A GENERAL APPROACH TO TOTAL REPAIR COST LIMIT REPLACEMENT POLICIES

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1 67 ORiON, Vol. 5, No. /2, pp ISSN X A GENERAL APPROACH TO TOTAL REPAIR COST LIMIT REPLACEMENT POLICIES FRANK BEICHELT Deparmen of Saisis and Auarial Siene Universiy of he Wiwaersrand Johannesburg Souh Afria ABSTRACT A ommon replaemen poliy for ehnial sysems onsiss in replaing a sysem by a new one afer is eonomi lifeime, i.e. a ha momen when is long-run mainenane os rae is minimal. However, he sri appliaion of he eonomi lifeime does no ake ino aoun he individual deviaions of mainenane os raes of single sysems from he average os developmen. Hene, Beihel [2] proposed he oal repair os limi replaemen poliy: he sysem is replaed by a new one as soon as is oal repair os reahes or exeeds a given level. He modelled he repair os developmen by funions of he Wiener proess wih drif. Here he same poliy is onsidered under he assumpion ha he one-dimensional probabiliy disribuion of he proess desribing he repair os developmen is given. In he examples analysed, applying he oal repair os limi replaemen poliy insead of he eonomi lifeime leads o os savings of beween 4% and 3%. Finally, i is illusraed how o inlude he reliabiliy aspe ino he poliy.. INTRODUCTION Only a few sysems are required o operae wihou mainenane. They operae in environmens suh as ouer spae and high radiaion fields, where arrying ou mainenane is dangerous, very expensive or simply no possible. Usually, sysems are subjeed o mainenane, boh o prevenive and orreive mainenane. In prevenive mainenane, sysems or heir omponens are exhanged, inspeions sheduled, lubrians applied and so on before a failure ours. In his way, sysem reliabiliy is preserved or inreased by saving off aging effes aused by orrosion, wear, faigue and oher influenes. In orreive mainenane, he onsequenes of sysem failures are removed by eiher repairs or replaemens. The purpose of prevenive mainenane is o redue he osly and dangerous effes of sudden sysem

2 68 failures. On he oher hand, prevenive mainenane gives rise o oss as well. Hene, he problem is o find a os opimal ompromise beween prevenive and orreive mainenane. The main subje of he mahemaial heory of mainenane is o onribue o he soluion of his problem, globally and in numerous speial siuaions. Probably he firs os-based approah owards mainenane opimisaion onsiss in replaing a sysem by a new one afer is eonomi lifeime, i.e. a ha momen, when is long-run oal mainenane os rae (inluding replaemen os) is minimal (Clapham [4]; Eilon, King, Huhinson [5]). However, he sri appliaion of he eonomi lifeime does no ake ino aoun individual deviaions of mainenane os raes of sysems from he average os developmen. Hene, he repair os limi replaemen poliy has been proposed: A sysem is replaed afer failure by a new one if he orresponding repair os exeeds a erain level. Oherwise, a minimal repair is arried ou. By definiion, a minimal repair does no affe he failure rae of he sysem, bu enables he sysem o oninue is work. (For a survey and disussion, see Beihel []) Thus, even if he oal repair os rae migh jusify a replaemen, he deision o repair or o replae a sysem depends only on he os of a single repair. Hene, furher os savings seem o be possible if he whole hisory of he repair proess is aken ino aoun. This leads o he following replaemen poliy: Poliy The sysem is replaed as soon as is oal repair os reahes or exeeds a given level. In omparison wih he repair os limi replaemen poliy, poliy has wo major advanages: ) Applying poliy does no require informaion on he underlying lifeime disribuion of he sysem. 2) Apar from he pure repair os, oss due o oninuous monioring, serviing, sok keeping, personnel os, loan repaymen (inluding ineres raes) e. an be aken ino aoun. Hene, from he modelling poin of view and wih respe o is praial implemenaion, poliy is definiely superior o boh he eonomi lifeime approah and he repair os limi replaemen poliy. Therefore, in view of is simple sruure, and he fa ha mainenane os daa are usually available, poliy seems o be a suiable basi sraegy for planning osopimal replaemen yles of omplex, wear-subjeed ehnial sysems suh as ruks,

