Evaluation of maintenance policies for equipment subject to quality shifts and failures

Transcription

1 Evaluaion of mainenance policies for equipmen subjec o qualiy shifs and failures Sofia Panagioidou, George Tagaras To cie his version: Sofia Panagioidou, George Tagaras. Evaluaion of mainenance policies for equipmen subjec o qualiy shifs and failures. Inernaional Journal of Producion Research, Taylor Francis, 00, (0), pp.-. <0.00/000000>. <hal-00> HAL Id: hal-00 hps://hal.archives-ouveres.fr/hal-00 Submied on Sep 00 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Inernaional Journal of Producion Research Evaluaion of mainenance policies for equipmen subjec o qualiy shifs and failures Journal: Inernaional Journal of Producion Research Manuscrip ID: TPRS-00-IJPR-00.R Manuscrip Type: Original Manuscrip Dae Submied by he Auhor: -Aug-00 Complee Lis of Auhors: Panagioidou, Sofia; Arisole Universiy of Thessaloniki, Deparmen of Mechanical Engineering Tagaras, George; Arisole Universiy of Thessaloniki, Deparmen of Mechanical Engineering Keywords: STOCHASTIC MODELS, PREVENTIVE MAINTENANCE, MAINTENANCE PLANNING, PROCESS MONITORING Keywords (user):

3 Page of Inernaional Journal of Producion Research Evaluaion of Mainenance Policies for Equipmen Subjec o Qualiy Shifs and Failures SOFIA PANAGIOTIDOU (span@auh.gr) and GEORGE TAGARAS * (agaras@auh.gr) Arisole Universiy of Thessaloniki Deparmen of Mechanical Engineering Thessaloniki, Greece el.: fax.: Revised Augus 00 * Corresponding auhor

4 Inernaional Journal of Producion Research Page of Evaluaion of Mainenance Policies for Equipmen Subjec o Qualiy Shifs and Failures Absrac We develop an economic model for he opimizaion of mainenance procedures in a producion process wih wo qualiy saes. In addiion o deerioraing wih age, he equipmen may experience a jump o an ou-of-conrol sae (qualiy shif), which is characerized by lower producion revenues and higher endency o failure. The imes o qualiy shif and failure are allowed o be generally disribued random variables. We consider wo ypes of mainenance: minimal mainenance (MM) ha upgrades he qualiy sae of he equipmen wihou affecing is age and perfec prevenive mainenance (PM) ha fully upgrades he equipmen o he as-good-as-new condiion. We derive he expression for he expeced profi per ime uni and we invesigae, hrough a large number of numerical examples, he ype of he opimal soluion. I is concluded ha in pracically every case he opimal mainenance policy is an exreme one: i eiher calls for immediae MM as soon as a qualiy shif occurs (acive policy) or i allows operaion in he ou-of-conrol sae unil he ime of a scheduled PM acion (passive policy).

5 Page of Inernaional Journal of Producion Research Inroducion Equipmen mainenance is a very imporan operaion in almos every producion sysem. An appropriae prevenive mainenance (PM) policy no only reduces he probabiliy of equipmen failure bu also improves he working condiion of he equipmen resuling in lower producion coss and/or higher produc qualiy. The failure rae of he equipmen is ypically assumed o increase wih ime or usage (afer he iniial infan moraliy period) and consequenly age-based PM models have been widely sudied in he lieraure. These models usually assume ha he operaing condiion of he equipmen remains sable hroughou producion and no deerioraion mechanism exiss oher han complee failure. Someimes, hough, he equipmen may deeriorae o a less desirable working condiion before failing alogeher. This inferior condiion may be associaed wih boh higher operaing or qualiy coss and increased failure raes. For example, consider he cuing process of meal frames using power saws. From ime o ime, he cuing disc loses is balance and consequenly is abiliy o produce perfecly fla bur-free cus on he meal pars. In addiion o he lower qualiy of he produced meal pars, he misbalance leads o higher cenral axle faigue and higher probabiliy of breaking down (failure). Similarly, misbalance is a common malfuncion in many elecric and elecronic devices where a fan is used for he cooling process of he device. When his happens, he consequences are poor cooling, higher elecriciy consumpion, and higher proneness o failure of he elecric moor due o overheaing. In cases like he above condiion-monioring can provide useful informaion regarding he operaing sae of he equipmen and consequenly can lead o PM acions ha proec he equipmen more effecively agains failures. Several condiion-based PM models have been developed o address he problem of properly mainaining

6 Inernaional Journal of Producion Research Page of equipmen wih muliple operaing saes; see he surveys by Pierskalla and Voelker () and Valdez-Flores and Feldman (). However, he grea majoriy of hese models focus on he invesigaion of replacemen or perfec PM policies ha resore he equipmen o an as-good-as-new sae, while relaively few conribuions have been made o he field of imperfec PM. Imperfec PM is assumed o improve he working condiion of he equipmen bu no necessarily o an as-good-as-new sae. Various imperfec PM models have been suggesed in he lieraure, some of which are applicable o muli-sae producion processes/equipmen. See he survey by Pham and Wang () for more deails. Here we are mainly ineresed in cases where an imperfec PM upgrades he operaing sae of he equipmen by reducing is failure rae wihou affecing is age. A special case of his ype of imperfec PM is minimal mainenance (MM), which improves he equipmen condiion by one sae only. The objecive of his paper is o sudy a mainenance policy, including boh perfec and imperfec mainenance acions, which is appropriae for a producion process (equipmen) wih wo operaing saes, namely an in-conrol sae and an ou-ofconrol sae in Saisical Process Conrol erminology, and a failure sae. The wo disinc operaing saes are characerized by differen operaing and qualiy-relaed coss and by differen failure raes; he in-conrol sae has generally lower operaing/qualiy cos per ime uni and lower failure rae for he same equipmen age compared o he ou-of-conrol sae. Producion processes wih muliple operaing saes (qualiy saes) are ypically encounered in he conex of saisical process conrol, where a common pracice agains qualiy deerioraion is o bring he process back o is in-conrol sae afer he deecion of a qualiy shif o an ou-of-conrol sae. In his paper, we combine qualiy

