Journal of KONES Powerrain and Transpor, Vol.14, No. 3 2007 PEVENTIVE MAINTENANCE WITH IMPEFECT EPAIS OF VEHICLES Józef Okulewicz, Tadeusz Salamonowicz Warsaw Universiy of Technology Faculy of Transpor Koszykowa 75, 00-662 Warsaw, Poland el.: +48 22 234 7930, +48 22 234 8247 e-mail: jok@i.pw.edu.pl, sa@i.pw.edu.pl Absrac A vehicle abiliy o realise ranspor asks may be resored by repairing only failed elemens. This is called imperfec repair as he vehicle is no as good as new afer such a repair. Prevenive replacemen is an example of imperfec repair as well. The advanage of such mainenance is ha i enables conrolling a reliabiliy level of a flee of vehicles. Ses of vehicles elemens which should be replaced in ha aim are derived on a base of saisical diagnosing wih use of daa abou elemens failures. The accepable level of a failure risk while execuing ransporaion asks has been aken as a crierion. An algorihm for selecing elemens for prevenive replacemen has been developed. I was shown ha a level of a flee reliabiliy can be conrolled by changing an order of a quanile funcion in coordinaion wih a number of redundan objecs. A compuer simulaion model of he vehicle flee was used as an example o illusrae derived dependencies. In paricular algorihm for selecing elemens for prevenive replacemen, graph of model saes, graphical inerpreaion of calculaing new quanile orders, simulaion experimens resuls for sysem n ou of n, simulaion experimens resuls for d = 2,5 are presened in he paper. Keywords: ranspor, mainenance, prevenive replacemen, imperfec repair, saisical diagnosis 1. Inroducion Prevenive replacemens are used o mainain demanded reliabiliy of vehicles in a ranspor firm. They enable avoiding failures of individual vehicles in a flee. However a need for high reliabiliy of flee of vehicles being used can effecs in grea amoun of elemens replaced during prevenive acions. As i can no be considered full resoraion of he vehicle reliabiliy afer he service, so only elemens of he vehicle should be replaced. This can be called as imperfec repairs of he vehicle. High reliabiliy is achieved in pracice by services when specific elemens are replaced by new ones. A crierion of selecing elemens depends on level of reliabiliy ha is expeced. In a case of a homogeneous se of vehicle, a range of prophylacic aciviies depends on a reliabiliy level of he whole flee and on is reliabiliy srucure. This akes in accoun redundancy, ha enables firs off all o replace failed objecs enabling execuion of ransporaion asks. A number of redundan objecs depend on he accepable probabiliy of failure during he ask implemenaion period. In order o minimize he size of redundancy one should, on he one hand, be using objecs of high reliabiliy, and keep heir reliabiliy in he operaing process a possibly high level, on he oher hand. Insead of known from he lieraure mehod of replacing objec in a given rae [6], he mehod of replacing ses of chosen elemens ha enables achieving demanded level of he flee reliabiliy is proposed. This mehod uses saisical characerisics of he vehicles insead of applying measurable parameers of is elemens.
