MSEC MODELING OF DEGRADATION PROCESSES TO OBTAIN AN OPTIMAL SOLUTION FOR MAINTENANCE AND PERFORMANCE

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1 Proceeding of the ASME 9 International Manufacturing Science and Engineering Conference MSEC9 October 4-7, 9, West Lafayette, Indiana, USA MSEC MODELING OF DEGRADATION PROCESSES TO OBTAIN AN OPTIMAL SOLUTION FOR MAINTENANCE AND PERFORMANCE Seungchul Lee University of Michigan Ann Arbor, MI, USA Lin Li University of Michigan Ann Arbor, MI, USA Jun Ni University of Michigan Ann Arbor, MI, USA ABSTRACT This paper presents an approach to represent equipent degradation and various aintenance decision processes based on Markov processes. Non-exponential holding tie distributions are approxiated by inserting ultiple interediate states between the two different degradation states based on a phase-type distribution. Overall syste availability then is nuerically calculated by recursively solving the balance equations of the Markov process. Preliinary siulation results show that the optial preventive aintenance intervals for two repairable coponents syste can be achieved by eans of the proposed ethod. By having an adequate odel representing both deterioration and aintenance processes, it is also possible to obtain different optial aintenance policies to axiize the availability or productivity for different configurations of coponents. INTRODUCTION Since odern anufacturing systes have becoe highly autoated and echanized, the ipact of unplanned downtie caused by syste failures are worse than ever. Unplanned downtie of equipent ight not only reduce line productivity but also affect the quality control of the products. Another consequence of syste failures is the escalation of aintenance expenses due to unpredictable aintenance. Therefore, identifying a cost effective aintenance progra is becoing one of the key objectives in the production line [-4]. It has been recognized that aintenance is not an isolated technical discipline but an integral part of the copetitive plant operations [5]. To exaine the trade-offs between aintenance and operation costs, a atheatical odel should be developed to estiate an appropriate aintenance policy and relevant syste perforance easureents. A set of Markov processes has been used to iic the effect of non-exponential holding tie distributions that are often needed in the aintenance policy. Even though sei- Markov processes have been eployed to odel ulti-state deteriorating systes by allowing the holding tie distributions to be non-exponential, it is well-known that the atheatical forulations of sei-markov odels [6, 7] are so coplicated that they are not analytically tractable. Non-exponential holding tie distributions can be approxiated by inserting ultiple interediate states between the two degradation states [8-]. In general, the syste consists of ore than a single unit. If all units in the syste are stochastically independent of one another, a aintenance policy for the single unit odel [, 3] ay be applied to the ulti-unit aintenance probles. On the other hand, if any units in the syste are stochastically dependent on each other, then an optial decision on aintenance of one unit is not necessarily the optiu for the entire syste [4]. A decision ust be ade to iprove the whole syste rather than any single subsyste. Therefore, we ust also investigate optial aintenance policies for a ultiunit syste, which ay or ay not depend on each other. Although the coplexity of a ulti-unit syste poses challenges in finding the optial aintenance policies, this ay introduce an opportunity for a group replaceent of several coponents provided that a joint replaceent cost is less than that of the separate replaceents of the coponents [5]. For a syste consisting of only two identical coponents which are subject to exponential failure, Berg [6, Copyright 9 by ASME

