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1 Coyright 2015 IEEE. Rerinted, with ermission, from Huairui Guo, Ferenc Szidarovszky, Athanasios Gerokostooulos and Pengying Niu, On Determining Otimal Insection Interval for Minimizing Maintenance Cost, 2015 Reliability and Maintainability Symosium, January, This material is osted here with ermission of the IEEE. Such ermission of the IEEE does not in any way imly IEEE endorsement of any of ReliaSoft Cororation's roducts or services. Internal or ersonal use of this material is ermitted. However, ermission to rerint/reublish this material for advertising or romotional uroses or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to By choosing to view this document, you agree to all rovisions of the coyright laws rotecting it.

2 On Determining Otimal Insection Interval for Minimizing Maintenance Cost Huairui Guo, PhD, ReliaSoft Cororation Ferenc Szidarovszky, PhD, ReliaSoft Cororation Athanasios Gerokostooulos, ReliaSoft Cororation Pengying Niu, ReliaSoft Cororation Key Words: maintenance lan, insection interval, reairable systems SUMMARY & CONCLUSIONS The majority of system failures do not occur without any warning signs. This is esecially true for failures caused by degradation. By eamining the failure-critical indees during scheduled insections, actions can be taken to address degraded comonents and revent big losses due to failures. In this aer, we assume that an imminent failure can be noticed when an insection is conducted during a short time eriod right before the failure. Clearly, by conducting frequent insections, failures can always be detected and revented. However, the total insection cost will be very high if the insection interval is too short. On the other hand, if the insection interval is too long, a coming failure may not be effectively detected and the total cost due to failures will be high. Therefore, an otimal insection interval balancing these two costs needs to be identified. A model for determining the otimal insection interval to minimize the maintenance cost is roosed in this aer. The analytical solution of the model is rovided and comared with simulation results. The roosed method is esecially useful for rocess industries such as oil and gas refineries, food rocessing and harmaceutical manufacturing. By conducting otimal insections for comonents with degradation characteristics, failures will be revented, maintenance cost will be reduced and the rocess throughut can be imroved. 1 INTRODUCTION Preventive relacement and eriodical insections are two major aroaches used for imroving system availability and reducing maintenance cost. Preventive relacement relaces a system/comonent/unit at a certain time interval or at a failure, deending on which one is earlier. A great deal of work has been done on insection and reventive maintenance strategies [1, 2, 3]. Most of the ublished aers on insection strategies focus on finding hidden failures. Hidden failure refers to the case where a failure remains undiscovered unless an insection or a test is erformed [4]. The interval between two successive insections is therefore called the failure finding interval. There are two tyes of hidden failures in general: Tye I: Protective devices or standby unit. The function of these devices is to rotect the main system in case of failures. Safety devices, emergency devices, standby units are this tye of hidden failure devices [5]. Their failure will not cause direct loss if they are not needed. Tye II: Oerating devices. Underground ies and underwater equiment are this tye of devices. They are oerating systems, and their failure will cause direct loss. Different from the above mentioned hidden failures, some failures can be detected even before they haen. For eamle, a long crack length on a shaft or a thinning connector line is an early sign of an imminent failure. This tye of failures is called revealed failures. In this aer, we will focus on reventing revealed failures using insection, rather than finding hidden failures as in the ublished aers [1, 5, 6-10]. This insection strategy is esecially useful for failures caused by degradation. Units that are going to fail soon will be relaced or reaired at insection if there is evidence of an oncoming failure. For eamle, if a failure is going to haen at time 100, an insection at time 20 may not be able to detect it since the signs of the coming failure may not be strong enough. However, if an insection is conducted at time 95, then we should see strong evidence that the comonent is going to fail soon. Therefore, a failure detection