7 Page of Inernaional Journal of Producion Research adjusmens o upgrade he process qualiy sae wih convenional mainenance acions o deal wih failures. More specifically, he proposed mainenance policy comprises he following ypes of mainenance acions: Perfec correcive mainenance (CM) upon failure: i resores he failed equipmen o he as-good-as-new sae. Perfec prevenive mainenance (PM) a some criical age: i resores he working equipmen o he as-good-as-new sae. Minimal mainenance (MM) applied only when he process is ou-ofconrol: i upgrades he equipmen from he ou-of-conrol o he in-conrol sae wihou affecing he equipmen age. The main conribuion of he presen paper is ha by allowing for generally disribued imes o qualiy shifs and o failures (wih non-decreasing failure raes) i grealy expands he model s realism and applicabiliy o acual producion sysems compared o exising models ha require a leas one of he above ime disribuions o be exponenial. An addiional conribuion lies in he resuls; i is documened ha in pracically all cases i suffices o consider and compare only exreme qualiy mainenance policies (acive or passive), which are easy o undersand and implemen. Secion presens a brief lieraure review while Secion describes he problem in deail and inroduces he necessary noaion. In secion we develop he mahemaical model while in secion we discuss he form of he opimal policy. Secion provides numerical examples and a sysemaic discussion of he effec of several model parameers on he opimal mainenance policy. The las secion summarizes he basic resuls and suggess direcions for fuure research.

8 Inernaional Journal of Producion Research Page of Lieraure Review The academic lieraure is replee wih models of mainenance policies comprising various ypes of acions (imperfec mainenance, minimal repair ec.). In his brief lieraure review we resric our aenion o models for producion processes (equipmen) wih muliple disinc operaing saes and a leas one failure sae, which assume perfec failure resoraion and include he noion of imperfec PM. More specifically we consider imperfec PM ha upgrades he operaing sae of he equipmen by reducing is failure rae wihou affecing is age. One of he mos imporan disinguishing feaures among hese models is he ype of he failure mechanism. We herefore sar he review wih models based on purely Markovian deerioraion mechanisms (exponenial disribuion of he ime o failure) and nex we proceed o he presenaion of more general non-markovian models. Purely Markovian deerioraion The pioneering work of Derman () concerns a repair-replacemen policy for a muli-sae Markovian deerioraing producion process. The model allows several alernaive mainenance decisions a every operaing sae of he process, which do no necessarily resore he equipmen o an as-good-as-new sae bu hey affec he sae ransiion probabiliies. In addiion, Derman () considers muliple inoperaive saes where replacemen is he only feasible acion. The objecive is o find he mainenance policy ha maximizes he expeced ime beween replacemens. I is shown ha his kind of problem can be expressed hrough a linear programming formulaion. Özekici and Günlük () sudy a similar problem bu wih deerminisic mainenance effecs (every mainenance acion leads o a cerain process sae) and a more general cos funcion. The objecive is o selec he mos appropriae mainenance

9 Page of Inernaional Journal of Producion Research acion for each process sae so as o minimize he expeced cos. Özekici and Günlük () provide srucural properies of he opimal mainenance policy under various cos srucures. A slighly differen approach o modeling of imperfec mainenance (repair) for producion processes wih muliple operaing saes and an absorbing failure sae has been proposed by Chiang and Yuan (00). The process is periodically inspeced and: a) nohing is done if he equipmen is found o be wihin a group of good saes, or b) he equipmen is repaired o a beer sae if found o be wihin a group of inermediae saes, or c) he equipmen is replaced if found o be wihin a group of inferior saes including he failure sae. The opimal policy is derived by minimizing he expeced long-run cos rae. A Markovian model which explicily combines qualiy conrol wih mainenance procedures has been developed by Tagaras (); i concerns a producion process subjec o qualiy shifs and failures, assuming muliple ou-of-conrol saes. The objecive is o opimize boh qualiy conrol schemes and PM procedures. In addiion o CM (resoraion) following a failure and periodic PM, qualiy adjusmens are carried ou whenever he process is found o operae in an ou-of-conrol sae. Non-Markovian deerioraion Mousafa e al. (00) sudy a muli-sae semi-markovian deerioraing sysem, allowing boh replacemen and minimal mainenance. They assume ha minimal mainenance upgrades he equipmen condiion by one sae. They use a conrol limi policy wih wo hreshold saes in a way similar o Chiang and Yuan (00), bu hey show ha his ype of policy is no always opimal.