3 69 ranes, aerpillars, bel onveyors e. and for deermining os-opimal overhaul yles of whole indusrial plans. For obvious reasons, i makes sense o all poliy a oal repair os limi replaemen poliy if he erm repair os inludes all mainenane oss apar from replaemen oss. Poliy was inrodued in Beihel [2]. There repair os developmens are modelled by funions of he Wiener proess wih drif. This approah allows o generae desired rends of he underlying os proess {), `ZLWK) being he oal repair os in [, ]. Moreover, he expeed firs passage imes of {), `QHHGHGIRUHVWDEOLVhing he long-run oal mainenane os rae under poliy, an easily be deermined. The main resul proved in ha paper is ha applying poliy insead of he ommon eonomi lifeime approah leads o subsanial os savings. An, a leas formal, disadvanage of he Wiener proess assumpion is ha he sample pahs of he orresponding os proess {), `ZLOOGerease in some ime inervals, even if funions are hosen in suh a way ha he rend funion of he proess {), ` M()E()), is fas inreasing in. (However, simulaion sudies indiae ha he sample pahs of proesses {), ` PRGHOOHG LQ WKLV ZD\ VKRZ D TXLWH UHDOLVWLF EHKDYLRXU 7KLV Saper pursues he same goal as Beihel [2], namely o show ha, from he os poin of view, poliy is superior o he eonomi lifeime approah. However, i is based on a more adequae heoreial framework. I makes explii use of he fa ha he sample pahs of he proess {), ` are non-dereasing. Hene, given he one-dimensional probabiliy disribuion of he proess {), ` LWV ILUVW-passage ime disribuion is given as well. Two models of onedimensional probabiliy disribuions are analysed. They indiae he superioriy of poliy o he eonomi lifeime approah. (Noe ha os omparisons beween poliy and he repair os limi poliy make lile sense sine he laer assumes knowledge of he underlying lifeime disribuion of he sysem.) The example disribuions (Weibull disribuion and a relaed one) are hosen suh ha simple explii formulas for boh he repair os rend funion and he expeed firs passage ime exis. Less raable disribuions require he use of numerial mehods. For praial appliaions i is imporan o poin ou ha, given a se of empirial repair os daa, his paper provides he heoreial basis for deermining opimal oal repair os limi replaemen poliies by simulaion. Finally i has o be menioned ha he repair os proesses {), ` FRQVLGHUHG LQ WKLV Saper need no be sohasi proesses in he srily mahemaial sense. Apar from he problem of is exisene, a sohasi proess is usually no fully haraerized by is one-dimensional probabiliy disribuion and he fa ha i has non-dereasing sample pahs.

4 7 Assumpions ) The planning horizon is infinie. 2) Replaemens and oher mainenane aions ake only negligibly small imes. 3) The lenghs of replaemen yles (imes beween wo neighbouring replaemens) are independen, idenially disribued random variables wih finie expeaion. 4) ) does no involve replaemen oss. 2. CST CRITERIA To deermine he long-run oal mainenane os per uni ime under poliy, we need he firs passage ime L( of he proess {), `), wih respe o a fixed, bu arbirary posiive level : L ( inf{, ) > } Sine he sample pahs of {), `DUHQRQ-dereasing, P() P(L(>) for any > and > () Hene, if P( ) x) denoes he probabiliy disribuion funion of ), hen he F expeed value of L( is E( L( ) F ( d (2) By he elemenary renewal heorem, using assumpions o 4, he long-run oal mainenane os rae under poliy is easily seen o be a + K(, E( L( ) where a, he os of a replaemen, is assumed o be onsan. Noe ha a+ is he oal mainenane os wihin a replaemen yle and E(L() is he mean yle lengh. The problem onsiss in finding a mainenane os limi * whih is opimal wih respe o K(. In wha follows, poliy is ompared wih poliy 2. Suh a omparison makes sense, sine poliies and 2 have he same os inpu and do no require informaion on he probabiliy disribuion of he sysem lifeime. Moreover, poliy 2 is a very ommon replaemen sraegy. Poliy 2 The sysem is replaed by an equivalen new one afer ime unis. The orresponding long-run oal mainenane os rae is

5 7 a + M ( ) K( ) and he opimal replaemen inerval * saisfies equaion K()dM()/d. The ime span given by * is alled he eonomi lifeime of he sysem. Obviously, i only makes sense o onsider probabiliy disribuions of ) wih properies ) F for all x 2) lim M ( ) / Propery 2 guaranees a suffiienly fas growh of he repair os rend funion M()E()). Oherwise, a leas wih respe o poliy 2, no replaemen a all is he os-opimal behaviour. 3. DISCUSSION OF SPECIAL CASES Example For any posiive, le he probabiliy disribuion funion of ) be given by F x P( ) x) exp ; λ >, >, > ; x λ (Weibull disribuion). The orresponding probabiliy densiy is f λ x x exp λ ; λ >, >, > ; x Poliy Aording o (), he probabiliy disribuion funion of he firs passage ime L( is P( L( > ) f dx Hene, he expeed value of L( is given by E ( L( ) f dxd f ddx (3) Changing he order of inegraion is allowed sine he inegrand is a oninuous funion for all x, > and bounded a. Inegraion yields E ( L( ) k / wih k / λ Γ