2 7] proposed a preventive replaceent policy for a syste. The proble was forulated as a sei-markov process and proved that the control liits exist. In the case of the aintenance policy, however, nonnegligible repair ties still need to be taken into account to ake this policy ore realistic. Therefore, we present a ore realistic Markov odel and its optial preventive aintenance policy with two independent coponents. NOMENCLATURE S the states in Markov process the failure rate between states PM Preventive Maintenance RM Reactive Maintenance (= ) the nuber of interediate states T the tie interval between consecutive PMs T the tie for PM T the tie for RM (= /T) the transition rate (= /T ) the transition rate (= /T ) the transition rate MODELING OF MAINTENANCE POLICIES FOR A SINGLE MACHINE A syste that has one unit is the starting point in analyzing the siple degradation process because of its siplicity. We will then broaden our scope to the ultiple unit syste. The Markov process with three discrete states is used to odel the degradation process under the following assuptions: ) A syste will degrade gradually so that there are only sequential transitions between the initial and final states. ) Only three states (S : fully operational S : degraded but still operational and S 3 : failed) will be used to represent the degradation process. However, the nuber of states can be easily changed, depending on degree of specificity. 3) The failure rates ( and are constant between states. Q FIGURE : A THREE STATE SYSTEM AND ITS TRANSITION RATE MATRIX A degraded syste will eventually fail, requiring repair or replaceent. In general, aintenance actions cannot be odeled by a siple exponential transition in the Markov process. For exaple, a fixed tie repair or periodic inspection/repair does not follow the exponential distribution. The constant tie repair assuption ay be useful when only the Mean Tie to Repair (MTTR) inforation is provided fro a coponent or if the cause of coponent failure is well known so that repairing tie can be assued to be alost constant. PM is perfored for T after T tie units of continuous operation. If the syste fails before T tie units have elapsed, we perfor RM for a required tie T fro the tie of failure. Then, PM is rescheduled. For this odel, it is assued that the syste is as good as new after any type of aintenance is conducted. However, in general, the sae ethodology can be extended to any type of aintenance actions such as the perfect, inial, and iperfect aintenance. This PM policy is illustrated in Figure. FIGURE : ILLUSTRATION OF PM POLICY The syste and PM have been assued: ) The unit is repairable and the failure is self-announcing ) The unit will be as good as new after repair. (i.e., perfect aintenance) 3) The unit will be inspected after T tie units of operation. 4) PM is perfored for the constant tie T, and RM is conducted for the constant tie T. Thus, the Markov process for the above PM policy can be odeled as illustrated in Figure 3. The states here are representing: S i : Fully operational ( i ) S : Degraded but still operational ( i ) i S : PM ( i ) 3i S : RM ( i ) 4i T T T FIGURE 3 : MARKOV PROCESS FOR THE ABOVEMENTIONED PM POLICY Copyright 9 by ASME

3 The concept of a phase-type distribution [8-] is used to approxiate a constant tie delay in aintenance. It is also known that the Erlang process iniizes the variance aong any phase type distributions []. This approxiation of the constant tie delay in Markov process enables one to incorporate various aintenance activities into the equipent deterioration. The Markov process is created by jointly staking up the Erlang processes on top of the degradation odel in Figure. The siulation result of a saple path is illustrated in Figure 4. PM is perfored after the first inspection because the achine is in a degraded condition. RM is then conducted iediately after achine failure. MODELING OF MAINTENANCE FOR A TWO UNIT SYSTEM Without aintenance, the Markov process of the degradation process for two identical unit systes can be odeled as shown in Figure 5 with the following assuptions and states. However, the Markov odel of a two-unit syste will be uch ore coplex if we take aintenance policy into account. Saple Path of CTMC States RM 8 PM T 5 T 5 6 Degraded 4 Good tie FIGURE 5 : MARKOV MODEL FOR A TWO UNIT SYSTEM WITHOUT MAINTENANCE It is assued that the tie spent on aintenance depends on the achines condition at the oent of inspection. For instance, the tie T is required to repair if one of units is degraded. On the other hand, it will take the tie T if both are degraded. Since two different configurations (parallel and serial) are possible with two coponents as shown in Figure 6, both configurations are exained. FIGURE 4 : SAMPLE PATH FOR PM POLICY One advantage of using Markov process is that one is