Estimation of component redundancy in optimal age maintenance

Transcription

1 EURO MAINTENANCE 2012, Belgrade May 2012 Proceedings of the 21 st International Congress on Maintenance and Asset Management Estimation of comonent redundancy in otimal age maintenance Jorge ioa CDRP, Polytechnic Institute of Leiria Morro do Lena - Alto do Vieiro Leiria, Portugal jorge.sioa@ileiria.t José Garção IDMEC/IT, Centro de Engenharia Mecatrónica,Universidade de Évora Colégio Luís Verney, Rua Romão Ramalho Évora, Portugal jesg@uevora.t Júlio ilva DEM, Instituto uerior Técnico Avenida Rovisco Pais, Lisboa, Portugal julio.montalvao@ist.utl.t ABTRACT The classical Otimal Age-Relacement defines the maintenance strategy based on the equiment failure consequences. For severe consequences an early equiment relacement is recommended. For minor consequences the reair after failure is roosed. One way of reducing the failure consequences is the use of redundancies, esecially if the equiment failure rate is decreasing over time, since in this case the reventive relacement does not reduce the risk of failure. The estimation of an active comonent redundancy degree is very imortant in order to minimize the life-cycle cost. If it is ossible to make these estimations in the early hase of system design, the imlementation is easier and the amortization faster. This work rooses an adatation of the Otimal Age-Relacement method in order to simultaneously otimize the equiment redundancy allocation and the maintenance lan. The main goal is to rovide a simle methodology, requiring the fewer data ossible. A set of examles are resented illustrating that this methodology covers a wide variety of oerating conditions. The otimization of the number of reairs between each relacement, in the cases of imerfect reairs, is another feature of this methodology. KEYWORD: redundancy; age-relacement; maintenance strategy; life-cycle cost. 1 INTRODUCTION The classical Otimal Age-Relacement defines the maintenance strategy based on the equiment failure consequences, as function of the failure rate variation over the oerating time. The consequences of a failure are measured by the difference between the cost of reair, before the occurrence of failure and the cost of reair after a failure in service. For severe consequences an early equiment relacement is recommended. For minor consequences the reair after failure is roosed. One method of reducing the failure consequences is by increasing the system reliability. This can be achieved by using more reliable comonents, by using the comonents just when their failure rates are minimal or by the use of redundancies. Unlike the first two strategies, which lead to small system reliability increases, the use of redundancies rovides an exonential growth. Therefore, when the consequences of failure are relevant, it is necessary to examine the ertinence of using redundancy

2 The estimation of an active comonent redundancy degree is very imortant in order to minimize the life-cycle cost. If it is ossible to make these estimations in the early hase of system design, the imlementation is easier and the amortization faster. This work rooses an adatation of the Otimal Age-Relacement method, in order to simultaneously otimize the equiment redundancy allocation and the maintenance lan. The main goal is to rovide a simle methodology, requiring the fewer data ossible. Because reliability gains are difficult to access when using standby redundancy, since a large quantity of data about equiment erformance over the time is necessary, which is very hard to get in the early hase of system design, the alication of standby redundancy is not covered by this aroach. 2 MATHEMATICAL FORMULATION For a redundant system of n equal equiment oerating simultaneously and in arallel, the system robability of failure function F (t), the system reliability function R (t), and the system failure robability density function f (t), are obtained from the corresondent equiment reliability functions by: ( ) = ( ) n F t F t (1) ( ) 1 ( ) n R t = F t (2) ( ) ( t) df n 1 = = ( ) ( ) (3) dt f t n F t f t Where f(t) and F(t) are the failure robability density and robability functions for each equiment. The basis for the develoment of this work is the well-known Otimal Age-Relacement formula of Jardine /7./ (also considered in/1./ - /9./), which for only one equiment is given by: ( ) 1 ( ) C R t + C ( ) = R t C t f ( ) t t R t + t f ( t) dt where: C(t ) Total exected relacement cost er unit of time. t - Preventive relacement age. C Cost of a reventive relacement. C f Cost of a failure relacement. The oeration time, can be exchanged by other counting unit: cycles, energy consumtion, roduction units, etc. Let C A denote the equiment acquisition cost including all the fixed costs that result from the ossession of an equiment: urchase rice, oerating ermits, sace, etc. For a system the total acquisition cost is: C = n C A A imilarly, for n equiment the system exected reventive relacement cost er relacement cycle is: ( ) = ( ) = 1 ( ) n EC P t C R t n CP F t (6) The reventive relacement cost is the average cost value due to reairing a defect rior to failure occurrence, including all materials and labour costs. At the reventive relacement age, all of the n equiments are relaced. (4) (5) 042-2