J. Okulewicz, T. Salamonowicz The abiliy of he objec of fulfilling given asks wih demanded probabiliy could be saisically measured by quanile of given order. For his measure, a mehod of saisical diagnosis was developed. I poins ou a given momen o a se of elemens ha should be replaced by new ones o achieve demanded reliabiliy of he whole flee of vehicles. 2. Prevenive replacemens A mehod ha is known from lieraure and used for defining of a scope and deadlines of prevenive replacemens is o include he coss of aenive replacemens and he coss generaed by he occurring failures [1, 5]. As a resul of applicaion of he mehod, minimum average coss per uni of ime relaed o mainenance of objecs in a proper reliabiliy saus are achievable. However, in order o benefi from ha effec here is a need o replace individual elemens in various ime inervals, usually uncoordinaed wih he performance of asks, which may wipe ou advanages effecing from he implemened opimisaion. Therefore, a possibiliy should be considered o make prevenive replacemens of seleced elemens of objecs in he assumed ime inervals whose scope is defined on he basis of assessmen of reliabiliy of he elemens and he assumed reliabiliy level of he enire flee [3]. The flee mainained in such a way preserves is abiliy o realize ransporaion asks wih a given probabiliy. In case of complex objecs, a failure appears whenever an elemen, which creaes a series reliabiliy srucure wih he ohers, has failed. A repair usually involves a replacemen of he elemen for a brand new one. However, he replacemen of he damaged elemen for he new one does no effec in recovery of such a reliabiliy level as ha before occurrence of he failure. This is because he value of he reliabiliy funcion of he damaged elemen before he failure was less han 1, and following he replacemen i was equal o 1. In effec, he condiion of he objec afer he repair is and mus be slighly beer han before he failure. So, pracically here are no possibiliies o recover such a saus of he objec following he repair, as he one righ before he failure. Boh he objecs and heir componens are considered when developing he prevenive replacemens sraegy. Properies of he componens are more predicable han hose of objecs which hey are par of. Dynamic deerminaion of a scope of prevenive replacemens could be based on a saisical assessmen of presen saus of objecs elemens. The saisical diagnosis is a mainenance mehodology in he area of mainaining objecs wih non-exponenial disribuions. I idenifies prevenive mainenance asks o realise he inheren reliabiliy of equipmen a a minimum expendiure of resources. In order o do ha, daa is required abou a disribuion of ime o failure and is parameers as well as abou is operaional use so far (since being new or from he momen of is replacemen). The saisical diagnosis uses daa gahered during normal uilizaion of objecs. They concern failures, repairs and replacemens of objec elemens. Nex, he probabiliy disribuion funcion of ime or mileage o failure for each of hese elemens is deermined. I can be done eiher wih he use of daa colleced in he pas or by relying upon expers' opinions a he sar. On ha basis, defined is a se of hose elemens he replacemen of which will effec in a siuaion ha a failure probabiliy will no exceed is assumed value in he duraion of he scheduled ask. The main arge is increasing he availabiliy of he equipmen by reducing he amoun of echnical asks o is minimum, and i reaches his arge by subsiuing he echnical inspecion wih he saisical diagnosis. The procedure saisically predics failures a par level by calculaing he mean residual lifeime o failure (ML). However, he ML compared o required work period effecs in ha abou half of objecs would be serviced before failure and he res would fail wihou any reamen. Thus, i is beer o apply a quanile funcion of residual lifeime insead of he ML o enlarge he probabiliy of 486