10 Inernaional Journal of Producion Research Page of Makis and Fung (, ) sudy he inegraed problem of deermining he opimal qualiy conrol schedule and he opimal producion quaniy in a producion process wih wo qualiy saes and a single failure sae. The model allows for periodic prevenive replacemen as well. In boh models he ime o qualiy shif is assumed o be exponenially disribued while he ime o failure is assumed o be generally disribued. As soon as a qualiy shif is deeced he process is resored o he in-conrol sae wih he same equipmen age. I is worh noing ha he failure ime disribuion is assumed o be independen of he acual qualiy sae of he process and consequenly resoraions of qualiy shifs do no consiue PM acions agains failures. The producion process sudied in his paper is similar o ha of Makis and Fung (wo qualiy saes and a single failure sae) bu we focus on mainenance procedures raher han on he deerminaion of producion quaniies. In addiion, we consider general (no necessarily Markovian) deerioraion mechanisms no only for failure bu for qualiy shifs as well. Furhermore, we allow he failure ime disribuion o depend no only on he equipmen age bu also on is sae. Before concluding his brief lieraure review i should be noed, for he sae of compleeness, ha here exis a number of papers combining saisical process conrol (SPC) wih prevenive mainenance procedures. Alhough hese papers share some degree of similariy wih our work, hey approach he problem mosly from a qualiyoriened poin of view, since hey only consider qualiy deerioraion mechanisms. One of he earlies models in his field is ha of Rahim and Banerjee (). Their objecive is he inegraed opimizaion of he SPC parameers and he PM ime in producion processes subjec o qualiy shifs, which resul in an inferior-qualiy, ye operaing, sae. Oher relevan models have been developed more recenly by Cassady e al.

11 Page of Inernaional Journal of Producion Research (000), Lee and Rahim (00) and Linderman e al. (00). Ben Daya and Rahim (000) sudy a similar problem incorporaing he noion of imperfec prevenive mainenance. However, hese papers do no ake ino accoun he possibiliy of a complee failure ha would enforce an immediae cease of operaion, which is a ypical elemen of pracically all mainenance problems and models.. Problem definiion, assumpions and noaion We consider a producion process ha may operae in one of wo possible qualiy saes; in-conrol sae or sae 0 and ou-of-conrol sae or sae. Regardless of he acual qualiy sae of he process, he equipmen may suffer a failure a any ime, resuling in complee soppage of operaion. The failure rae of he process in boh qualiy saes is a non-decreasing funcion of he equipmen age. The failure rae in he ou-of-conrol sae is assumed o be higher han ha of he in-conrol sae for he same equipmen age. Apar from ha, he wo qualiy saes also differ in erms of producion revenues; he ou-of-conrol sae is assumed o be less profiable han he in-conrol sae. Equal failure raes or equal producion revenues are easily reaed as special cases. Qualiy shifs from he in-conrol o he ou-of-conrol sae may occur a any ime bu hey are immediaely observed and consequenly he acual qualiy sae of he process is always known wih accuracy. Noe ha he assumpion of coninuous and accurae knowledge of he acual process qualiy sae is no characerisic of ypical SPC problems. However, i is ofen realisic since here are many cases where he qualiy shif is direcly observable or even self announced, e.g. hrough some disinc noise. Besides, even when a qualiy shif has o be deeced hrough inspecion, he inspecion process is someimes performed on a coninuous basis and is very accurae; e.g. inspecion by means of on - line sensors.

12 Inernaional Journal of Producion Research Page 0 of Since he failure rae increases wih he age of he equipmen due o physical deerioraion even in he in-conrol sae, i is reasonable o prevenively mainain he equipmen when i reaches some criical age ( m0 ), beyond which he probabiliy of failure is unaccepably high. PM upgrades he equipmen in he as-good-as-new condiion and oally renews he process. Alhough he frequency of failures can be radically decreased hrough PM, he equipmen will ineviably fail occasionally and producion will be inerruped. CM is implemened as soon as a failure occurs and he equipmen is resored again o he as-good-as-new condiion. From he above descripion i is clear ha he producion-mainenance process consiss of a series of independen and sochasically idenical cycles. Each cycle begins wih he process in he as-good-as-new condiion (in-conrol sae and zero equipmen age) and erminaes eiher wih a PM a m0 or wih a CM following a failure, whichever occurs firs. Wihin each cycle he process may shif o he ou-of-conrol sae leading o decreased producion revenues and/or increased failure rae. To avoid producion under hese poor condiions i is possible o sop he process and bring he equipmen back o he in-conrol sae wihou affecing he equipmen age. In oher words, an MM can ake place o improve he qualiy sae of he process. I is worh noing ha such MM acions are usually assumed o be performed immediaely afer he process is deeced o operae a or above some hreshold sae. Our model is more flexible han ha, since he process may inenionally be allowed o operae for some ime in he ou-of-conrol sae before MM is performed. In general, MM is no necessarily preferable o operaion in he ou-of-conrol sae. Wheher MM is worhwhile or no depends on he radeoff beween is cos on one hand and he lower