3 The system exected failure relacement cost er relacement cycle is: ( ) = ( ) = + ( 1) ( ) EC f t C f F t CF CP n F t (7) The system failure relacement cost is the sum a comonent failure relacement cost, with (n-1) comonents reventive relacement costs. It is assumed that the equiment is visited only when the time for reventive reair arrives, therefore it is time to make reairs in all working equiment and the ossible failures are also detected only at this time. The failure reair costs are the average costs of in service failure occurrence, includes all costs for materials, labour loss of roduction, loss of image, etc. The mean time of good oeration, denoted MTGO, is the exected length of a good oeration cycle. It is a function given by summing the equiment oerating time, with the already failed equiment mean time of oeration: ( ) ( ) MTGO t = t R t + t f ( t) dt (8) 0 t The MTGO(+ ) corresonds to the MTBF - Mean Time Between Failures. imilar to Eq.(4) above, the system total exected relacement cost er unit of time is: ( t ) EC f ( t ) ( t ) ( t ) n ( ) ( ) C + EC + C + C R t + C F t C ( t ) = = MTGO t R + t f ( t) dt A A f t The reventive relacement time t and the number of equiment n that minimize Eq. (9) are the otimal relacement age and redundancy. Please note that the cost values considered must be chosen with caution, because all the costs that are roortional to the oeration time can be neglected, since their induced costs er unit time are constants, which are rocess or equiment deendent only, and do not influence the t or the redundancy. In the Eq.(9) it is assumed that any reair is erfect, in the sense that it is assumed that the comonent returns to an as good as new condition. Therefore the failure robability function is always the same. However, Eq.(9) can be generalized to the case of non-refect reairs or different costs er reair, by considering different functions of reliability before and after each reair, and different costs. In this case, between each relacement, there will be imerfect reairs, where each can have different costs and failure rates. Therefore, the system total exected reair cost er unit of time, is now the ratio between the sum of the costs and the sum of the oeration times of each one of these oerations: C k ( ) ( ) C + C R t + C F t A i i i fi i i i= 1 ( t) = k i= 1 MTGO i ( ti ) where: t Vector of reventive reair ages k Number of reventive reairs (k-1 imerfect reairs andone erfect reair). Note that at the otimal k th reventive reair, the equiment will be relaced instead of reaired, since the k+1 intervention will have a higher cost, because it is not the otimal. Therefore, between the otimal k reventive reairs there will be effectively only k-1 imerfect reairs. The local minimums of Eq.(10) give, for each redundancy n, the otimal reventive reair ages for each oeration t i (note that after reair i time is reset to 0). The global minimum corresonds to the otimal redundancy n and the corresonding t i, with lower cost. 0 (9) (10) 042-3

4 3 NUMERICAL IMPLEMENTATION The resented methodology allows the use of any arametric or nonarametric robability distribution. However, in the examles shown below the Weibull robability function is used, because of its simlicity and the ossibility to have increasing and decreasing failure rates. In ractice, the robability function that best fits the equiment failure times should be selected. In the resence of exerimental data, a first good otion for a robabilistic model is to consider the adjustment to a Weibull distribution by arametric estimation, because of its flexibility. β t λ R( t) = e (11) F ( t) = 1 R( t ) (12) β 1 β t β t λ f ( t) = e (13) λ λ Where: λ - is a scale arameter (sometimes termed characteristic life) β - is a shae arameter (associated with the variation of the failure rates). The comutation rocess is easily imlemented in sreadsheet software, roviding a quick, chea and standard way of using it and integrating it with other maintenance software. Only the numerical integration for calculating the mean time of good oeration MTGO(t) can involve some rogramming deending on the accuracy required. The following examles were all generated using a sreadsheet imlementation, however any other mathematical software can be used. 4 EXAMPLE AND DICUION For simlification, in the next examles all the costs and times were nondimensionalised, by taking the reventive reair cost to be C =1 monetary unit and the characteristic life of the Weibull model λ=1 time unit. The results of the otimal values for oeration are resented in tables. Each table is built for one tye of equiment and each tye of equiment is characterized by the acquisition cost C A. and the Weibull shae arameter β. Deending on the equiment oerating conditions there are different failure reair costs. The best degree of redundancy for each situation (column resented in bold) is the one which resents the minimum value of the system total exected reair cost er unit of time C(t )(reresented by the underlined value). For each equiment oerating condition, the values of otimal reventive reair age t, the corresondent ercentage of failure reair F(t ), mean time of good oeration MTGO (t ) and the system total exected cost of a run-to-failure decisionc(+ ), are also resented. 4.1 Examle 1: Low-riced equiment (C A =C =1) with β=2 (increasing failure rate) Consider a system with low-riced comonents where the acquisition cost is equal to the reventive relacement cost and the failure rates increase. For the failure reair costs 100C, 18C, 6C, 3C, the best redundancy configuration is analysed using Eq. (9) and the results resented in Table