Prevenive Mainenance wih Imperfec epairs of Vehicles prevenive service. This measure direcly relaes o prediced work period and he reliabiliy of he sysem. For any momen he following condiions have o be me: d asks implemenaion period, q p () quanile of residual lifeime funcion, order p. Funcion q p () shall be defined as follows [2]: q p () d, (1) p 1 q () F inf x : F (x) p, (2) p ( x) 1 F (x) (x), x, 0, (3) () F (x) cumulaive disribuion funcion of he residual lifeime, (x) condiional reliabiliy funcion. Saisical conrol can be performed a any momen because i rerieves daa gahered in he informaional area of he means-of-ranspor mainenance managemen sysem. I could be done eiher in a consan period of ime or during planned service or during curren repair. The disribuion parameers are modified when eiher repair or replacemen of he elemen has been done. The acual echnical condiion of he objec is no aken ino consideraion here as ha would require for he objec o be excluded from is operaional use. Having daa, reliabiliy characerisics of elemens, updaed working ime of individual elemens, a period for execuion of he ransporaion ask, i is possible o define elemens ha require prevenive replacemen in order for he projec implemenaion probabiliy no o decline below is assumed value. I can be applied as well o elemens as o complex objecs (i.e. funcional ses or he whole vehicle). In a case of complex objec, is reliabiliy srucure is o be considered as well as special procedure of choosing elemens o replace, which enables achieving demanded probabiliy of proper work of he objec [3, 4]. A paricular example of complex objec is a flee of vehicles in a ranspor firm. I could be characerized by reliabiliy and as well as by a reliabiliy srucure. Probabiliy of a failure during a ask period can be deermined in boh cases, ha is, when he replacemens eiher have or have no been made. Addiionally, he assessmen may refer o he enire flee of objecs ha have been assigned for execuion of he ransporaion asks. 3. Imperfec repair Majoriy of heoreical conclusions are derived wih assumpion of perfec (ideal) objec resoring. However such processes wih use of models of full renewal are adequae only when objec is replaced by a new one or in a case of a general repair. In he case of correcive repairs made afer failing of any vehicle elemen, a model of minimal repair is ofen used [1]. This means ha he objec is o be resored o he condiion jus before failure. However i is no possible pracically, as objec reliabiliy saus afer repair of is elemen is beer han before failure. Those are reasons ha heoreical models of eiher perfec or minimal repairs have limied applicabiliy. eal repair resores objec reliabiliy o an inermediae value. Thus i is called an imperfec (incomplee) repair [4]. However a degree of objec resoraion by replacing one or more is elemens can be esimaed only afer repair. 487
J. Okulewicz, T. Salamonowicz Modelling exploiaion process wih use of he imperfec repairs means defining characerisics of random variable X k concerning ime of proper work afer (k-1) h repair. Objec s reliabiliy funcion afer repair is given by he following formula: 2 x x x 1 1 1, (4) 1 1 (x), 2 (x) reliabiliy funcions of he objec before and afer he repair respecively, degree of he objec resoraion. The formula for he failure rae funcion relaion before and afer he repair is as follows: hence x x (1 ) x 2 1 1, (5) x 2x x x 1, (6) 1 1 (x), 2 (x) are he failure rae funcions before and afer he repair respecively. 4. Algorihm for selecing elemens o be replaced A prevenive replacemen of elemens is made if he value of funcion (11), which has been calculaed for he enire se of objecs, is lower han he duraion of he scheduled ask planned for ha se of objecs. In order o selec a se of elemens o be replaced a given momen, an updaed value of he reliabiliy funcion is calculaed including operaional ime of each and every one of hem. Then a quanile of a given order is calculaed for a disribuion of he residual lifeime of each elemen. The elemens are pu in order according o he growing quanile value. sar d = ime inerval beween saus conrols p = order of quanile of residual lifeime funcion in he inerval from o + d 1 i () = reliabiliy of i-h elemen, i= 1, n q pi i ( x) () inf x : (1 p) i (), i= 1, n r = 0 () q p ( x) inf x : (1 p) () YES q p() < d NO q pj() = min {q pi (), i= 1, n - r} j-h elemen replacemen q pj ou of sequence r = r + 1 Fig. 1. Algorihm for selecing elemens for prevenive replacemen 488