able to calculate any probabilities of interest in a closed for [8]. Therefore, we can exaine how the optial aintenance policy can be handled with the Markov process. The perforance of syste will depend on the value of inspection interval T given other syste paraeters such as,, T, and T. In other words, the controllable variable T can change the syste perforance. Let us assue that the perforance criterion of interest is the availability of the syste A(t), defined as the probability that the syste functions at tie t. Steady-state syste availability is then equal to A( ) li P P i i i in the Markov process of Figure 3. Then, the optial PM inspection interval T which axiizes the steady-state availability of a given syste will satisfy the following equation: FIGURE 6: PARALLEL (A) AND SERIAL (B) CONFIGURATION Parallel Configuration This parallel syste in Figure 6a can run a production line unless both of units fail. Therefore, RM will be perfored only when both of coponents are down (S 33 ) as shown in Figure 5. The units will be inspected after T tie units of operation and group repair of two units will then perfored. The corresponding Markov odel of Figure 8 is illustrated in Figure 7. T T T 4 5 T ( T T) T T T TT T e TT T e TT () The equation () is derived by solving the balance equations of the Markov process (see Appendices). FIGURE 7: MARKOV MODEL FOR A TWO UNIT PARALLEL SYSTEM 3 Copyright 9 by ASME

4 Parallel Configuration Serial Configuration FIGURE 8: MAINTENANCE IN PARALLEL CONFIGURATION Availability optia r = r = r = r = 3 Availability r = r = r = r = 3.. Serial Configuration For the serial connected syste, RM will be conducted whenever one of the coponents is down (S 3, S 3, S 3, S 3, and S33) as shown in Figure 5 because failure of one coponent will stop the entire production line. The corresponding Markov odel of PM policy in Figure 9 is illustrated in Figure. FIGURE 9: MAINTENANCE IN SERIAL CONFIGURATION T 3 T 4 T 5 T PM Interval PM Interval FIGURE : OPTIMAL PM INTERVAL WITH DIFFERENT CONNECTIONS By axiizing A( ) with respect to T, the optial tie interval between consecutive PMs is deterined. Let r be the ratio T T. Figure shows the effect of r on the availability and corresponding optial intervals for PM. For exaple, the optial value of interval is 93 tie units for r =. The availability declines only slowly as T exceeds its optial value; the decrease is uch faster if T is less than its optial value. If the duration for RM is not penalized enough (i.e., r ), the optial value for interval between consecutive PMs will go to infinity. In other words, for this case, running up to failure is the best policy. In the parallel case, axiization of productivity, N(t) rather than the availability of the syste is of interest because the syste is twice as productive when both coponents are functional. The productivity of the syste can be calculated by the equation (4). p ( ) ijk i3k 3jk i j k i k j k N P P P.6.4. (4) 64 Parallel Configuration r = r = FIGURE : MARKOV MODEL FOR A TWO-UNIT SERIAL SYSTEM Optial PM Intervals Maxiizing the availability of the syste can also be the objective of finding the optial PM inspection interval in twounit syste. The availabilities of the syste are given by the following equations: Parallel: Serial: A p ( ) Pijk (, i j) (3,3) k () A ( ) P (3) s ijk i j k Instead of finding a closed expression for steady-state probabilities P ijk and P ij, a nuerical ethod can be used to solve linear equations. N p () optia r = r = PM Interval FIGURE : OPTIMA PM INTERVAL TO MAXIMIZE PRODUCTIVITY OF SYSTEM Figure suggests that the optial interval values for PM are strongly dependent on r. For r =, the optial interval between consecutive PMs is 64 tie units, which is less than the 93 tie units that corresponds to the case of axiizing availability. In other words, aintenance will be conducted ore frequently. This suggests that we want to keep both of the units in either good state or degraded state in order to achieve axiu productivity. 4 Copyright 9 by ASME