13 Page of Inernaional Journal of Producion Research revenues and higher failure rae associaed wih ou-of-conrol operaion on he oher. We consider he following hree alernaive siuaions/policies:. There is a criical age of he equipmen, m (0< m < m0 ), beyond which operaion in he ou-of-conrol sae is uneconomical due o he unaccepably high failure rae. Tha is, if a qualiy shif occurs a ime > m hen MM is performed immediaely and he process coninues is operaion in he in-conrol sae. If a qualiy shif occurs a ime < m, he process is allowed o coninue in he ou-of-conrol sae unil m and i is only hen resored o he in-conrol sae (unless a failure occurs before m ).. Operaion in he ou-of-conrol sae is so cosly ha he process is no allowed o operae a all in his sae. In such cases MM is always performed as soon as a qualiy shif occurs ( m =0). Borrowing from he auomaic conrol erminology, a policy wih m =0 will be called Acive Qualiy Mainenance policy, or simply AQM.. The cos of MM is oo high relaive o is benefis and consequenly he equipmen is no resored o he in-conrol sae earlier han m0 ( m = m0 ). A policy wih m = m0 will be called Passive Qualiy Mainenance policy, or simply PQM. Noe ha when m < m0 (cases and above) he process may shif o he ou-ofconrol sae more han once in each producion cycle and as a resul MM will be performed several imes in a cycle. To summarize, he proposed mainenance policy is characerized by wo criical imes (equipmen ages) m and m0 ( 0 m m0 ) and he objecive is o find he

14 Inernaional Journal of Producion Research Page of opimal values of m0 and m ha maximize he expeced profi per ime uni. The noaion ha will be used o develop he opimizaion model is presened below: f() densiy funcion of he ime o qualiy shif F() cumulaive disribuion funcion of he ime o qualiy shif; F() = F( ) h( ) φ i () = f () F() (qualiy shif rae) densiy funcion of he ime of failure (equipmen age) if he process is in sae i (i=0,) a =0; noe ha he densiy funcion of he ime o failure if a qualiy shif occurs a ime s is φ () Φ ( s ) for > s Φ i () cumulaive disribuion funcion of he ime of failure in sae i; hi = i i m0 m Z Z P Z M R i W W P W M i i Φ () = Φ φ () Φ () (failure rae in sae i) equipmen age; =0 a he beginning of each cycle scheduled prevenive mainenance ime in he in-conrol sae scheduled minimal mainenance ime if a qualiy shif occurs a < m expeced ime o perform correcive mainenance expeced ime o perform prevenive mainenance expeced ime o perform minimal mainenance expeced ne revenue per ime uni of operaion in sae i cos of correcive mainenance cos of prevenive mainenance cos of minimal mainenance 0

15 Page of Inernaional Journal of Producion Research n Expeced number of MM acions in a cycle E(T) expeced cycle lengh E(P) expeced cycle profi EPT expeced profi per ime uni Noe ha R 0 and R ake ino accoun he poenial cos of low qualiy iems and he operaing cos. Consequenly i is reasonable o assume ha R R 0.. Model developmen In order o formulae he EPT funcion we firs develop he expressions for he expeced cycle lengh E(T) and he expeced cycle profi E(P). Each cycle begins wih zero equipmen age (=0) and erminaes eiher wih repair afer failure before m0 or wih prevenive mainenance a m0. In boh cases he process may never shif o he ou-ofconrol sae, may shif jus once or may shif several imes. Thus, he oal duraion of a cycle consiss of he following sub-periods: operaing ime in he in-conrol sae (random variable T 0 ) operaing ime in he ou-of-conrol sae, if a qualiy shif occurs prior o failure and before m (random variable T ) repair ime, if a failure occurs before m0 prevenive mainenance ime, if he equipmen reaches age m0 wihou failure ime for minimal mainenance acions. Similarly, he expeced profi per cycle consiss of he following componens: producion ne revenue in he in-conrol sae

16 Inernaional Journal of Producion Research Page of producion ne revenue in he ou-of-conrol sae, if a qualiy shif occurs prior o failure and before m repair cos, if a failure occurs before m0 prevenive mainenance cos, if he equipmen reaches age m0 wihou failure cos of minimal mainenance acions. Operaing ime in he in-conrol sae Once he equipmen reaches age m (eiher in he in-conrol or in he ou-of-conrol sae) i is no allowed o operae in he ou-of-conrol sae in he remainder of he cycle, alhough qualiy shifs may sill occur (bu will be immediaely removed hrough MM). Consequenly, he expeced in-conrol period during he cycle can be divided ino wo pars; one before m and one afer m. Regarding he firs par, he in-conrol period lass unil m only if neiher a failure nor a qualiy shif occurs by ha ime; oherwise, i lass unil some ime < m. Thus, he expeced duraion of he in-conrol period before m is m. Ε(Τ before ) = Φ F + φ F d+ f Φ d 0 m m 0 m m Noing ha φ ( ) d= -dφ ( ) and inegraing F( ) dφ ( ) 0 0 m 0 by pars yields 0 m Ε(Τ before ) = Φ F d. () 0 m 0 0 The in-conrol period afer m lass eiher unil m0 or unil an equipmen failure before m0. Thus, he expeced lengh of he in-conrol period afer m provided ha he equipmen survives unil m is m