6 72 Hene, he orresponding oal mainenane os rae is a + K( (4) / k The opimal limi * and he orresponding oal mainenane os rae are ( ) / a a *, K( *) (5) k Poliy 2 The oal expeed repair os is E( )) x f dx k2 (6) wih k 2 λ Γ + Hene, when applying he onsan replaemen inerval, he oal mainenane os rae beomes a K ( ) + k2 The orresponding opimal values of and K() are * k / a / a, ( *) 2 2( ) K k ( )/ (7) Comparison of poliies and 2 The inequaliy K(*)<K(*) is easily seen o be equivalen o < Γ + Γ (8) Noe ha [ Γ( / )] of inequaliy (8) beomes is an inreasing funion in, >. For, he righ-hand side

7 73 Γ + The funion Γ Γ Γ π π sin f x / sin x assumes is infimum in (, π] a x. Moreover, i is f ( + ). Hene, inequaliy (8) holds for all, > so ha poliy is superior o he eonomi lifeime approah. In pariular, if. 5 and 2, hen average os savings of beween 3% and 4.2% are ahieved by applying he opimal os limi * insead of he eonomi lifeime *. Example 2 For any nonnegaive, le he probabiliy disribuion funion of ) be given by F λ P( ) x) exp ; λ >, >, > ; x > x f λ x ( + ) λ exp x ; λ >, >, > ; x > Poliy Using (3), he expeed value of L( is seen o be / E ( L( ) k, where / k Γ + (9) λ Hene, he oal mainenane os rae again has sruure (4) so ha he opimal values of and K( are given by (5) wih k given by (9). Poliy 2 The expeed mainenane os is E( )) k2, where k 2 λ Γ () Wih k 2 given by (), he opimal values of and K() are again given by (7). Comparison of poliies and 2 The inequaliy K(*)<K(*) is equivalen o < Γ + Γ

8 74 Analogously o example, i an be shown ha his inequaliy holds for all,, so ha poliy is again superior o poliy 2. In pariular, if 5 and 2, hen average os savings beween 33% and 6.2% are ahieved when applying poliy insead of poliy COMBINED AGE-TOTAL REPAIR COST LIMIT REPLACEMENT POLICY Sheduling replaemens on he basis of a oal repair os limi does no ake ino aoun, a leas no expliily, reliabiliy requiremens imposed on he sysem. There are several possibiliies for inluding he reliabiliy aspe ino he model. A simple way onsiss in limiing he lengh of a replaemen yle by a onsan. By suiably hoosing severe breakdowns of he sysem an be avoided wih a given probabiliy. This leads o he following replaemen poliy. Poliy 3 The sysem is replaed as soon as he oal repair os reahes level or afer ime unis, whihever ours firs. Noe ha, nowihsanding a formal analogy from he modelling poin of view, his poliy srongly deviaes from he ommon life-ime based age replaemen poliy. Following poliy 3, he probabiliy disribuion funion of he yle lengh, Y min (L(, ), is, wih he noaion inrodued in he previous seions, F (, P( Y ), > so ha E Y ) F ( ( d In view of assumpions o 3 of seion, he long-run mainenane os rae has sruure K(, ) [ a + E( ) L( )] P( L( + [ a + ] Leing F F, E( Y ) E ) L( ) E( ) ) < ) F ( ( F dx Hene, he long-run oal mainenane os rae beomes P( L( < )

9 K(, ) a + F ( + F dx F ( d 75 Sine arises from meeing speifi reliabiliy requiremens, i is a fixed parameer. Generally, numerial mehods have o be applied o minimize K(,) wih respe o 5. CONCLUSIONS The sohasi models for he repair os developmen analysed in in his paper as well as some more models no disussed here (Beihel [3]), give srong argumens in favour of sheduling replaemens on he basis of oal repair os limis insead of he eonomi lifeime. However, muh more heoreial and experimenal work, in pariular Mone-Carlosimulaion, needs o be done o ge a deeper insigh ino he relaionship beween poliies and 2. By inluding he age replaemen onep ino he oal repair os limi replaemen poliy, reliabiliy requiremens an be aken ino aoun. REFERENCES [] Beihel, F. (993). A Unifying Treamen of Replaemen Poliies wih Minimal Repair, Naval Researh Logisis, Vol. 4, No., pp [2] Beihel, F. (997). Toal Repair Cos Limi Replaemen Poliies, ORiON, Vol. 3, No /2, pp [3] Beihel, F. (2). A Replaemen Poliy Based on Cos Resriions, Leure a he 2 nd In. Conf. On Mahemaial Mehods in Reliabiliy Theory, Bordeaux, 3 rd -7 h July, 2. [4] Clapham, J. C. R. (957), Eonomi life of equipmen, Operaions Researh Quarerly, Vol. 8, No. 2, pp [5] Eilon, S.; King, J. R.; Huhinson, D. E. (966), A sudy in equipmen replaemen, Operaions Researh Quarerly, Vol. 7, No., pp ACKNOWLEDGEMENT The auhor is graeful o a referee and he Edior for many valuable ommens on he original draf of he paper.