criterion for a device needs to be established based on engineering knowledge. It could be a fied time eriod such as an insection that is conducted within 10 hours before the occurrence of failure. It also can be a fied ercentage of life consumtion such as an insection that is conducted after 90% of the failure times. In this aer, we will use the ercentage criterion. The value of the ercentage can be determined based on engineering knowledge. Past failure information and warning signs such as degraded erformance of a comonent, increasing temerature or vibration level can hel engineers determine a reasonable ercentage value. To better understand the failure detection criterion based on ercentage of life, let s define the failure time as t for the /15/$ IEEE 214

3 unit under consideration, and as the fied ercentage. If an insection is conducted in interval [ t, t ], then the failure will be noticed before it occurs. Therefore interval [ t, t ] is called failure detection zone. If an insection is conducted within a failure detection zone, then the coming failure will be noticed. Otherwise, a failure will not be noticed during insection and it will occur later. These two scenarios are illustrated in Figure 1, where, 2, and k are the scheduled insection oints. For a given insection interval, the first question arising is the robability of noticing a coming failure. If the robability is very low, then we need to select a smaller to increase the chance of detecting coming failures. 2.1 Calculate the Probability of Noticing a Coming Failure From Figure 1, it is clear that if t k t, or in other words, if a failure time meets the following condition: k k t, (1) then failure will be noticed in advance. Equation (1) can be illustrated in Figure 2. Figure 1 Insection for Detecting a Coming Failure In Figure 1, the red star is the failure time t. The yellow shaded area is the failure detection zone. The aer is organized as follows. In section 2, two models will be roosed. One model is for determining the otimal insection interval for minimizing the long term maintenance cost er unit time. The other model is for the otimal insection interval to minimize the cost er unit time in one insection renewal cycle, called short term maintenance cost. Section 3 gives eamles of using the roosed method. Simulation results will be used to validate the analytical solutions. Section 4 concludes the aer. 2 OPTIMAL INSPECTION FOR DETECTING COMING FAILURES Two objectives are usually considered when determining an otimal insection interval. They are: 1) imroving the system availability, and 2) minimizing the maintenance cost. In this aer, our urose is to minimize the maintenance cost. The following assumtions are used: Insection intervals are based on system (or comonent) age (such as mileages, hours of oeration), not calendar time. Failure time is a random variable following a cumulative distribution function Ft (). The time required for insection is negligible. If an insection is within a certain ercentage (0 < < 1) of a coming failure time, then the system will be relaced and the failure is revented. A failure will be noticed right away when it occurs. System is relaced either at failure or at insection if the insection time meets the failure detection criterion. The time for relacing a system under insection is negligible. An insection will not affect the age of the system. A new cycle starts (the system is renewed) when the system is relaced. Insections do not introduce failures. Figure 2 Failure Zones for Detectable Failures The shaded area in Figure 2 is called failure zone for detectable failures. There are two cases for failure zones: Case 1: This case is shown in Figure 2, where k < ( k + 1). It means that there is no overla between the kth and the (k+1)th failure zones for noticeable failures. Failures that fall in the failure zones will be noticed. For case 1, we have: k < ( k + 1). That is: k < (2) 1 Therefore, insections with numbers less than ( 1 ) belong to case 1. Case 2: This is the case where k ( k + 1). It means that the kth and the (k+1)th failure zones are overlaed and so do all later failure zones. Therefore, any failures occurring after k will always be noticed through insection, and the system will be relaced at insection. Similar to case 1, insections with numbers larger than or equal to ( 1 ) belong to case 2. In case 1, the robability of noticing a failure before it occurs is: k P1 = F F ( k) (3) where F() is the cumulative distribution function for the failure time of the system. It is assumed to be known. Define M as the largest integer less than ( 1 ) : 1 if is an integer 1 1 M = (4) if is not an integer 1 1 The robability of noticing a failure before it occurs for case 2 215

