Availability Analysis with Opportunistic Maintenance of a Two Component Deteriorating System

Transcription

1 Analysis with Maintenane of a Two Component Deteriorating System Joel P. Varghese and Girish Kumar Abstrat This paper desribes the opportunisti maintenane model for availability analysis of two omponent series system using the Markov and the Monte Carlo Simulation methods. The system model is developed inorporating four states for eah omponent onsidering their level of degradation. Separate models are developed for three ases of orretive repair suh perfet, imperfet, and minimal repair with and without ondition based opportunisti maintenane. These models are solved using analytial Markov approah. The solutions are validated by the Monte Carlo Simulation. Sine the Markov approah is limited to exponential failure and repair rates, the Monte Carlo Simulation based odes an be further extended to aommodate systems whih follow non-exponential failure and repair rates. Index Terms, markov, monte arlo simulation, opportunisti maintenane. I. INTRODUCTION Maintenane has been defined as a ombination of tehnial and assoiated administrative ations intended to retain an item or system in, or restore it to, a state in whih it an perform its required funtion [1]. The nasene of modern high tehnology omponents has resulted in an inrease in omplexity and ost of industrial omponents and systems. This makes it important to maximize the utilization of these omponents and systems throughout its lifetime. In reent times, the total maintenane ost is largely beoming a substantial portion of the total operating ost of the system or omponent. Thus the development of a judiious and effetive maintenane poliy is a major onern for industrial and manufaturing engineers. Conventional maintenane poliies enompass simple orretive and preventive maintenane for omponents. Corretive maintenane is effetively illustrated by the philosophy if it isn t broken, don t fix it [2]. Preventive maintenane, on the other hand, onsists of ertain sheduled ativities that are performed while the system is still funtioning. The underlying aim behind preventive maintenane is to redue the number of unplanned system downtime due to system or omponent failure. In the urrent industrial senario of inreased demand and surfeit global ompetition, suh primitive maintenane poliies will not suffie and provide an advantageous edge over ompetitors. From the maintenane point of view, this advantageous edge an be failitated by inreasing system or Manusript reeived November 2, 2013; revised February 15, The authors are with Delhi Tehnologial University, Shahbad Daulatpur, Main awana Road, Delhi , India ( joelprine25@gmail.om, girish_kumar_ar@yahoo.om). DOI: /IJMMM.2014.V omponent availability whih means reduing downtime without ompromising on the system reliability. This has resulted in the adoption of muh more effiient maintenane poliies suh as ondition based maintenane and opportunisti maintenane. maintenane an be visualized as a fruitful ombination of orretive and preventive maintenane. maintenane makes maximum utilization of the time to perform orretive maintenane on a omponent or system by using this time to arry out preventive maintenane on omponents whih may require maintenane in the near future. maintenane is not a maintenane poliy for a single omponent, but for a olletion of omponents in a prodution line or plant [3]. The work in the field of opportunisti maintenane first began by [4]. The authors implemented the opportunisti poliy for unmonitored omponents/systems. Referene [5] shows how this poliy was investigated for two omponents in series, but no mathematial treatment for availability measure was proposed. An imperfet maintenane onept in this ontext was applied by [6]. Sine then, many extensions of opportunisti maintenane have been introdued and studied in the literature. The Markov and Semi-Markov approahes have been used in literature to analyze various maintenane poliies. Referene [7] shows how a Markov renewal proess was used to onstrut a maintenane ost model for a simple multi-unit system. A ontinuous time Markov hain was used by [8] to desribe the degradation of a omponent. A ost model was developed inluding the presene of resoure onstraints. Dynami programming was used by [9] to analyze a ondition based and opportunisti model for a two unit deteriorating system involving a ost model. The transient behavior of a multi-unit system was studied by [10] using renewal theory. The instantaneous availability for a fixed time was alulated and a methodology was developed to find times to preventive and opportunisti maintenane suh that ost is minimized. His omponents had binary levels of degradation, i.e. operating or failed. In the existing literature, the Markov model has been used to analyze systems undergoing various orretive and preventive repairs. The appliation of Markov for systems undergoing opportunisti maintenane has not been explored so far. Also, the existing literature doesn t take into aount multi-state degradation of omponents and onsiders omponents with binary states. The existing literature has resorted to a ost model to analyze and ompare various maintenane poliies without giving muh signifiane in quantifying the gains of opportunisti maintenane. This paper overomes these limitations by firstly developing a system model onsidering multi-state