Prevenive Mainenance wih Imperfec epairs of Vehicles Subsequen elemens, saring from an elemen of he lowes quanile value unil he quanile of he enire flee of objecs calculaed by having included he replacemen of assigned elemens for brand new ones is no lower han he duraion of he scheduled ask (algorihm in Fig.1), are assigned for replacemen. The replacemen of elemens ha have been assigned in ha way ensures he assumed probabiliy ha he objec will no fail during implemenaion of he ranspor ask. 5. edundancies in a flee of vehicles Susaining a high reliabiliy level of echnical objecs in heir operaional use process served by prevenive replacemens of componens being hreaened by a failure can be accompanied by adding redundan objecs o he flee. Le us assume ha n objecs are essenially required for carrying ou ransporaion asks. If he enire flee consiss of n objecs, hen an assumpion can be made ha reliabiliy srucure of he flee is in series. This imposes large requiremens on reliabiliy of each objec, which is ofen no achievable. Then, in order o keep reliabiliy of he flee a is required level, redundan objecs can be inroduced ino he flee. Adding k redundan objecs o he flee allows for considering he flee reliabiliy srucure as a hreshold srucure, in his case n ou of n+k. The flee reliabiliy model depends on he way he redundan objecs are operaing in. edundan objecs may play a role of he cold (unloaded) reserve, ha is, hey passively wai for one of he objecs o fail, or he ho (loaded) reserve, hus increasing he enire flee capaciy unil one of he objecs has failed. In case of srucure n ou of n+1 wih he cold reserve, he flee reliabiliy funcion n+1 () will be a sum of probabiliies for occurrence of he following siuaions: 1) unil momen no objec will fail ou of n objecs esablishing a series reliabiliy srucure, 2) a any momen < one ou of n objecs shall fail and will be replaced wih a reserve objec ha will no fail along wih he remaining objecs in a range of (, ). Probabiliies for occurrence of he above siuaions are respecively: f n P 1 = n (), (7) P2 f n n, d, (8) 0 n, n1 n n1 1 n f, (9) d, (10) d f n1 P 2 n d, (11) 0 () n1 n1 P1 P2 n f d, (12) 0 objec s reliabiliy funcion, 489
J. Okulewicz, T. Salamonowicz f n () probabiliy densiy funcion of a failure of one ou of n idenical objecs esablishing a series reliabiliy srucure, n (, ) probabiliy of a non-failure in he range of (, ) of he flee consising of (n-1) objecs aged and one new objec. Probabiliy densiy funcion of a failure of he flee wih a srucure of n ou of n+1 wih he cold reserve is expressed by relaion: n 1 f f n 1 n f - n -1 0 df, (13) and no recurrence formulas are known. In case of he srucure n ou of n+2, he analyical descripion becomes more complex, as here is he second reserve objec. This means ha in he flee, esablished a he momen and consising of (n-1) objecs aged and one new objec, one of he objecs may fail and be replaced by he second reserve objec before he momen. In case of he flee wih srucure of n ou of n+k wih he ho reserve, we may use he following relaion: and he recurrence formula: n k (14) nk i nki (n, nk) (1 ) in i n, 1. (15) n k n 1,n k1 n,n k1 Complexiy of he analyical descripion, regardless of simplifying assumpions ha have been made (i.e. idenical objecs, omission of he reliabiliy srucure of objecs alone), indicaes ha here is a need for using a compuer simulaion for issues being considered here. If k vehicles works as he ho reserve, he sysem can be reaed as n ou of n+k srucure and he order p represens demanded level of reliabiliy. However, in he case of k redundan vehicles working as he cold reserve wih n vehicles presening a series reliabiliy srucure here is a need for calculaions new value for he level of reliabiliy. 1 p 1 - k, (16) k i k n k n k 1 n i1 n i 1 k probabiliy of failure one of n vehicles ( 0 = p), p accepable probabiliy of sysem failure, reliabiliy of a single vehicle, n number of vehicles needed for ransporaion asks execuion, k number of redundan vehicles. This order is less hen ha for he whole flee according o his formula, derived from he formula (14). The relaion beween orders (p, k ) is shown in Fig.2. For he order p i can be calculaed quanile for he srucure n ou of n+k. Then he reliabiliy of a single vehicle for he same quanile is calculaed for he srucure 1 ou of 1. 490