4 is: P2 = 1 F ( M + 1) (5) From Equations (3) and (5), we can obtain the combined robability of noticing a coming failure: M k P = F F ( k) + 1 F [( M + 1) ] k = 1 (6) From Equation (6), we can see that if 0, the robability of noticing a failure before it occurs is 1. This is true because if the insection interval is 0, then the system is being insected continuously. If, then no insection is conducted and the robability of noticing a coming failure is Otimal Insection Interval Criteria There are two commonly used criteria for determining an otimal insection interval. One is for minimizing the average cost within one insection cycle, and the other is for minimizing the long term average cost. Let s use an eamle to elain the differences between these two. Assume a failure of a comonent can be detected any time after 90% of its life has been consumed. Insections are scheduled every 20 hours of oeration. A failure will cost $50. An insection will cost $5 and a reventive relacement will cost $20. Now assume a new comonent failed after 18 hours of oeration. Since the failure occurs before the first insection which is at 20 hours, the failure cannot be revented. Therefore, the length of this relacement cycle is 18 hours, and the cost is $50. The failed comonent is relaced with a new one and a new cycle started. Assume this newly installed comonent will fail at 21 hours. 90% of its life is 18.9 hours. Since the scheduled insection is at 20 hours of oerating, which is between 18.9 and 21, the failure will be noticed before it occurs. The comonent is relaced at 20 hours. The cost of this insection is $25 (insection cost + reventive relacement cost) and the length of this cycle is 20 hours. The average cost er unit time in one cycle is calculated as follows. The cost er time for the first relacement cycle is $50/18 = $2.7778/hour; the cost er time for the second relacement cycle is $25/20 = $1.25/hour. The average cost er unit time in one cycle is the average of these two. It is ($ $1.25)/2 = $2.0139/hour. The long term average cost is the total cost divided by the total oeration time. This is ($50+$25)/(18+20) = $1.974/hour. This is the same as the average cost er cycle divided by the average length of one cycle, if you divide both the numerator and denominator by 2. The formulas for the two cost rates are given net. For a given renewal (relacement) cycle, the eected average cost rate in one cycle is: Cost in one cycle CS ( ) = Eected (7) Time of one cycle Equation (7) is the eected one cycle cost rate, or the short term cost rate. It is used when different comonents are eected to be used over an oeration eriod, and its urose is to minimize the cost er unit time over one relacement cycle only. The eected long term cost rate with an insection time interval of can be calculated based on the renewal reward rocess theory, and it is Eected cost in one cycle CL( ) = (8) Eected time of one cycle The long term cost rate in Equation (8) is used when the same otimal insection interval and the same tye of comonents are lanned to be used for a long time. Our objective is to find an otimal insection interval to minimize either the long term cost rate or the short term cost rate. The following three cost tyes are considered in the calculation: C P is the cost if a coming failure is noticed before it occurs. This is the cost of a reventive relacement. C F is the cost if a failure occurs without being noticed during revious insections. This is the cost of a failure relacement. C I is the cost of conducting one insection. Clearly, the cost of reventing relacement should be less than the cost in case of a failure, so C P should be less than C F. The cost of conducting an insection should also be less than the cost of a failure relacement, so C I should be less than C F. In the following sections, we will resent models for minimizing the short term cost rate and a model for minimizing the long term cost rate. 2.3 Otimal Insection Interval for Minimizing Short Term Cost Rate If a failure occurs before the first insection, it will never be noticed through insection, so the cost rate for this failure is: C F C ( ) =, when 0 t< (9) t From section 2.1, we know that if a failure occurs in the first detection zone, it can be noticed by the first insection, so the cost rate for this failure is: CP + CI C ( ) =, when t< (10) We also know that if a failure occurs between the first and the second detection zone, then it cannot be noticed by insection, so the cost rate for this failure is: CF + CI C ( ) =, when t< 2 (11) t The insection cycle restarts whenever the system is relaced. The general formulas for the cost rate within a renewal cycle are given net. Case 1: When k M, the detection zones do not overla. Some of failures will be detected before they occur but not all. When a failure is in the kth detection zone, it will be detected. The cost rate is: 216