5 CONCLUSIONS We have presented a ethod of obtaining an optial PM policy in a single unit syste as well as a two unit syste. The difference fro the previous works lies on the fact that we consider non-negligible repair tie and fixed tie periodic inspections for PM in a transient anner. A constant tie repair odel will be useful if a ean tie to repair inforation is available or tie for repair is alost constant. This is realized via the liit of the Erlang process. With a ore realistic aintenance policy, we have deonstrated the optial interval for PM in ters of availability of the syste. However, any assuptions are ade thus odifications on assuptions are still desired. First of all, it will be ore reasonable if the repair tie is a rando variable with a certain distribution rather than a fixed value. Secondly, there ay be ultiple aintenance tasks even within a single coponent syste. Even though the proposed Markov process can take the transient behavior of the syste into account, we have only used the steady state characteristics to find an optial aintenance policy. Therefore, transient characteristics should also be added in the aintenance optiization process. The required nuber of states needed to indicate ultiple degraded systes increases so rapidly that there is a coputational liit. Having ore than two units in a syste ight be difficult to solve analytically as the coplexity of the proposed ethod grows exponentially according to the nuber of coponents. REFERENCES [] Mccall, J. J., 965, "Maintenance Policies for Stochastically Failing Equipent: A Survey," Manageent Science, (5), pp [] Willia P. Pierskalla, J. A. V., 976, A survey of aintenance odels: The control and surveillance of deteriorating systes, Naval Research Logistics Quarterly, 3(3): pp [3] Ciriaco Valdez-Flores, R. M. F., 989, A survey of preventive aintenance odels for stochastically deteriorating single-unit systes, Naval Research Logistics, 36(4): pp [4] Scarf, P. A., 997, On the application of atheatical odels in aintenance, European Journal of Operational Research, 99(3): pp [5] Dekker, R., 996, Applications of aintenance optiization odels: A review and analysis, Reliability Engineering & Syste Safety, 5(3): pp [6] Toasevicz, C. L. and Asgarpoor. S., 6, Preventive Maintenance Using Continuous-Tie Sei-Markov Processes, in Power Syposiu, NAPS 6. 38th North Aerican. [7] Berenguer, C., Chu, C. B., and Grall, A., 997, "Inspection and Maintenance Planning: An Application of Sei- Markov Decision Processes," Journal of Intelligent Manufacturing, 8(5), pp [8] Assaf, D., Langberg, N. A., Savits, T. H., and Shaked, M., 984, "Multivariate Phase-Type Distributions," Operations Research, 3(3), pp [9] Osogai, T. and Harchol-Balter, M., 6, Closed for solutions for apping general distributions to quasiinial PH distributions, Perfor. Eval., 63(6): pp [] O'Cinneide, C. A., 99, Phase-Type Distributions and Majorization, The Annals of Applied Probability, (): pp [] David, A. and Larry, S., 987, The least variable phase type distribution is Erlang, Stochastic Models, 3(3): pp [] Si, S. H. and Endrenyi, J., 988, Optial preventive aintenance with repair, Reliability, IEEE Transactions on, 37(): pp [3] Si, S. H. and Endrenyi, J., 993, A failure-repair odel with inial and ajor aintenance, Reliability, IEEE Transactions on, 4(): pp [4] Cho, D. I. and Parlar, M., 99, A survey of aintenance odels for ultiunit systes, European Journal of Operational Research, 5(): pp. -3. [5] Sherif, Y. S., and Sith, M. L., 98, "Optial Maintenance Models for Syste Subject to Failure - a Review," Naval Research Logistics, 8(), pp [6] Berg, M., 976, Optial Replaceent Policies for Two- Unit Machines with Running Costs I, Stochastic Processes and their Applications, 4: pp [7] Berg, M., 977, Optial Replaceent Policies for Two- Unit Machines with Running Costs II, Stochastic Processes and their Applications, 5: pp [8] Ross, S. M., 996, Stochastic Processes, Second Edition ed. Probability and Statistics, John Wiley & Sons Inc 5 Copyright 9 by ASME

6 APPENDIX The steady-state syste balance equations are: ( ) Pi Pi (5) ( ) P P ( ) P i Pi P i P ( P P ) 3 P P 3i 3i (6) (7) P4 ( P P P ), i (8) P4i P4i Then, the steady-state availability in Markov process is equivalent to A( ) li P P i i i T T e e T T T T e e T TT e e By definition, an optial PM Policy is one which axiizes the steady-state availability of a given syste. Hence we wish to axiize the equation (9) by appropriate selection of T, which is the tie interval between PMs. Taking the derivative with respect to T, and setting it equal to zero, we obtain da( ) dt d dt T e T T e e T T T e T T T e e T T T T T e T T T e T T (9) 6 Copyright 9 by ASME