17 Page of Inernaional Journal of Producion Research m 0 Φ0 m0 φ0 0 m ( m0 m) ( m) Φ0 m Φ 0 m Ε(Τ afer ) = + - d. Again, using φ ( ) d -dφ ( ) 0 0 = and inegraing by pars yields m m 0 m 0 Φ Ε(Τ0 afer m) = d. () Φ 0 m The probabiliy ha he equipmen will survive unil m wihou a failure is Φ = +. () m m p Φ m 0 m F m f Φ0 d Φ 0 The firs erm in he righ hand side of () expresses he probabiliy ha neiher a failure nor a qualiy shif occurs before m (he process remains in he in-conrol sae unil m ), while he second erm expresses he probabiliy ha a qualiy shif occurs prior o m, ye no failure occurs unil m (he operaing ime in he ou-of-conrol sae is sricly posiive). The oal expeced ime ha he process operaes in he in-conrol sae in a cycle is given by m E T = Ε(Τ before ) + p Ε(Τ afer ), 0 0 m 0 m which, combining () hrough () and simplifying resuls in = + + m m 0 m m 0 Φ m Φ0 0 0 m 0 0 Φ 0 m 0 Φ m 0 m. () E T Φ F d F Φ d f Φ d d Operaing ime in he ou-of-conrol sae The process spends some ime in he ou-of-conrol sae if and only if a qualiy shif occurs prior o m. Then, he operaing ime in he ou-of-conrol sae lass unil m if no failure occurs by ha ime or unil an equipmen failure before m. Thus, he

18 Inernaional Journal of Producion Research Page of expeced period ha he process operaes in he ou-of-conrol sae in a cycle is given by: ( - )φ Φ Ε(Τ ) f Φ d d - f ()Φ d m m m m = 0 + ( m ) 0 Φ 0 Φ 0. () Inegraing by pars he second inegral of he firs erm in he righ hand side of () yields ( ) Φ m m Ε(Τ ) f( Φ ) 0( ) d d Φ 0 =. () Probabiliy of prevenive mainenance and probabiliy of failure Prevenive mainenance is only performed whenever he equipmen reaches age m0 wihou a failure. The probabiliy ha he equipmen will reach age m wihou a failure, p m, is given by (), while he probabiliy ha he equipmen will no fail in he ime inerval from m o m0, provided ha i survived unil m, is Φ0( m0) Φ0( m). The fac ha he process may shif (once or more) o he ou-of-conrol sae afer m has no effec on he failure probabiliy of he process since he equipmen is immediaely resored o he in-conrol sae. Thus, he probabiliy of prevenive mainenance in a cycle is p p Φ F f Φ d m Φ0 m0 Φ m Φ0 m0 PM = = m 0 m m + 0 Φ0 m Φ 0 Φ0 m = Φ Φ Φ F + f Φ d. () m 0 m0 m 0( m0) ( m) 0 Φ0 m Φ 0 The probabiliy of failure in a cycle is simply -p PM.

19 Page of Inernaional Journal of Producion Research Expeced number of minimal mainenance acions The earlies ime ha a minimal mainenance can ake place is m ; his happens if he equipmen reaches age m in he ou-of-conrol sae. Therefore, he expeced number of minimal mainenance acions by m, denoed n, is equal o he probabiliy of ha even Φ =. () m m n f( Φ ) 0( ) d Φ 0 The equipmen is also minimally mainained every ime a qualiy shif occurs afer m. The expeced number of minimal mainenance acions afer m, provided ha he equipmen survived unil m, is denoed n and compued by he following lemma, he exensive proof of which is presened in he Appendix. Lemma The expeced number of minimal mainenance acions afer m in a cycle, provided ha he equipmen survives unil m, is n 0 m0 = h d. () m Φ Φ 0 m The expeced oal number of minimal mainenance acions hroughou a cycle is n= n + p n. Combining equaions (), () and () and simplifying we obain m Φ Φ Φ n= f Φ d+ F Φ h d+ f Φ d h d m m 0 m m0 m m 0 0 ( m) 0 0 Φ Φ 0 m 0 Φ m 0 m. Expeced profi per ime uni The oal expeced cycle lengh is given by Ε(Τ) = Ε(Τ ) + E(T ) + Z p + Z - p + Z n, 0 P PM PM M while he oal expeced profi per cycle is given by (0)

20 Inernaional Journal of Producion Research Page of Ε(P) = R Ε(Τ ) + R E(T ) - W p - W - p W n. 0 0 P PM PM M Finally, since he process is a renewal reward process, he expeced profi per ime uni can be expressed as he raio of he expeced profi per cycle o he expeced cycle lengh: EPT (, ) m0 m E(P) =. E(T) Noe ha he model holds for any value of m in he inerval [0, m0 ] including he special cases m =0 (immediae MM afer a qualiy shif; acive qualiy mainenance policy) and m = m0 (he equipmen is allowed o operae in he ou-of-conrol sae unil m0 wihou any MM inervenion; passive qualiy mainenance policy). In he laer case, if no failure occurs and he equipmen succeeds o survive unil m0 operaing in he ou-of-conrol sae, he scheduled PM a m0 is combined wih an MM, wih oal cos W M + W P and oal ime Z M + Z P, so ha he nex producion cycle begins wih he process in a perfec condiion (equipmen as-good-as-new and in-conrol).. Wha is (usually) he ype of he opimal policy? The mainenance model developed in he previous secion allows m o be anywhere in he range [0, m0 ]. For given m0, he model essenially compares he cos of MM o upgrade he process qualiy o he in-conrol sae agains he benefis of such an acion and reurns he opimal value of m. However, he mainenance managemen of mos acual producion sysems eiher adops an Acive Qualiy Mainenance (AQM) policy, whereby he equipmen is never allowed o operae in he ou-of-conrol sae ( m =0) or i employs a Passive Qualiy Mainenance (PQM) policy, whereby he equipmen is reaed he same regardless of is qualiy sae ( m = m0 ). The choice beween AQM and PQM is ypically made empirically; if he ou-of-conrol sae is considered