5 CP + kci k C( ) =, when k t < (12) k When the failure is between the kth and the (k+1)th detection zone, it will go unnoticed, so the cost rate is: CF + kci k C ( ) =, when t< ( k+ 1) (13) t Case 2: When k M + 1, the detection zones are overlaed. All failures occurring in or after the (M+1)th detection zone will be detected in advance. When a failure occurs in the (M+1)th detection zone, the cost rate is: CP + ( M + 1) CI C ( ) =, ( M + 1) (14) when ( M + 1) t < ( M + 1) When a failure occurs between the end of the (M+1)th detection zone and the end of the (M+2)th detection zone, the cost rate is: CP + ( M + 2) CI C ( ) =, ( M + 2), (15) when ( M + 1) t < ( M + 2) and so on for all larger values of k. Combining the formulas of case 1 and case 2, the final cost rate function in one cycle becomes: k M ( k+ 1) CF CP + kci CF + kci CS ( ) = f () t dt + f () t dt f () t dt 0 t + k = 1 k k k t ( M+ 1) ( k+ 1) CP + ( M + 1) CI CP + ( k+ 1) CI + f () t dt + f () t dt ( M + 1) ( k + 1) ( M+ 1) k= M+ 1 k (16) The only unknown in Equation (16) is the insection interval. The value of that minimizes Cs ( ) can be found numerically. However, the simlest way for finding an aroimated solution is to lot the cost vs. curve. The otimal can then be identified from the curve. 2.4 Otimal Insection Interval for Minimizing Long Term Cost Rate The long term cost rate is calculated from Equation (8). From the revious section we know that when the insection interval is, the eected length for a renewal cycle is: E 0 ( Cycle Length ) k M ( k+ 1) tf () t dt + ( k) f () t dt + t f () t dt k = 1 k k ( M+ 1) ( k+ 1) ( M+ 1) [ ] = k= M+ 1 k [ ] + ( M + 1) f ( t) dt + ( k + 1) f ( t) dt (17) Equation (17) takes the denominator of each term in equation (16). The eected total cost in a renewal cycle is: E(Total Cost in One Cycle )= C f ( t) dt k M ( k+ 1) + [ CP + kci] f () t dt + [ CF + kci] f () t dt k = 1 k k ( M+ 1) P ( M+ 1) ( k+ 1) k= M+ 1 k [ ] + C + ( M + 1) C f ( t) dt [ ] + C + ( k + 1) C f ( t) dt P I I 0 F (18) Equation (18) takes the numerator of each term in equation (16). So the long term cost rate is: E(Total Cost in One Cycle ) CL( ) = E( Cycle Length ) (19) The only unknown in Equation (19) is the insection interval. The value of that minimizes CL( ) can be found either numerically or grahically, similar to the revious case. 3 EXAMPLES Two eamles will be given in this section. The first is for the short term cost otimization, and the second is for the long term cost otimization. 3.1 Eamle for Otimal Insection Interval for Minimizing Short Term Cost Rate Problem Statement: Assume a comonent is used in a manufacturing rocess and it needs to be relaced eriodically. The following information is rovided: the average cost er insection is $10. If an imminent failure is noticed at insection, the relacement cost for the comonent is $200. However, if a failure haens, the cost will be $1,000. The failure detection threshold is 90% of the failure time. It is also known that the failure time distribution of this comonent is a Weibull distribution with β = 2 and η = 500 hours. Find the otimal insection interval that will minimize the short term (one cycle) cost. Analytical Solution: Alying Equation (16) with different insection interval and conducting calculations using Mathcad, we get the results shown in Table

6 Table 1 Results for Short Term Insection Cost and Failure Detection Probability Insection Interval (hour) The first column is the insection interval while the second column is the long term cost rate for the corresonding value of. It is clear that when the insection interval is about 20 hours, the eected long term cost rate is minimal. The results in Table 1 also are resented in Figure 3. Figure 3 Plot for the Short Term Insection Cost Figure 3 shows that the eact otimal insection interval is hours long and the corresonding cost is $1.865 er hour (the red dot with a circle on the lot). Since the cost at 20 hours is almost the same as at the otimal solution, we can consider 20 as the otimal insection interval. Equation (6) gives the robability of detecting an imminent failure by insection. Since the detection threshold for the consuming life is 90%, we have: and from Equation (6), we get: Short Term Cost Rate ($/hr) 0.9 M = 1= 1= Detection Probability M k P = F F k + F M + k = k = F F( 20k) + 1 F[ 180] k = = ( ) 1 [( 1) ] This is the robability of noticing an imminent failure when the insection interval is 20 hours. The detection robabilities for different insection intervals are also given in the third column of Table 1. Simulation Results: The above analytical solutions are validated using simulation. 