Maintenance free operating period an alternative measure to MTBF and failure rate for specifying reliability?

Transcription

1 Reliability Engineering and System Safety 64 (1999) Technical note Maintenance free operating period an alternative measure to MTBF and failure rate for specifying reliability? U. Dinesh Kumar a * J. Knezevic a J. Crocker b1 a Centre for Management of Industrial Reliability Cost and Effectiveness Faculty of Engineering University of Exeter Exeter EX4 4QF UK b Rolls Royce Military Aero Engines Limited Customer Logistic Support Bristol BS12 7QE UK Received 16 December 1997; accepted 2 June 1998 Abstract The paper analyses the concept of maintenance free operating period (MFOP) the reliability requirement driven by the Ministry of Defence (UK) for the next generation of future aircraft to be included in the fleet. Since the traditional reliability requirement MTBF (mean operating time between failure) has several drawbacks the immediate reaction would be to analyse the credibility of the new measure MFOP against MTBF. The paper discusses various issues associated with MFOP. Two mathematical models are developed to predict the maintenance free operating period survivability (MFOPS) one using mission reliability approach and the other using alternating renewal theory. The paper also analyses cost implications of MFOP to the customer and to the producer Elsevier Science Ltd. All rights reserved. Keywords: Failure rate; Maintenance free operating period (MFOP); Maintenance recovery period (MRP); Mean operating time between failure (MTBF); Reliability; Renewal theory 1. Introduction Billions of dollars are spent by commercial and defence industries every year as a direct consequence of the unreliability of their systems. The cost of unscheduled maintenance by civil airline operators is in the order of one million pounds per aircraft per year [1]. Industries have started thinking whether they are specifying the reliability requirement in the right way. For example US Air Force [2] operating command prefer reliability requirements to be based on mission and operational requirement rather than mean time between maintenance mean down time availability etc. They argue that the old way relies on probabilistic design based measure that are more meaningful to statisticians. The preferred way forces the technical community to consider operational based requirements. In the past MTBF (mean operating time between failure) or its reciprocal the failure rate have been used by many customers as a reliability specification without realising that in most cases it is almost impossible to demonstrate. Knowles [3] makes the point that specifying reliability in terms of MTBF * Corresponding author. Tel.: ; Fax: ; d.k.unnikrishnan@exeter.ac.uk 1 The views expressed in this paper are those of the author and do not necessarily reflect the opinion of the Rolls Royce Military Aero Engines Limited. is not beneficial either to customers or suppliers. Neither of these measures adequately describe the reliability of the system. The main drawbacks of MTBF are: 1. it is almost impossible to predict MTBF if the time-tofailure distribution is not exponential; 2. the methodology most widely used to predict MTBF and failure rate is based on the exponential distribution. This distribution is used to model failure times (and supported by Military Standards [4 6] like MIL-HDBK-217 MIL-STD-1388 and British Defence Standard 00-41) primarily because of its mathematical friendliness and the belief that Drenick limit theorem [7] is universally applicable rather than any scientific reason. Since MTBF has several drawbacks The Royal Air Force [8] are considering a new reliability metric maintenance free operating period (MFOP) as the prime reliability and maintainability requirement for their future generation aircraft. The SBAC s (Society of British Aerospace Companies) ultra reliable aircraft project specifies MFOP as one of its important objectives. Now the ultimate goal for many British Aero Space [8] industries is to reduce the dependence on MTBF and specify reliability as a probability of time in service before failure. It is interesting to note that the Hockley and Appleton [8] have acknowledged the concept of MFOP as a positive step for specifying reliability /99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S ( 98)