21 Page of Inernaional Journal of Producion Research unaccepable hen AQM is adoped, while if process inerrupions are considered cosly and undesirable hen PQM is preferred. To invesigae he acual ype of he opimal policy and evaluae he qualiy of AQM and PQM policies we solved a large number of numerical examples wih differen problem parameers. Specifically, we solved 0 examples wih all process parameers randomly seleced from a wide range of values. In addiion, we have allowed boh Weibull and Gamma disribuions o describe he process failure and qualiy shif mechanisms. The densiy funcions of hese disribuions are presened in Table. Noe ha ensuring ha he failure rae when operaing in he ou-of-conrol sae wih equipmen age is a leas as large as he failure rae in he in-conrol sae wih he same equipmen age, requires c 0 =c and λ0 λ for boh disribuion ypes. The allowable values of all process parameers are shown in Table. [Inser Tables and abou here] The opimizaion is performed by means of an exhausive search over all possible m0 and m values, under he consrain 0 m m0, o ensure ha he global opimum of EPT is obained. For compuaional simpliciy we resric our numerical invesigaion o ineger values of m0 and m wih iniial values equal o zero. The opimizaion algorihm compues he expeced profi for all m0 and m values and he search procedure sops as soon as m0 and m reach some hreshold values beyond which heir effec on he expeced profi is insignifican. The convergence of EPT is assured by he fac ha he probabiliy of failure will evenually reach uniy for large values of m0. The compuaional ime required for finding he opimal soluion ypically varies beween 0 and 0 minues on a Penium IV. GHz personal compuer, depending on he parameers of he example.

22 Inernaional Journal of Producion Research Page 0 of The main findings are summarized below: In all 0 cases, he opimal policy is eiher AQM ( m =0) or PQM ( m = m0 ). In he vas majoriy of he cases examined ( ou of 0) he opimal policy is AQM. Among he cases where AQM is no opimal, he percenage loss ha would resul from using he bes possible AQM insead of he opimal PQM is.% on average, ranging beween 0.0% and.%. Among he cases where PQM is no opimal, he percenage loss ha would resul from using he bes possible PQM is.% on average, ranging beween 0.0% and.%. When he failure imes follow Gamma disribuions, he opimal soluion usually dicaes ha he equipmen should no undergo PM a all ( m0 ). This can be explained by he fac ha he failure rae of a Gamma disribuion is sabilized as he equipmen age grows large (similar o he memoryless exponenial disribuion) and i may be beer o coninue operaion from some poin on raher han mainain he equipmen. I is imporan o menion here ha alhough he opimal soluion in all numerical examples solved in he course of his research (including he 0 examples of his secion and he examples of he nex secion) is obained for an exreme value of m (eiher 0 or m0 ), in general he opimal soluion is no necessarily unique. In some cases he expeced profi funcion EPT is so fla wihin a range of m values (from m =0 o some criical m value) ha pracically all m in his range can be considered opimal. In hese cases i may be preferable o choose he larges m in he opimal range so as o minimize downimes due o MM acions.

23 Page of Inernaional Journal of Producion Research Also noe ha our exensive numerical invesigaion of he behaviour of EPT has shown ha an exreme value of m (eiher m =0 or m = m0 ) is always opimal, even when here are muliple opimal soluions wih 0< m < m0. Alhough a formal proof of such a propery, if i is indeed rue in general, remains elusive, he pracical implicaion is ha i is sufficien o search for he opimal soluion only beween he wo exreme policies AQM and PQM. In his way he compuaional requiremens are significanly reduced since i suffices o opimize wo single-variable funcions insead of searching over a wo-dimensional decision space.. The effec of process parameers on he opimal policy To invesigae sysemaically he effec of he process parameers on he opimal policy we have solved problems, differing subsanially in key model feaures such as he relaive coss of operaion and mainenance aciviies and he equipmen proneness o qualiy shifs and failures. Specifically, we express he qualiy shif mechanism and he failure mechanism in boh qualiy saes (i=0,) by Weibull disribuions of he equipmen age. The parameers λ, λ, R and W P are examined a levels each as shown in Table, while he pair W M, Z M is examined a hree levels as follows: (a) W M =0.W P and Z M =0.Z P (b) W M =0.W P and Z M =0.Z P (c) W M =0.W P and Z M =0.Z P The remaining parameers are se equal o he following values in all cases: c 0 =c =, c=., λ 0 =0.00, R 0 =00, Z=Z P =.0, W=00. The cases are numbered a, b, c o a, b, c where a, b or c indicaes he W M, Z M combinaion. [Inser Table abou here]