2 degradation and providing an analytial method based on Markov to analyze the availability. maintenane poliies for different types of orretive repairs suh as perfet, imperfet, and minimal repairs have been onsidered. Aknowledging the limitation of the Markov method to be useful for omponents having exponential failure and repair rates, a Monte Carlo Simulation method is also presented whih has been orretly validated by the Markov model to extend this availability analysis to systems having non-exponential failure and repair distributions. The gains of an opportunisti maintenane poliy has been quantified in terms of availability of system and this provides us with a suitable deision making tool to hoose the best maintenane poliy and also ompare different ombinations of omponents in terms of availability for a partiular maintenane poliy. Software suh as RAPTOR (ARNIC, Annapolis, MD), loksim (Reliasoft, Tuson, AZ), et al., are widely used for availability analysis of repairable systems. These software programs do not yet have a module for opportunisti maintenane. The proposed methodology an be used as an extension for these software programs. The remaining paper is organized as follows. Setion II will enumerate the approahes used. Setion III will desribe the system modeling to inorporate multi-level degradation and orretive repair i.e. perfet, imperfet, and minimal, with and without opportunisti maintenane. The solutions of the models based on Markov and Monte Carlo Simulation will be desribed in Setion IV. Setion V will use an example to illustrate the methodology, and disuss the results. Finally, Setion VI onludes the work. 0 λyo (1) 1 λy1 (2) y0 y1 1 (3). Overview of Monte Carlo Simulation The Monte Carlo Simulation is a powerful and versatile tool based on random sampling whih has a wide range of appliations in various fields. One underlying priniple of Monte Carlo Simulation is the law of large numbers, whih states that the larger the sample, the more likely the sample mean will be a good estimate of the population mean [12]. The Monte Carlo Simulation an be used to model the availability of systems. It makes use of the failure and repair statistial distributions of omponents or systems to analyze the system behavior over time [13]. Initially, the simulation time to failure and repair time of the omponent or system are generated. Random numbers are reated using an uniform distribution (U) in the interval (0, 1), and a suitable onversion method is used to onvert these random numbers into the times for failure and repair. The frequently used onversion methods inlude the inverse transform method, omposition method and aeptane rejetion method. We shall use the inverse transform method in this artile due to its simpliity. However, its use is limited to when the repair and failure distributions an be inverted analytially. For an exponential distribution, with parameter λ, the expression for the random variable T whih an be the time to failure or time to repair using an inverse transform method is given by (4). II. APPROACHES 1 T =- ln(1 U ) A. Overview of Markov Method Markov models are frequently used in reliability and maintainability work where events, suh as the failure or repair of the module, an our at any point in time. The Markov method evaluates the probability of jumping from one known state into the next logial state. This ontinues until the system being onsidered has reahed the final or totally failed state or until a partiular mission time is ahieved. The basi assumption of a Markov Proess is that the behavior of the system in eah state is memory less [11]. For any given system, a Markov model onsists of a list of the possible states of that system, the possible transition paths between those states, and the rate parameters of those transitions. Let the symbol λ denotes the rate parameter of the transition from State 1 to State 2 and be yi(t) the probability of the system being in State i at time t. Fig. 1 shows the Markov model for a simple omponent with binary states. (4) Also, sine the (1-U) is distributed in the same way as U, T an be given by (5). 1 T =- ln U (5) Similarly, for a Weibull distribution with parameters θ and β the expression is (6). T ln U 1 (6) After the generation of the failure and repair times of all omponents or systems aording to the appropriate distribution, the sequene of failures and repairs for the system are simulated for a speifi mission time. This proess is repeated for a large number of simulations. The system availability for a given mission time is alulated as the ratio of the total up time of the system for the total number of simulations to the total of mission time of all of the simulations. Fig. 1. Markov model for a simple omponent. The set of differential equations for this simple model are given by (1), (2) and (3). 156