5 Table 1: Otimal values of oeration for C A =C =1 and β=2. C f 100C 18C 6C 3C n=4 n=3 n=2 n=2 n=1 n=2 n=1 n=1 t - Otimal reventive 0,655 0,528 0,358 0,599 0,346 0,917 0,654 1,091 reair time [λ] C(t ) [C /λ] 14,48 14,10 15,22 9,45 11,78 6,67 6,54 4,36 77,06 81,37 89,89 18,33 21,44 7,85 7,90 4,51 C(+ ) [C /λ] F(t ) 34,9% 24,3% 12,0% 30,2% 11,3% 56,9% 34,8% 69,6% [%Failure reair] MTGO (t ) [λ] 0,654 0,527 0,357 0,587 0,333 0,843 0,572 0,777 It can be seen from Table 1 that for very small failure relacement costs (C f =3C ) the exected run-to-failure cost is only slightly higher than the otimal cost with reventive relacement. The otimal reventive reair time (1.091λ) and the otimum ercentage of failures F(t)=69,6% (equiment that fail before being reventively relaced), are relatively high. Without the use of redundancy, n=1, as the costs of failure increase, the ercentage of failures and otimal time of reventive relacements decrease quickly. The use of an=2 redundancy is only useful for failure reair costs C f >7C and a n=3 redundancy only comensates for C f >36C. Therefore the redundancy can increase the otimal reventive relacement time and the otimal ercentage of failures, without increasing the costs. 4.2 Examle 2: Low-riced equiment (C A =C =1) with β=0,9 (decreasing failure rate) Consider now the system from Examle 1 but with decreasing failure rates. The results from Eq.(9) are resented in Table 2. Table 2: Otimal values of oeration for C A =C =1 and β=0,9. C f 100C 18C 6C 3C n=9 n=8 n=7 n=5 n=4 n=3 n=2 n=2 t - Otimal reventive 1,343 1,219 1,081 2,072 1, reair time [λ] C(t ) [C /λ] 17,86 17,86 18,03 10,12 10,31 5,48 5,56 3,71 36,44 37,42 38,70 10,62 10,86 5,48 5,56 3,71 C(+ ) [C /λ] F(t ) 72,9% 69,7% 65,8% 85,4% 81,2% 100% 100% 100% [%Failure reair] MTGO (t ) [λ] 1,329 1,205 1,070 1,753 1,494 2,006 1,617 1,617 For the case of decreasing failure rate, the equiment becomes more reliable with the age. Therefore, reventive relacements are not recommended since they increase the equiment failure robability. The as good as new state is the worst situation in terms of reliability. In this case, the otimal time of reventive reair only makes sense when we have redundancies, and this maintenance oeration is just an insection to substitute the failed comonents. For this situation the run-to-failure strategy becomes convenient, whenever no redundancy is used and for small failure reair costs C f <

