case A, the optimal solution is obtained for index s equals to 7 with a win probability equals to 40.2%. This percentage may seem low, but in fact it is excellent, as far as a general decision making algorithm is concerned. In relation to case B, the optimal solution is obtained for index s equals to 13 with a win probability equals to 46.5%.
1 2 3 4 5 6 7
Stop Number s A B 7 13 6 11 13 8 4 7 3 6 2 4 1 3
Win Probability A B 0.4020 0.4650 0,4028 0.4349 0,3997 0,4285 0,4012 0,4564 0,4098 0,4640 0,4024 0,4124 0,4081 0,4129
Decision Optimal Degraded Degraded Degraded Degraded Degraded Degraded
Only the first solution (s=7 in case A; s=13 in case B) can be defined as ‘optimal’ because calculated by the odds algorithm according to the initial problem statement. For the other stops, the decision is necessary a non optimal decision (degraded decision). Some discussions can be done according to the Table 2 results. In relation to case A, stop 7 (s=7) is the last stop enough long (duration = 4h) before the stop concerned by the scheduled maintenance action (stop 13). Stop 1, 2, 3 and 6 have a win probability higher than stop 7 but they cannot be chosen as optimal because the odds algorithm is looking for the stop which must be the last success. Nevertheless stop 2 belongs to the list even if the stop duration is lower than MTTR (in average) because its probability of survival (Figure 2) is high. Stop 13 is not selected too because its win probability is lower than those affiliated to stop 7 but it appears on the list because its duration is higher than those required. The probability of survival of stop 11 is too weak for selecting this stop in the list (even if its duration is adapted). In relation to case B, a more global view is adopted by focusing on the process success according to its finality (its goal). On the Figure 2, the curve Prob(F(t) 600N) is always above the reliability curve R(t) leading to consider a optimistic vision compared to case A. This fact is also materialised in Table 2 with the win probabilities which are higher in the case B. As the stop durations do not change, the two visions (A & B) must be in agreement but with a temporal interval. Following the stops 13, 11, and 8, the stops proposed in the case B are similar to those proposed in the case A. In both cases, the algorithm is making a compromise between the two points of views in regards with Maintainability. Therefore the role of the expert is to use all these proposals for decision making but by integrating the uncertainties and constraints related to the industrial context. One advantage concerns the possibility for dynamically taking into account each new production stop in the future. If new information occurs, it is
integrated in the opportunity list and the ‘odds algorithm’ is applied again. For example, at time T=249h an information from the production staff indicates that a new production stop will be done at T=380h for 4h (named stop 8b). An emergent question is for maintenance point of view: is this stop an opportunity to optimize decision or should we keep our previous decisions? Table 3 underlies that if the reliability of the component cylinder is only considered, it is possible to move the maintenance task forecasted at Tp=310h to Tn=380h. Table 3 Results for both approaches
1
Stop Number s A B 8b 13
Win Probability A B 0.4041 0.4650
Decision Optimal
The win probability is almost the same. If economical and logistic criteria are better than for Tn then the new decision is selected. If we focus at the finality performance view, this opportunity doesn’t replace the previous decision. 5
CONCLUSIONS
In this paper, an odds algorithm-based approach is proposed to select a production stop in order to perform a just-in-time maintenance action preserving finality performances and not only component conditions. This approach is proved to be optimal and in face with system evolution by taking into account the results of a prognosis process. An application on a specific process (one component) of TELMA platform has been carried out to show the feasibility of this approach. Based on these results, further developments should be implemented to demonstrate the real efficiency and added value of this approach. It should consist mainly in: - Developing the approach to a global system (a lot of components) at different abstraction levels highlighting the impact on the real product delivered by the system. In this case the European IP DYNAMITE project can provide an experimentation site, for example, in FIAT or VOLVO plants. - Integrating the logistic and costs aspects in the algorithm, - Combining the local optimal decisions (for each component) to deliver a global optimal decision (for the system). REFERENCES Berk, E. and K. Moinzadeh (2000). Analysis of maintenance policies for M machines with deteriorating performance. IIE Transactions, 32, pp. 433–444. Bruss, F.T. (2000). Sum the odds to one and stop. Annals of Probability, 28, pp. 1384–1391 Bruss, F.T. (2003). A note on bounds for the oddstheorem of optimal stopping. Annals of Probability, 31, pp. 1859–1861.
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