2 128 U. Dinesh Kumar et al./reliability Engineering and System Safety 64 (1999) Fig. 1. The operating profile of the system. of future combat aircraft. The main objective of this paper is to develop mathematical models for predicting maintenance free operating period for a system. The paper analyses the cost implications of MFOP and when one can use MFOP as a reliability specification. To our best knowledge this is the first attempt to develop a mathematical model for maintenance free operating period. An example problem is used to illustrate the mathematical model. 2. Maintenance free operating period (MFOP) The concept of maintenance-free operating period (MFOP) is not new; it is essentially same as the warranty period. What is new is that the operators are considering extending this concept throughout the life of the system. In practice the contractor/manufacturer will be expected to guarantee that no unscheduled maintenance activities will be required during each defined period of operation. To achieve this with the required level of confidence will almost certainly require full parts life tracking and an increased level of part and module exchange. Maintenance-free operating period (MFOP) allows a period of operation (say t life units) during which an item will be able to carry out all its assigned missions without the operator being restricted in any way due to system faults or limitations with the minimum of maintenance. Maintenance-free operating period survivability (MFOPS) is then defined as the probability that the item will survive for the duration of the MFOP. In other words it is the probability of not having any unscheduled maintenance for a period of t life units given the current age of the item. It is also the probability that the item maintains its functionality at least for a period of t life units without the need for corrective maintenance due to failure of a component of the system which results in an overall critical failure of the system. However it should be noted that during MFOP the system is allowed to undergo any planned minimal maintenance. Also the redundant components can fail during an MFOP without forcing any corrective action. The idea here is to find the probability of any unplanned maintenance. Hockley and Appleton [8] make a point that during MFOP the necessity for maintenance should be kept to a minimum. They further went on to say that specifying reliability through MFOP will force the supplier to analyse various failure mechanism which will further help to improve the overall design of the system. However Crocker [9] points out that giving a warranted maintenance-free operating period of so many flying hours miles or days resolves into deciding on an acceptable probability of survival offset against an increased cost. The manufacturer will need to determine the expected cost of providing the MFOP and hence increase the price to minimise the risk of losing money whilst keeping the price competitive. A MFOP (or cycles of MFOP) is usually followed by a maintenance recovery period (MRP). This is defined as the period during which the appropriate scheduled maintenance is carried out. Consider a repairable system. The time to failure and the time to repair follow some arbitrary distribution. The operating profile of the system can be represented as shown in Fig. 1. In Fig. 1 it can be noted that the system fails at time points {T 1 T 3 T 5 } and the system is repaired and put into operation at time points {T 2 T 4 T 6 }. Assume that it is required to have a MFOP of t life units and a MFOPS of 95% (i.e. the probability of the system surviving t life units without needing corrective maintenance). A necessary condition for the system to have a MFOP of t life units with probability of 0.95 is given by: MFOPS(t ¼ Pr d i ¼ 0 (T 2i þ 1 ¹ T 2i ) t 0:95: (1) If the random variables {T 1 ¹ T 0 ; T 2 ¹ T 1 ; T 3 ¹ T 2 ; } are independent and identically distributed then the operating profile of the system can be modelled using alternating renewal process. 3. MFOP prediction mission reliability approach In this section we develop a mathematical model based on a mission reliability approach to predict MFOPS. Let us consider a system with n components connected in series. If the reliability requirement is MFOP of t life units then the corresponding probability given that all the components of the system are new is given by: MFOPS(t 1) ¼ R k (1 t ) (2) R k (0) where R k (t ) is the reliability of the kth component for (the first) t life units. Eq. (2) gives the probability for the system to have MFOP of t life units during the first cycle. For the second cycle the expression for MFOPS is given by: MFOPS(t 2) ¼ n R k (2 t ) R k (1 t ) : (3)

3 U. Dinesh Kumar et al./reliability Engineering and System Safety 64 (1999) In general for ith cyclethe probability the system will have MFOP of t life units is given by: R k (i t ) R k ([i ¹ 1] t ) : (4) In Eq. (4) MFOPS(t i) represent the probability that the system will survive ith cycle of MFOP given that it survives (i ¹ 1) cycles. Let us assume that it is required to achieve a MFOP of t life units with MFOPS ¼ a. The following procedure can be used to find the number of cycles the system satisfies MFOP with probability a. Note that in deriving the above mathematical expressions we assume independence between time-to-failure distribution of various items of the system. This assumption may not be valid in some cases. In that case one has to derive alternative mathematical model Procedure to calculate the number of cycles the system satisfies the required MFOPS Step 1. Set i ¼ 1. Step 2. Calculate Step 3. If i