6 Comaring Table 1 and Table 2 it is observed that the otimum degree of redundancy, the otimal reventive relacement time and the minimum costs are higher for decreasing failure rates. 4.3 Examle 3: Medium-riced equiment (C A =5C =5) with β=2 (increasing failure rate) Let s assume a 5 times increase in the acquisition cost and a dulication on the failure costs for the system from Examle 1. Table 3 is obtained in this situation. Table 3: Otimal values of oeration for C A =5C =5 and β=2. C f 200C 36C 12C 6C n=4 n=3 n=2 n=2 n=1 n=2 n=1 n=1 t - Otimal reventive 0,701 0,573 0,400 0,675 0,420 1,042 0,774 1,219 reair time [λ] C(t ) [C /λ] 40,81 39,16 41,06 25,50 29,40 18,29 17,02 12,17 160,60 168,17 184,15 41,02 46,26 20,07 19,18 12,41 C(+ ) [C /λ] F(t ) 38,8% 28,0% 14,8% 36,6% 16,2% 66,2% 45,1% 77,4% [%Failure reair] MTGO (t )[λ] 0,699 0,571 0,398 0,655 0,397 0,920 0,644 0,811 Comaring Table 3 with Table 1, it is verified that the dulication on the failure costs comensates the 5 times increase in the acquisition cost in the otimal degree of redundancy, suggesting only a small increase in the otimal reventive reair time. As exected, more exensive equiment imly less redundancy and require a longer run between reventive relacements. 4.4 Examle 4: High-riced equiment (C A =20C =20) with increasing reventive reairs costs This examle assumes erfect reairs f i (t)=f(t), but considers the increase of the reventive reair cost with the number of reairs C Pi =C P α i-1. In this case, the system total exected reair cost er unit of time must be calculated by Eq. (10). Consider an equiment for which the acquisition cost is 20 times the cost of reventive reair (C A =20C =20) and whose failure rate follows the Weibull model of increasing risk (β=2). In table 4 are resented the results for each one of the otimal reventive reair times, of a redundant system with 3 equiment oerating simultaneously in arallel, with a (C f =100C =100) failure reair cost and a successive 50% increase in the reventive reairs costs. Table 4: Values of otimal oeration for C A =20 C =20, β=2, C f =100, n=3 and α=1,5. Reair i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i t Otimal reventive reair time [λ] 0,911 0,778 0,728 0,706 0,696 0,706 0,728 0,767 t i [λ] 0,911 1,689 2,417 3,123 3,819 4,525 5,253 6,020 C( t ) [C /λ] 91,71 57,40 45,67 40,47 38,51 38,84 41,24 45,86 C( t i t =+ ) [C /λ] 125,54 84,64 67,76 58,75 53,76 51,50 51,49 53,76 F( t i ) 56,4% 45,4% 41,1% 39,3% 38,4% 39,3% 41,1% 44,5% [%Failure reair] MTGO i ( t ) [λ] 0,880 1,646 2,365 3,064 3,754 4,453 5,172 5,

7 Table 4 indicates that for archiving the minimum system total exected reair cost er unit of time of 38,51 C/λ, the k=5 reventive reairs have to be erformed at the ages of 0,911λ, 1,689λ, 2,417λ and 3,123λ, and 0,696λ after the fourth reventive reair (k-1 imerfect reairs) it is better to relace the system for a new one, instead of executing de fifth reair which would cost 5,0625C (1,5 4 C P ) instead of just C. Note that the otimal k* in Eq.(10) corresonds to having (k-1) imerfect reairs and a substitution at k, since k+1 interventions will be subotimal. The values in Table 4 are not the global minimums, since the number of redundancies n was not otimized, as Figure 1 below will also indicate. In Table 5 the number of redundancies is also otimized and it is found that the global minimum cost for 100C corresonds to n=2 redundancies and k*=5 reventive reairs (k-1=4 imerfect reairs and k=5 is a relacement). Additional otimal values are resented for other C. Table 5: Values of otimal oeration for C A =20 C =20, β=2 and α=1,5. 200C 100C 36C 18C n=3 n=5 n=4 n=3 n=2 n=1 n=2 n=1 k*=5 k*=5 k*=5 k*=5 k*=5 k*=6 k*=5 k*=+ - Otimal reventive 0,595 0,931 0,826 0,696 0,524 0,292 0,755 >5 reair time [λ] C( t ) [C /λ] 44,29 44,47 41,15 38,51 38,35 53,92 28,40 <21 C( t N t =+ ) [C /λ] 84,95 56,33 54,38 53,76 57,04 74,75 31,40 <21 N t F( N t ) 29,8% 58,0% 49,5% 38,4% 24,0% 8,2% 43,5% 100% [%Failure reair] MTGO ( t o )[λ] 3,209 4,919 4,406 3,754 2,869 1,912 3,935 + The system total exected reair cost er unit of time for a range of redundancy from n=1 to n=5 and for a set of more than 10 successive reairs with increasing costs, is resented in Figure Examle 5: High-riced equiment (C A =20C =20) with increasing failure rates. In this examle it is analyzed the introduction of imerfect reairs f i (t) f(t). Comaring to Examle 4, now the Weibull scale arameter is successively decreasing with each reair according to λ i = λ/ε i-1. Using Eq.(10), Table 6 resents results for each one of the otimal reventive reair times, of a redundant system of 3 equiment oerating simultaneously and in arallel, with a (C f =100C =100) failure reair cost and a successive decreasing of the Weibull scale arameter. Table 6: Values of otimal oeration for Ac=20 Pc=20, β=2, Fc=100, K=3 and ε=1,3. Reair i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9 i t Otimal reventive 0,595 0,425 0,311 0,230 0,173 0,131 0,099 0,075 reair time [λ] t [λ] 1,505 1,930 2,241 2,471 2,645 2,775 2,874 2,949 i C( t ) [C /λ] 62,23 52,30 47,62 45,17 43,87 43,26 43,09 43,21 C( t i t =+ ) [C /λ] F( i t ) [%Failure reair] 95,54 84,30 78,75 75,80 74,25 73,53 73,34 73,51 42,47% 35,66% 30,77% 27,10% 24,41% 21,95% 19,97% 18,15% C f 042-7