3 III. SYSTEM MODELING The system studied in this researh is a two omponent series multi-unit system shown in Fig. 2. The term unit ould be a omponent, a set of omponents, or even a subsystem whih an be part of a greater robust or intriate industrial system. Fig. 2. System model. A. Modeling Multi-Level Degradation Pratially any omponent/unit whih is part of an industrial system undergoes multiple levels of degradation before reahing the failed state. Multi-level degradation, therefore, implies that the omponent will not have a onstant failure rate up to the failed state, but the failure rate will vary as the omponent reahes the failed state. Eah omponent is assumed to have four levels of degradation with are given by, State 1(Initial State / Good as new), State 2 (Middle State), State 3(etter than worst), and State 4(Failed State).. Ation Component A Sine a series system is onsidered, one a omponent fails the system will shut down. Corretive repair must be performed on the failed omponent. The orretive maintenane an be of three types namely perfet, imperfet, or minimal repair. Perfet repair implies that the omponent is repaired from the failed state to the good as new state (State 1). Imperfet repair implies that the omponent is repaired from the failed state to the middle state (State 2). Imperfet repair implies a repair of lesser intensity as ompared to perfet repair but of a quiker duration. Minimal repair implies that the omponent is repaired from the failed state to the better than worst state (State 3). Minimal repair is of least intensity and quikest duration. This downtime during the orretive repair proess provides an opportunity to perform preventive maintenane on the non-failed omponent. C. Model Development Component The proposed model of two omponent series system inorporating multi-level degradation an be developed onsidering the above mentioned multi-level degradation and repair ation. The system ontains two omponents eah having a possibility of being in one of the four states at a partiular instant of time. Their series ombination gives a total of sixteen possible states the entire system an exist throughout its lifetime. This is inlusive of the system state in whih both omponents are in a failed state. In a series system, this state an our only if both the omponents fail simultaneously. The probability of this event is onsidered to be negligible and an be negleted. The generalized assumptions made for the model are as follows. 1) The opportunisti maintenane ativity is assumed to be ompleted before the failed omponent gets repaired. 2) The omponent an funtion perfetly in all states exept the failed state. IV. SOLUTION TO THE SYSTEM MODEL A. Analytial Markov Approah Solution of the system model with the analytial Markov approah involves the following steps. Step 1: Construt a state diagram of the system representing all possible states of the system. The transition from one state to another is speified by an arrow whose diretion indiates the diretion of transition. Transition rates from one state to another an also be speified in the state diagram by overlaying it with the arrow. Step 2: Model the differential equation for a partiular system orresponding to eah state in the similar lines as (1), (2) and (3). Step 3: Solve the differential equations simultaneously with suitable initial ondition and the required mission time. The availability of the system at the speified mission time is alulated by summing the probabilities of the non-failed system states at a partiular mission time.. Monte Carlo Simulation The Monte Carlo Simulation odes whih simulate the funtioning of the system throughout its mission time are developed, and availability is omputed for a large number of trails. The Monte Carlo Simulation odes are based on the following algorithm. Step 1: Generate random numbers with uniform distribution orresponding to eah possible transition of omponent A and (failure and repair). Step 2: Convert these numbers into a value of operating time using the inverse transform method based on the exponential distribution or a non- exponential distribution based on the required ondition. Step 3: At eah state ompare the times of the possible transition and proeed to the state with the minimum transition time while reduing this time from the transition time of the state not hosen. Step 4: When a failed state is reahed, add the repair time to the system downtime and reset the transition times and the repair time for the failed omponent. Step 5: Repeat the steps 1-4 till the