MFOPS(t i) a; then Go To Step 5. Step 4. i ¼ i þ 1 Go To Step 2. Step 5. Number of cycles is i ¹ 1. Step 6. Stop. R k (i t ) R k ([i ¹ 1] t ) : 3.2. MFOP of items with Weibull distributed failure times For a component with a failure mode which can be modelled by the Weibull distribution the probability of surviving t units of time given that the item has survived t units of the time is given by: MFOPS(t ) ¼ exp ¹ tb ¹ (t þ t ) b h b! (5) where h is the scale parameter and b is the shape parameter of the Weibull distribution. The MFOP period for a given level of confidence can be calculated by rearranging the above equation as follows: n t ¼ t b ¹ h b o 1=b ln MFOPS(t ) ¹ t: (6) 4. MFOP prediction renewal theory approach After a certain number of cycles of MFOP there will be a maintenance recovery period (MRP) [3]. MRP is related to MFOP s and is defined as the down time during which appropriate maintenance actions are carried out. In this section we derive MFOPS for a repairable item using renewal theory allowing a maintenance recovery period after an MFOP. Assume that it is required to have a MFOPS of t life units. Then it will be interesting to find MFOPS during a stated period of time T along with the maintenance recovery period. Consider a repairable item assume that: 1. The time to failure distribution of the item follows arbitrary distribution with density function represented by f(t). 2. Maintenance recovery time of the item follows some arbitrary distribution with density function represented by g(t). The item can be in two states {10}. Where 1 is up state and 0 is the down state. Let P 1 (T) be the probability that the item will have t hours of maintenance free operating period through out the mission T. Maintenance is carried out as soon as the item fails. The expression for P 1 (T) can be written as: P 1 (T) ¼ R(t þ P 0 ¼ T 0 T 0 f (u l t )P 0 (T ¹ u) du (7) g(v)p 1 (T ¹ v) dv (8) where f(u l t ) is the probability that the system fails at time u given that it has survived up to time t. The above system of integral equations can be solved by numerical approximation [10] for any time to failure distribution like exponential normal Weibull lognormal etc. to evaluate the value of P 1 (T). Here P 1 (T) gives the probability that the system will operate for at least t hours before it fails during T hours of operation. 5. Relationship of MFOP duration to degree of ageing Fig. 2 shows how the 95% confidence level of MFOP varies with age of the item for Weibull distributions with different shapes (b) but same MTTF of 1000 h. The important point is to note with these lines are that for b 1 they are convex starting from zero for b ¼ 1 it is horizontal and for b 1 they are concave tending towards zero (as the age of the item tends to infinity). This means for items with constant failure rate the MFOP probability will remain same throughout the life of the component. For components which wear out the probability of achieving the desired MFOP will decrease as they age almost certainly to a point where these components will have to be replaced prematurely to maintain the desired MFOP.

4 130 U. Dinesh Kumar et al./reliability Engineering and System Safety 64 (1999) Fig. 2. MFOPs for various shape parameter values of Weibull distribution. 6. Cost factors associated with MFOP To maintain a specified MFOP probability it may be required to replace the components prematurely (or insert inspection). Discarding components before they fail will inevitably cost money (however it reduces the cost due to unscheduled maintenance). Besides the increase in the number of spares of these particular components there is likely to be an increase in the number of LRI (line replaceable item) removals. In general each component will have different failure distribution parameters and may have different ages. Thus the times when they will become due for replacement (to maintain the required confidence level) will be different. This will effectively generate an LRI removal every time an item becomes critical (i.e. required to be replaced). In practice this number could be reduced if all the components which will become critical in a given period are replaced at the same time but this will inevitably reduce their useful life. The optimum length of this period will be a compromise between the cost of an LRI removal and the cost of the lost life. Each such removal will require the LRI to be stripped rebuilt and probably tested. Depending on the time to recover an LRI the fact that there is an increase in the number may lead to a need for more spare LRIs to maintain the system availability. Thus use of MFOP will require careful analysis of all these factors. 