25,000 simulations were conducted using an insection interval of 20 hours. One simulation ends whenever a system is relaced, either at a failure or at an insection if the imminent failure is detected. In each simulation, a failure time is generated first. Assume a generated failure time is 51.44, then we know its failure detection zone is [46.296, 51.44] because = Since the insection interval is 20, no multiles of 20 are included in this interval, therefore no insection is scheduled within this detection zone. This simulation (insection cycle) ends at 51.44, the failure time. The cost for this cycle is the cost of the failure lus the cost of the two insections conducted at time 20 and 40. It is $1, $10 = $1, 020. The cost er unit time for this cycle is $1, 020 / = $ If for another simulation the generated failure time is 318.5, then the failure detection zone is [286.65, 318.5]. A scheduled insection at time 300 is in this detection zone, so this simulation (insection cycle) ends at 300. The unit is relaced at time 300, and the failure at is revented. The cost for this cycle is the cost of the relacement lus the cost of the 15 insections conducted during this cycle. It is $ $10 = $350. The cost er unit time for this cycle is $350 / 300 = $ Reeating the above rocess 25,000 times by generating 25,000 failure times, 25,000 cost er unit time in one cycle will be obtained. Taking the average of them, the final simulated average short term cost er hour is $ To get the simulated failure detection robability, we can divide the number of simulations that end at insection by the total number of simulations of 25,000. The obtained value is Both the cost and the detection robability are very close to the analytical results of $1.867 and The simulation results suort that our roosed method is correct. 3.2 Eamle for Otimal Insection Interval for Minimizing Long Term Cost Rate For the same eamle as in the revious section 3.1, we also can calculate the long term cost for different insection intervals, if the same comonent is lanned to be used for a long eriod of time. Analytical Solution: Alying Equations (17-19) with different insection interval and conduct the calculation in Mathcad, we get the average long term insection costs, which 218

7 are given in Table 2. Table 2 Results for Long Term Insection Cost Insection Interval (hour) We can see that when the insection interval is about 30 hours, the eected long term cost rate is minimal. The results in Table 2 are also illustrated in Figure 4. Figure 4 Plot for the Long Term Insection Cost Simulation Results: The above analytical solutions are validated using simulation. 25,000 simulations were erformed using an insection interval of 30 hours. The simulation rocess is the same as the one used for the short term cost. The only difference is how the cost er unit time is calculated, as elained at the beginning of section 2.2. The calculated long term cost er hour from the simulation results is $ This result is very close to the analytical result of $ The simulation result suorts that our roosed method is correct. 4 CONCLUSIONS Long Term Cost Rate ($/hr) Periodical insection and reventive maintenance are two commonly used strategies for reducing maintenance cost and imroving system availability. Otimal relacement by eriodical reventive maintenance has been etensively studied in the ast [1-5]. Otimal insection interval for detecting hidden failures that have occurred is also studied [6-10]. However, little work has been done on otimal insection for detecting an imminent failure and reventing it. In this aer, we roosed two models for determining otimal insection intervals. Numerical eamles show that the roosed analytical methods are correct since the results match the simulation results very well. REFERENCES 1. T. Nakagawa, Maintenance Theory of Reliability. Sringer, London, S. Osaki, Stochastic Models in Reliability and Maintenance. Sringer, Berlin, H. Wang and H. Pham. Reliability and Otimal Maintenance. Sringer, London, M. A. Worthman, G. A. Klutke, H. Ayhan, A Maintenance Strategy for Systems Subjected to Deterioration Governed by