umulative time of the sequene is equal to or more than mission time. Step 6: Add the downtime for this trail to the total downtime. Step 7: Repeat steps 5-6 for the required number of trails. Step 8: Calulate the system availability by dividing the total downtime upon the total mission times after all the trails. V. ILLUSTRATION A power system onsisting of a power transformer (Component A) and a protetion and ontrol subsystem for power transformer (Component ) in series was seleted for the ase stu to demonstrate the methodology for availability assessment. The exponential failure and repair distribution is assumed for eah subsystem and the data for failure rate and repair rate is taken from the literature [14]. Table I shows the literature values for failure and repair rates of both the omponents. To inorporate multi- level degradation in the example, the 157

4 TALE I: LITERATURE VALUES FOR THE COMPONENTS assumption is made that the failure rates of eah of the omponent inreases as it deteriorates. This assumption is based on the disussions with experts from the industry who are working in the field of maintenane. The failure and repair rates for different degrading states are obtained by dividing the binary state data available from the literature in onsultation with the industry experts. Failure rates (λ) are in failures per hour and repair rate (μ) are in repairs per hour. The supersript and subsript represent the omponents and the state transition respetively. The respetive alulated values are , 12 = x 10-4, A 23 = x 10-4, A 34 = A 10-4, x 12 = Component Temporary Failure Rate Sustained Failure Rate Average Failure Rate Mean Time Power Transformers ( kv) 0.4 / ( / ( / ( hours Protetion and ontrol for power transformers ( kv) 0.9 / ( / ( / ( hours x 34 =5.136x10-4, A 41 =8.621x10-3, A 42 = x 10-3, A 43 = x 10-3, 41 = 6.76 x 10-2, 42 = x 10-2, 43 = 27.03x Fig. 4. Corretive maintenane (perfet repair) with opportunisti maintenane. Ation Fig. 3. Corretive Maintenane (Perfet ). One the transition parameters of failure and repair have been defined the state diagram is drawn. The state diagram for perfet repair is shown in Fig. 3. Fig. 4 shows the state diagram for perfet repair with opportunisti maintenane. Due to spae onsiderations the state diagrams for imperfet and minimal repair with and without opportunisti maintenane are not shown but an be developed along the similar lines. The failure rates and repair rates an also be expressed in terms of system state as λij or μij where i represent the state of the system before transition and j represents the state after transition. From Fig. 3 and Fig. 4 it is notieable that 12 A TALE II: SYSTEM AVAILAILITY without without with Maintenane Maintenane Maintenane using Monte using using Carlo Markov Markov Simulation with Maintenane using Monte Carlo Simulation Perfet Imperfet Minimal Similarly the set of differential equations an be developed for the ases of imperfet and minimal repairs with and without opportunisti maintenane. The sets of differential equations for eah model are solved using MATLA (Matlab ) with initial onditions as y1(0)=1, y2(0)=0, y3(0)=0, y4(0)=0, y5(0)=0, y6(0)=0, y7(0)=0, y8(0)=0, y9(0)=0, y10(0)=0, y11(0)=0, y12(0)=0, y13(0)=0, y14(0)=0, y15(0)=0 and for larger values mission time to get the stea state results. The system availability is obtained by summing the probabilities of = λ15 = λ26 = λ37. If yi represents the probability of the system being at the state i at the time t, then as speified in Setion II-A, the set of differential equations are derived for the models shown in Fig. 3 and Fig. 4. The set of equations for Fig. 3 are listed in Appendix I and for Fig. 4 in Appendix II. 158

5 operating states. The availability results based on Markov are shown in Table II. The solution of the model is also obtained using Monte Carlo Simulation Methods. The odes are developed based on the algorithm in setion IV-. The system availability results based on the Monte Carlo Simulation are shown in Table II. The results learly show that when the system undergoes opportunisti maintenane there is an inrease in the system availability for perfet, imperfet, and minimal repair. This implies that for the given system an opportunisti maintenane poliy would inrease the availability irrespetive of the maintenane ation. The Markov and Monte Carlo Simulation give us mathing values of system availability. Thus the Monte Carlo Simulation ode has been validated and an be further extended to omponents having a non-exponential failure and repair rates, suh as Weibull or Lognormal. Quantifying the gains of opportunisti maintenane using an availability measure helps to identify the maintenane poliy resulting in maximum availability. The above results for the illustrative example learly show that if the system follows an opportunisti maintenane poliy with imperfet repair, the system availability will be maximized. This is