8 MTGO i ( t ) [λ] 1,467 1,889 2,198 2,428 2,601 2,731 2,830 2,905 C(t) [Pc/λ] 120 Total Cost er Time (C A =20, C f =100) n=1 n=2 n=3 n=4 n= Σ ti [λ] Figure 1: Grah with C A =20 C =20, β=2, C f =100C and α=1,5. In Figure 2 are resented the system total exected reair cost er unit of time for a range of redundancy from n=1 to n=5 and for a set of more than 10 successive reairs with successive decreasing of the Weibull scale arameter. C(t) [Pc/λ] 100 Total Cost er Time (C A =20, C f =100) n=1 n=2 n=3 n=4 n= ,5 1 1,5 2 2,5 3 3,5 4 4,5 Σ ti [λ] Figure 2: Grah with C A =20 C =20, β=2, C f =100C and ε=1,3. It is observed form Table 4 and Table 6 (and from Figure 1 and Figure 2) that for the resent case (Examle 5, decreasing of the Weibull scale arameter) the successive decrease of otimal reventive relacement is more severe, the otimal number of imerfect reairs between 042-8

9 relacements is also much higher and the system total exected reair costs er unit of time are slightly higher. From the examles above some general areciations can be drawn. More exensive equiment require the least use of redundancy. The redundancy can increase the otimal reventive reair time and the otimal ercentage of failures, without increasing costs. In most cases involving small failure reair costs Cf<5C, the Run-to-Failure (F(t)=100%) is the most aroriate strategy. The degree of growth of the failure rate is a crucial arameter. If there are random failures or decreasing failure rates, the increase in redundancy is inevitable, and so is the alicability of a run-to-failure strategy. As the failure reair costs grow (increase of the failure consequences) the otimal ercentage of failures and the otimal time to reventive reair decreases, whereas the otimal number of redundancies increases. 5 CONCLUION The roosed methodology requires a small amount of data and the required otimization rocess can be easily imlemented in a sreadsheet as was done for the examles. The set of examles resented intent to demonstrate the alicability to a wide variety of systems oerating conditions, roviding quantitative estimations useful in assisting some maintenance decisions and otimization situations. One of the most imortant decisions in maintenance is how many imerfect (or increasing costs) reairs should be made between each relacement. The methodology resented allows this otimization according to the degree of imerfection (or the increasing costs rate) of each reair. For some alications it is convenient to reformulate the acquisition cost definition, incororating time-deendent values or fixed values indeendents of the redundancy degree. This can also be included in using the roosed methodology. REFERENCE /1./ Chien, Yu-Hung (2008), Otimal Age-Relacement Policy Under an Imerfect Renewing Free- Relacement Warranty, IEEE Transactions on Reliability, Vol. 57, no. 1, /2./ Chien, Yu-Hung, (2008). A general age-relacement model with minimal reair under renewing free-relacement warranty, Euroean Journal of Oerational Research, no 186, /3./ Chien, Yu-Hung, heu, hey-huei (2006). Extended otimal age-relacement olicy with minimal reair of a system subject to shocks, Euroean Journal of Oerational Research, no174, /4./ Coolen-chrijner, P. and Coolen, F.P.A. (2004). Nonarametric redictive inference for age relacement with a renewal argument, Quality and Reliability Engineering International, no 20, /5./ Coolen-chrijner, P., Coolen, F.P.A. (2007). Nonarametric adative age relacement with a onecycle criterion, Reliability Engineering and ystem afety, no 92, /6./ Coolen-chrijner, P., Coolen, F.P.A. and haw,.c. (2006). Nonarametric adative oortunitybased age relacement strategies, Journal of the Oerational Research ociety, no 57, /7./ Jardine, A. K.., Maintenance, Relacement and Reliability, Pitman Publishing (1998). /8./ Usher, John., Kamal, Ahmed H. and yed, WasimHashmi, (1998). Cost otimal reventive maintenance and relacement scheduling, IIE Transactions, no 30:12, /9./ Zhang, Fan and Jardine, Andrew K.. (1998). Otimal maintenance models with minimal reair, eriodic overhaul and comlete renewal, IIE Transactions, no 30:12,