24 Inernaional Journal of Producion Research Page of Table shows he opimal criical equipmen ages m, m0 and he corresponding maximum expeced profi per ime uni, EPT, for he cases. Noe ha, exacly as in he 0 examples of he previous secion, in all cases he opimal policy is eiher AQM ( cases) or PQM ( cases). In of he cases where a passive qualiy mainenance (PQM) policy is opimal, i is opimal o only use correcive mainenance upon failure and never resor o PM or MM acions ( m = m0 = ). The las columns of Table presen he opimal AQM and he opimal PQM policies along wih he percenage losses when using each policy. Obviously, when one of hese policies is opimal is respecive percenage loss is zero. [Inser Table abou here] The effecs of he parameers on he ype of he opimal policy as well as on he savings ha can be achieved by using he proposed model as opposed o blindly following he AQM or he PQM policy are summarized as follows: Large R ( R R 0 ) and/or small λ ( λ λ 0 ) implies ha sae (ou-ofconrol) is no ha inferior o sae 0 (in-conrol) in erms of profi and/or failure rae. Consequenly i may be more economical o allow operaion in sae raher han upgrade he process qualiy o sae 0. Thus, PQM ends o ouperform AQM in such cases. The percenage loss associaed wih he use of an AQM policy when PQM is opimal increases as R increases and/or as λ decreases. This is because increasing R or decreasing λ increases he expeced profi per ime uni for m = m0 (PQM policy), while he expeced profi per ime uni for m = 0 remains unalered (AQM policy); consequenly he wo soluions diverge. 0

25 Page of Inernaional Journal of Producion Research Combinaions b (W M =0.W P, Z M =0.Z P ) and c (W M =0.W P, Z M =0.Z P ) are characerized by a higher direc or indirec (downime) cos of MM han combinaion a (W M =0.W P, Z M =0.Z P ). Consequenly, a PQM policy is more likely o be opimal in combinaions b and c. The percenage loss due o incorrecly adoping an AQM policy (while PQM is opimal) increases when moving from combinaion a of W M and Z M o combinaion b or c, while he percenage loss associaed wih incorrec adopion of a PQM policy decreases. This is because he increased cos of MM in combinaions b and c ends o have a greaer negaive impac on he soluions wih m = 0 han on hose wih m = m0. Large W P and/or large λ signify more expensive or more frequen MM acions and consequenly a PQM policy is more likely o be opimal. The percenage loss associaed wih incorrec use of an AQM policy increases as W P and/or λ increase, due o he negaive effec of boh hese parameers on he oal cos of MM acions. However, he percenage loss associaed wih incorrec use of a PQM policy can eiher increase or decrease as W P and/or λ increase. We have also sudied he isolaed effec of W P, keeping W M consan; in all cases he ype of he opimal policy remained unalered. Finally, he effec of failure ime variabiliy has also been invesigaed using c0= c= insead of c0 = c = in all cases and modifying he values of λ 0 and λ, so as o keep he mean imes o failure unalered for boh qualiy saes. This invesigaion has no revealed any sysemaic effec of he failure ime variabiliy on he

26 Inernaional Journal of Producion Research Page of ype of he opimal policy nor on he percenage loss associaed wih consisenly adoping AQM or PQM.. Conclusions In his paper we have sudied mainenance procedures in a producion process subjec o boh qualiy shifs and failures. The equipmen deerioraes coninuously due o he ageing process and a he same ime i may experience a jump o an inferior qualiy sae (ou-of-conrol sae) upon he occurrence of an assignable cause. Transiions o he ouof-conrol sae have a dual impac on he process; hey resul in lower producion qualiy implying lower producion revenues bu hey also increase he failure rae of he process. The proposed mainenance policy is a combinaion of an age-based prevenive mainenance policy a some criical age m0, wih addiional minimal mainenance acions, which upgrade he process qualiy o he in-conrol sae. Such MM acions (qualiy adjusmens) are commonly used in qualiy conrol o eliminae he negaive effecs of assignable causes. In our model resoring he process o he in-conrol sae afer he occurrence of an assignable cause no only improves producion qualiy bu also decreases he failure rae of he process. In conras o ypical qualiy conrol models, he MM considered here is no necessarily implemened immediaely afer qualiy shifs bu may inenionally be posponed unil some criical equipmen age. Our invesigaion showed ha in pracically every case he opimal mainenance policy eiher calls for immediae MM as soon as a qualiy shif occurs (acive qualiy mainenance, AQM) or allows operaion in he ou-of-conrol sae and an MM is only implemened along wih prevenive mainenance a predeermined imes (passive qualiy mainenance, PQM). Thus, i suffices o consider only he opimal AQM and

27 Page of Inernaional Journal of Producion Research PQM policies, compare hem and use he one ha is mos effecive in each paricular case. Neverheless, using he wrong exreme policy may resul in significan loss. Our numerical invesigaion has shown ha such losses can be as high as abou 0% of he opimal expeced profi. There are some ineresing exensions of he proposed mainenance model ha are worh sudying, such as he case of incomplee informaion abou he qualiy sae of he process. In addiion, full inegraion of qualiy conrol procedures wih equipmen mainenance in deerioraing producion processes under general (non-resricive) assumpions would be of grea pracical ineres. Appendix: Proof of Lemma We firs deermine he probabiliy ha exacly n (n=,,..) qualiy shifs occur in he inerval ( m, m0 ) provided ha he equipmen survives unil ime/age m. We sar wih he derivaion for n= and hen generalize for higher values of n. The process shifs exacly once from he in-conrol o he ou-of-conrol sae (n=) if eiher one of he following wo scenarios maerializes: a) The process shifs o he ou-of-conrol sae a ime ( m < < m0 ) prior o failure, an MM is immediaely implemened upgrading he process qualiy o he in-conrol sae and operaion coninues wihou inervenion (neiher a failure nor a qualiy shif occurs) unil m0. b) The process shifs o he ou-of-conrol sae a ime ( m < < m0 ) prior o failure, an MM is immediaely implemened upgrading he process qualiy o he in-conrol sae and operaion coninues unil he occurrence of a failure a ime before m0. No oher qualiy shifs occur beween and.