valid for the urrent set of data but depending upon the failure and repair rates, the poliy may hange. This doesn t imply that it is the optimal maintenane poliy. The trade-offs in eah ase should be onsidered in terms of ost, effet on omponent life (as an imperfet or minimal repair results in more frequent system failure) and the opportunisti maintenane poliy seleted must be suh that the eonomi gain due to the inrease in system availability must ompensate for the ost aquired by performing opportunisti maintenane. VI. CONCLUSION In this artile an analytial Markov model is applied to find the system availability of a two omponent series system with opportunisti maintenane. Separate models are developed for different orretive ations namely perfet, imperfet, and minimal repairs with and without opportunisti maintenane. The results of system availability in all ases are validated using the Monte Carlo Simulation. The Monte Carlo simulation approah an be further extended for omponents having non-exponential distributions suh as Weibull and lognormal distributions. This approah an be used as an extension for the existing software whih does not have the opportunisti maintenane module. The suggested approah provides a realisti assessment of availability values of the repairable system due to the inorporation of multi-state degradation. This work an be further extended to more omplex systems with more omponents in various onfigurations having more levels of degradation. A ost funtion an also be developed to analyze the eonomi pros and ons of arrying out opportunisti maintenane. There is a sope for development of a model for an optimal maintenane poliy based ost and availability under the limited maintenane resoures. APPENDIX I y1 41y4 131 y y2 12 y1 41y y3 23 y2 153 y y3 41y y5 15 y1 85 y y6 26 y2 56 y y7 67 y6 37 y y7 85 y y9 59 y5 129 y y y9 610 y6 11 y y y y y y9 131 y y y y y APPENDIX II y1 41y4 131 y13 81y8 121 y y y y2 12 y y3 23 y y3 41y y5 15 y1 6 y y y 7 y y y y9 59 y

6 y y y y y y y y y y y y y y REFERENCES [1] Colletion and exhange of reliability and maintenane data for equipment, ISO 14224, [2] J. Huang, Preventive maintenane program development for multi-unit system with eonomi dependeny- stohasti modeling and simulation stu, Ph.D. Dissertation, Dept. of Industrial and Management Systems, University of South Florida, University of South Florida, Tampa, [3] R. P. Niolai and R. Dekker, Optimal maintenane of multi-omponent systems: A review, K. A. H. Kobbay and D. N. P. Murthy, eds, London, [4] R. Radner and D. W. Jorgenson, replaement of a single part in the presene of several monitored parts, Management Siene, vol. 10, issue 1, pp , [5] F. A. V. D. D. Shouten and S. G. Vanneste, Analysis and omputation of (n, N)strategies for maintenane of a two omponent system, European Journal of Operational Researh, vol. 48, issue 2, pp , [6] H. Wang, H. Pham, and A. E. Izundu, Optimal preparedness maintenane of multi-unit systems with imperfet maintenane and eonomi dependene, Reent advanes in reliability and quality engineering. New Jersey: World Sientifi, pp , [7]. Castanier, A Grall, and C. erenguer, A ondition-based maintenane poliy with non-periodi inspetions for a two-unit series system, Reliability Engineering and System Safety, vol. 87, pp , January [8] S. Ambani, L. Li, and J. Ni, Condition-ased maintenane deision-making for multiple mahine systems, Journal of Manufaturing Siene and Engineering, vol. 131, pp , [9] Z. Zhang, S. Wu, and. Li, A ondition-based and opportunisti maintenane model, in Pro. International Conferene on Quality, Reliability, Risk, Maintenane, and Safety Engineering, pp , Xi'an, [10] S. Jain, maintenane poliy of a multi-unit system under transient state, M.S. Thesis, Dept. of Industrial and Management Systems, University of South Florida, Tampa, Florida, [11] Relex Software Tehnial rief. Markov Analysis Aurately Models Dynami ehaviors. [Online]. Available: [12] MIL-HDK-338, Military Handbook, Eletroni Reliability Design Handbook, Washington, DC, USA: United States Department of Defense, [13] A. Dennis, Appliation of Monte Carlo simulation system reliability analysis, presented at the 20th International Pump Users Symposium, Houston, Texas, Marh 17 20, [14] F. Roos and S. Lindahl, Distribution system omponent failure rates and repair times - an overview, presented at the Nordi Distribution and Asset Management Conferene, Espoo, Finland, August 23-24, Joel P. Varghese is a third year engineering student. He is pursuing his bahelor of tehnology degree in mehanial engineering at Delhi Tehnologial University, Delhi, India. Girish Kumar is presently an assistant professor in Mehanial Engineering Department at Delhi Tehnologial University, Delhi, India. He is also pursuing his dotoral degree from the Indian Institute of Tehnology, Delhi, India. His urrent researh interest is in the RAM modeling and analysis of repairable mehanial systems. 160