Prevenive Mainenance wih Imperfec epairs of Vehicles Wih use of hese values he order k can be calculaed, which is he order for he srucure n ou of n. 6. Simulaion experimens The above consideraion was confirmed wih use of a compuer simulaion. In he model, objecs were applied, ha were fully replaced a seady inervals of mileage, according o resuls of saisical diagnosis. The planned process of replacemens was combined wih random process of failures and repairs. A graph of model saes is presened in Fig. 3. d work saisical diagnosing repair prevenive replacemen Fig. 2. Graph of model saes (work objec is working, saisical diagnosing se of objecs is seleced, prevenive replacemen seleced objecs are replaced by new ones, repair failed objec is replaced by a new one) The flee of n objecs was used for execuion of asks in he model. Each objec is composed of hree groups of differen elemens. The mileage o failure of a single elemen was Weibull disribuion wih a reliabiliy funcion: x a (x) exp. (17) b The accepable probabiliy of he flee unavailabiliy was p. The required reliabiliy was mainained by prevenive replacemens of objecs. Saisical diagnosing was done in inervals of lengh d. The following operaional sraegies were applied: n objecs used flee wihou redundancy, n+1 objecs used one redundan objec, ho reserve, n+2 objecs used wo redundan objecs, ho reserve. Parameers of he model were as follows: n = 50, p = 0.1, d = 3, a 1 = 2.5, b 1 = 65, a 2 = 2.5, b 2 = 80, a 3 = 2.5, b 3 = 100. The range of simulaion is T = 1000, and experimens were repeaed 10 imes. As a resul of simulaion, numbers of replacemens and failures of objecs and unavailabiliy of he whole flee were esimaed. Firs, he number of failures in he sysem n ou of n wihou any prophylacic and hen wih saisical diagnosing wih p = 0.1 and d = 2,5 was esimaed. The resuls show ha i is possible o achieve demanded reliabiliy wih significan decreasing he number of random brakes in work bu wih a very big number of prevenive replacemens (Tab.1). 491
J. Okulewicz, T. Salamonowicz Tab. 1. Simulaion experimens resuls for sysem n ou of n Number of: Wihou prevenive replacemens wih perfec objec repair Wihou prevenive replacemens wih imperfec objec repair Wih prevenive replacemens (d = 2,5) p = 0,10 prevenive replacemens - - 33568 flee unavailabiliy 1102 2065 42 objec failures 1102 2065 42 Imperfec repairs effec in greaer number of flee unavailabiliy wih compare o perfec objec repairs. This is obvious because when repairing only elemens he objec resources are no resored, and he objec is no good as new. Numbers of flee unavailabiliy wih perfec and imperfec repairs are equal only for exponenial disribuion. The aim of prevenive replacemens is o decrease number of random objec failures by avoiding hem wih assumed probabiliy, as hey break flee uilizaion and bring many unpredicable consequences. However o achieve his significan decreasing he number of unavailabiliy, up o 60 % of all opporuniies o replace elemens were realized (maximum possible replacemens of elemen was 60 000 in he example). Such a grea number of replacemens in his case were a resul of raher low reliabiliy of objec componens. The reliabiliy of he flee can be also enlarged by adding redundancy o he sysem. This decreases he number of prevenive replacemens in sysem n ou of n+k as - according o formula (15) - he resul is analogical o appropriae increasing he quanile order of he sysem n ou of n. Such modificaion of he quanile order can be designaed graphically as i is shown in Fig. 3. The modified orders of quanile calculaed wih use of formula (15) for sysems wih n+1 and n+2 objecs are as follows: 1 = 0.410, 2 = 0.660. Bu he resul is only valid for a perfec repair afer every saisical diagnosing, i.e. each objec is replaced by he new one. For complex objecs, i.e. composed of some elemens his condiion could be fulfilled when he inerval of saisical esing where long enough. However, such a long inerval is useless and pracically should be lower han quanile of given order a x = 0. The imperfec repair done by replacing seleced elemens of serviced objecs are useful only when he inerval beween saisical diagnosing is shorer han he iniial quanile a = 0. Afer a number of