7. Example In this section we consider an example to illustrate the mathematical model based on mission reliability approach. Consider a system with four items connected in series as shown in Fig. 3. The time-to-failure distribution and the Fig. 3. A series system with four items. parameter values of various items of the system are given in Table 1. Now using Eq. (4) the MFOPS(t i) that is the probability that the system will survive ith cycle of MFOP given that it survives (i ¹ 1) cycles is given by: 4 R k (i t ) R k ([i ¹ 1] t ) (9) where R 1 (i t ) ¼ exp( ¹ 0:001 i t ) R 2 (i t ) ¼ exp ¹ i t R 3 (i t ) ¼ F 1500 ¹ i t 200 R 4 (i t ) ¼ exp ¹ i t 2: where F( ) is the standard normal variate. Substituting the above expressions in Eq. (9) one can get the value of MFOPS(t i) for different values of i. Fig. 4 shows MFOPS(t i) for different cycles starting from the first cycle where t ¼ 50 h. It is easy to see from Fig. 4 that one has to carry out maintenance recovery after 4 cycles (after 200 h) if the required confidence level a is greater than 0.94 and one has to carry our maintenance recovery after 11 cycles if a 0.90 (after 550 h). Table 1 Time-to-failure distribution of items of the system described in Fig. 3 Item Distribution Parameter values Item 1 Exponential l ¼ h ¹1 Item 2 Weibull h ¼ 1200 h b ¼ 3 Item 3 Normal m ¼ 1500 h d ¼ 200 Item 4 Weibull h ¼ 1400 h b ¼ 2.1

5 U. Dinesh Kumar et al./reliability Engineering and System Safety 64 (1999) Fig. 4. MFOPS(50i) for different cycles. 8. Conclusion The concept of MFOP is acknowledged by many aerospace industries in UK as a large step for future reliability specifications. Major projects like the ultra reliable aircraft (to be implemented by the Society of British Aerospace Companies) and future offensive aircraft (FOA) have recognised MFOP as a suitable reliability measure for replacing MTBF as the latter has many drawbacks. The main advantage of MFOP is that it tracks behaviour of the system throughout the life of the system. Also MFOP will force the suppliers to analyse various failure mechanism which will further help to improve the reliability and overall design of the item. Though a MFOP requirement is a good way of specifying reliability but may prove expensive in certain types of operation (e.g. training). In this paper two mathematical models are developed one based on mission reliability approach and the other based on alternating renewal theory. To our best knowledge this is the first time an analytical expression for MFOP prediction is reported. The models will help reliability engineers and practitioners to predict and specify MFOP probability. In our models we assume that the time-to-failure of different items to be independent of each other. Future models should consider items with dependent time-to-failure distribution and also the impact of maintenance on reliability function. However Eq. (4) is still valid if the time-to-failure is dependent except that one has to derive R k (t i) considering dependence between time-to-failure distribution. If the majority of the failures are non-age related (or so assumed) then there will be very little chance of improving the MFOP probability or the duration of MFOP. In this case MFOP will not have any advantage compared to that of MTBF. To determine if any of the major causes of failure are age-related it will be necessary to keep a full record of the ages of each occurrence of each of these components and carry out an analysis of the times to failure. The actual economics of MFOP s would need to consider the above as well as the parts costs and maintenance costs. An MFOP policy can only work if the producer operator and suppliers recognise that some components wear out with use that not all failures are independent of the age of the component as MIL-HDBK-217 and MIL-STD-1388 imply. They will also need to recognise that nothing in this world is free such that to achieve the required MFOP the suppliers will inevitably have to increase their prices or the cost of support. Acknowledgements We thank Mr. Ian Knowles Ministry of Defence (UK) for his useful suggestions. We also thank the referee for his constructive comments. References [1] Ultra Reliable Aircraft Consortium: Ultra reliable aircraft Pilot phase report [2] USAF R&M 2000 Process. United States Air Force report ed. 1. Washington DC: USAF [3] Knowles DI. Should we move away from acceptable failure rate. Communications in Reliability Maintainability and Supportability 1995;2(1): [4] MIL-HDBK-217. Reliability prediction of electronic equipment ed. F. Department of Defence [5] MIL-STD-1388-B. Department of Defence requirement for a logistic support analysis record ed. B [6] British Defence Standard Part 2: Reliability apportionment modelling and calculation [7] Drenick RF. The failure law of complex equipment. Journal of Society Of Industrial Applied Mathematics 1960;8:680. [8] Hockley CJ Appleton DP. Setting the requirements for the Royal Air Force s next generation aircraft. Annual Reliability and Maintainability Symposium [9] Crocker J. Maintenance free operating period Is this the way forward? In: Proceedings of the 7th International M.I.R.C.E Symposium. Exeter [10] Gopalan MN Dinesh Kumar U. Approximate analysis of n-unit cold standby systems. Microelectronics and Reliability 1996;36(4):