Random Shocks, IEEE Trans. Reliability, vol. 43, 1994, A. K. Jardine, A. H. Tsang. Maintenance Relacement, and Reliability Theory and Alications. CRC Press, Boca Raton, FL, J. Sarkar, S. Sarkar, Availability of a Periodically Insected System under Perfect Reair, Journal of Statistical Planning and Inference, vol. 91, 2000, J. K. Vaurio, A Note on Otimal Insection Interval, International Journal of Quality Reliability Management, vol. 11, 1994, T. Tang, Failure Finding Interval Otimization for Periodically Insected Reairable Systems, Phd Dissertation, University of Toronto, F.G. Badia, M.D. Berrade, C. A. Camos, Otimal Insection and Preventive Maintenance of Units with Revealed and Unrevealed Failures, Reliability Engineering and System Safety, vol. 78, 2002, J. K. Vaurio, Availability and Cost Functions for Periodically Insected Preventively Maintenance Units, Reliability Engineering and System Safety, vol. 63, 1999, Huairui Guo ReliaSoft Cororation 1450 S. Eastside Loo Tucson, AZ, BIOGRAPHIES Harry. Guo@ReliaSoft. com Dr. Huairui (Harry) Guo is the Director of the Theoretical Develoment Deartment at ReliaSoft Cororation. He received both his PhD in Systems & Industrial Engineering and M. S. in Reliability & Quality Engineering from the University of Arizona. His research and ublications cover reliability areas, such as life data analysis, reairable system modeling and reliability test lanning, and quality areas, such as rocess monitoring, analysis of variance, and design of eeriments. In addition to research and roduct develoment, he is also art of the training and consulting arm 219

8 and has been involved in various rojects from the automobile, medical device, oil and gas, and aerosace industries. He is a certified reliability rofessional (C. R. P), ASQ certified CQE and CRE. He is a member of IIE, SRE, and ASQ. Ferenc Szidarovszky ReliaSoft Cororation 1450 S. Eastside Loo Tucson, AZ, Ferenc. Szidarovszky@Reliasoft. com Dr. Ferenc Szidarovszky received his education in Hungary, where he earned two PhD degrees, one in mathematics, the other in economics. The Hungarian Academy of Sciences has awarded him with the degrees of "Candidate in Mathematics" and "Doctor of Engineering Sciences." Between 1973 and 1986 he was an investigator in several joint research rojects between Hungarian and American researchers, and in 1988 he became a rofessor in the Systems and Industrial Engineering Deartment of the University of Arizona. In 2011 he retired from the university and joined ReliaSoft as a senior researcher. His research interests are decision making under uncertainty, multi-criteria decision making, game theory, conflict resolution and their alications in natural resource management, industry, economics and cyber security. He is the author of 18 books and over 300 refereed journal ublications in addition to numerous conference resentations and invited lectures. He is regularly invited to give short courses in Game Theory in several countries in Euroe and Asia. Athanasios Gerokostooulos ReliaSoft Cororation 1450 S. Eastside Loo Tucson, AZ, Athanasios. Gerokostooulos@ReliaSoft. com Athanasios Gerokostooulos is a Reliability Engineer at ReliaSoft Cororation. He is involved in the develoment of ReliaSoft s software roducts and the delivery of training seminars and consulting rojects in the field of Reliability and Quality Engineering. His areas of interest include Reliability Program Plans, Design for Reliability, System Reliability and Reliability Growth Analysis. Mr. Gerokostooulos holds an M. S. degree in Reliability Engineering from the University of Arizona and an MBA from Eller College of Management at the University of Arizona. He is a Certified Reliability Professional (CRP) and an ASQ Certified Reliability Engineer and Certified Quality Engineer. Pengying Niu ReliaSoft Cororation 1450 S. Eastside Loo Tucson, AZ, Pengying. Niu@ReliaSoft. com Pengying Niu is a research scientist at ReliaSoft Cororation. She is currently laying a key role in the develoment of Lambda Predict. Before joining ReliaSoft, she worked at Teas Instruments where she was involved in IC design and testing. She received her master s degree from the National University of Singaore, and M. E. in Electrical and Comuter Engineering from the University of Arizona. She has done etensive work on AC/DC and DC/AC converters. Her current research interests include reliability rediction and hysics of failure for electronic comonents such as MOSFET, IGBT and electronic systems. She is also an ASQ Certified Reliability Engineer (CRE) and a Certified Quality Engineer(CQE). 220