such replacemens of objecs elemens he flee do no consiss of new objecs. So he probabiliy of asks fulfilling by a single vehicle should be calculaed for sysems n ou of n+1 and n ou of n+2 separaely on he base of appropriae experimens (Tab. 2), insead of basing on reliabiliy funcion of a new objec. Then he modified orders for sysem n ou of n should be calculaed wih use of formula (15). Simulaion resuls in Tab.2 show ha he numbers of replaced elemens in sysems n ou of n wih p = 1 and n ou of n+1 wih p = 0.1 are similar, as well as in sysems n ou of n wih p = 2 and n ou of n+2 wih p = 0.1. The perfec repair means resoring he objec afer each repair. Considering complex objecs his also means replacing all of heir elemens o achieve objec saus as good as new. This can no be acceped in pracice. Therefore imperfec repairs are more appropriae for pracical applicaions. The resuls of simulaions confirmed a naural supposiion, ha elemens of he lowes reliabiliy consiued a dominaing group of replaced elemens. The share of such elemens in he whole number is greaer han wihou saisical diagnosing. These elemen where recognized and such a shifing was done by he algorihm guaranying he demanded level of he flee reliabiliy. 492
Prevenive Mainenance wih Imperfec epairs of Vehicles 1,1 (x) 1,0 =0,989 =0,979 1 ou of 1 0,9 p=0,100 n ou of n 0,8 0,7 0,6 1 = 0,410 n ou of n+2 0,5 n ou of n+1 0,4 2 = 0,660 0,3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fig. 3. Graphical inerpreaion of calculaing new quanile orders Tab. 2. Simulaion experimens resuls (d = 2,5) Number of: n ou of n+1 p=0.10 n ou of n 1 =0,346 Model n ou of n+2 p=0.10 n ou of n 2 =0,482 prevenive elemens replacemens 14405 14638 10886 9807 group 1 7434 10213 5340 6582 group 2 4516 2982 3435 2175 group 3 2455 1443 2112 1050 flee unavailabiliy 23 174 16 256 objec failures 154 174 232 256 eliabiliy of a single vehicle 0.992 0.991 0.989 0.987 7. Conclusions The imperfec repairs are naural way of mainaining vehicles abiliy o performing ranspor asks. They beer fi o real siuaions, since he perfec repair policy is quie unrealisic in case of vehicles. A kind of imperfec repair is a prevenive replacemen of vehicle elemens as i resores flee capaciy parially. This way a considerable reducion in a number of incidenal failures of vehicles, compared o a use wihou any prophylaxis, is achievable hrough applicaion of he saisical conrol. However, mainaining a high reliabiliy of a flee of vehicles is accompanied by a grea amoun of prevenive replacemens of vehicles elemens. This means ha here are many more prevenive replacemens han random failures of objecs because of relaively low reliabiliy of a single objec. 493
J. Okulewicz, T. Salamonowicz Thus, i would be easier o achieve he required flee availabiliy by adding redundan vehicles ha replace damaged ones han o mainain a high reliabiliy of he flee of vehicles wihou redundancy. So, by adding a redundan objec, more failures of vehicles can be acceped as well as a number of prevenive replacemens is reduced. I would be useful o combine redundancy and prevenive replacemen based on saisical diagnosing. equired level of flee reliabiliy could be achieved by adding surplus vehicles and properly maching hem wih he quanile order applied o he main par of he flee. By hose wo measures, random failures of he vehicles flee are significanly reduced in number of replaced elemens being much lower han hose wihou redundancy. eferences [1] Barlow,. E., Proschan, F., Mahemaical Theory of eliabiliy, SIAM Philadelphia 1996. [2] Joe, H., Proschan, F., Percenile residual life funcions, Operaions esearch, vol. 32, 3; pp. 668-679, 1983. [3] Okulewicz, J., Salamonowicz, T., Porównanie wybranych sraegii odnów profilakycznych, Maeriay XXXIV Zimowej Szkoy Niezawodnoci, pp. 218-227, Szczyrk 2006. [4] Salamonowicz, T., Model niepenej odnowy przy naprawach wymuszonych i profilakycznych, Maeriay XXXIII Zimowej Szkoy Niezawodnoci, pp. 464-469, Szczyrk 2005. [5] Smalko, Z., The basic mainenance sraegies of machines and equipmen, Archives of Transpor, vol.3, no 3, Warszawa 1991. [6] Wang, H., A survey of mainenance policies of deerioraing sysems, European Journal of Operaional esearch 139, pp. 468-489, 2002. 494