28 Inernaional Journal of Producion Research Page of Under hese scenarios he probabiliy of a qualiy shif a prior o failure provided ha he equipmen survives unil m is p = f Φ. (A) m 0 0 qs d F m m Φ0 m The probabiliy ha neiher a failure nor a qualiy shif occurs in (, m0 ) is p Φ F = 0 m0 m0 a Φ0 F, (A) while he probabiliy of failure in (, m0 ) provided ha he equipmen operaes coninuously in he in-conrol sae is p φ F m0 = 0 b d Φ 0 F. (A) Combining (A) hrough (A) he probabiliy ha he process shifs exacly once from he in-conrol o he ou-of-conrol sae during he inerval ( m, m0 ) is given by P = p p + p qs a b f Φ φ F f Φ Φ F = + m 0 m 0 m m0 m0 dd d F m m Φ0 m Φ 0 F F m m Φ0 m Φ0 F. Using h( ) f( ) F( ) = and simplifying yields m 0 m0 m 0 P = h( ) φ0( ) F( ) dd+ Φ0( m0) F( m0) h( ) d F( m) Φ0( m) m m. Reversing he order of inegraion in he double inegral resuls in m 0 m0 P = φ0( ) F( ) h( ) dd Φ0( m0) F( m0) h( ) d F( m) Φ0( m) + m m m (A)

29 Page of Inernaional Journal of Producion Research = m 0 m where H( ) + φ0 F H H m d Φ0 m0 F m0 H m0 H m = h x dx. 0 Φ ( ) F m 0 m Similarly, he process shifs exacly wice (n=) from he in-conrol o he ou-ofconrol sae during he inerval ( m, m0 ) whenever a qualiy shif occurs a ime ( m < < m0 ) prior o failure (probabiliy p qs ), an MM is immediaely implemened upgrading he process qualiy o he in-conrol sae and hen he process shifs exacly once more (a ime/age > ) o he ou-of-conrol sae during he res of he cycle (, m0 ). Since he probabiliy of ha las even is analogous o P of (A) bu refers o he inerval (, m0 ), i is denoed P ( ). Thus, properly adaping (A) o he inerval (, m0 ) leads o he following expression for he probabiliy P of he even n=: = we ge f Φ f Φ φ F P = p P = dd d + m 0 m0 m qs ( ) F m m Φ0 m F Φ0 Φ 0 F m 0 f Φ f Φ Φ F m m0 m0 F Φ F Φ Φ F m m 0 m 0 0 d d m 0 m0 m 0 m 0 m0 + h h φ F dd d Φ F h h d d 0 0 m0 m0 m m Φ ( ) F m 0 m Reversing he order of inegraion in boh inegrals and hen using b a b H H a H b H a h( ) H( ) H( a) d = = a.

30 Inernaional Journal of Producion Research Page of = P = m 0 m0 + φ F h h d d d Φ F h h d d 0 0 m0 m0 m m m m m Φ ( ) F m 0 m m 0 m 0 + φ F h H H d d Φ F h H H d 0 m 0 m0 m0 m m m m = m 0 m Φ ( ) F m 0 m Φ ( ) H H m H m0 H m φ0( ) F( ) d+ Φ0( m0) F( m0) F m 0 m Exending he preceding analysis o n> we arrive (by inducion) a he following general expression for he probabiliy P n ha he process shifs exacly n imes from he in-conrol o he ou-of-conrol sae during he inerval ( m, m0 ): = and m 0 m P n = m 0 m n Φ ( ) m 0 m H H m H m0 H m φ0( ) F( ) d+ Φ0( m0) F( m0) n! n! F The expeced number of MM acions in he inerval ( m, m0 ) is = npn n= n Φ ( ) n n H H m H m0 H m 0 + 0( m0) ( m0) n= n! n= n! φ F n d Φ F n F Using successively he facs ha e H = F n= n= m 0 m. n n x x n = x = xe n! n! ( ) x n..

31 Page of Inernaional Journal of Producion Research he expression for n becomes n = m 0 m H( ) H( m) H( m 0) H( m) 0 m + 0 m0 m0 m0 m φ F H H e d Φ F H H e Φ ( ) F m 0 m = + Φ m 0 φ0( ) H( ) H( m) d Φ0( m0) H( m0) H( m) 0( m). m Finally, using φ ( ) d -dφ ( ) expression of Lemma : 0 0 = and inegraing by pars resuls in he simple n m 0 m 0 Φ = h( ) d. Φ 0 m References Ben-Daya, M. and Rahim, M.A., Effec of Mainenance on he Economic Design of X - Conrol Char. European Journal of Operaional Research, 000, 0, -. Cassady, C.R., Bowden, R.O., Liew, L. and Pohl, E.A., Combining Prevenive Mainenance and Saisical Process Conrol: a Preliminary Invesigaion. IIE Transacions, 000,, -. Chiang, G. H. and Yuan, J., Opimal mainenance policy for a Markovian sysem under periodic inspecion. Reliabiliy Engineering and Sysem Safey, 00,, -. Derman, C., Opimal replacemen and mainenance under Markovian deerioraion wih probabiliy bounds on failure. Managemen Science,,, -. Lee, B.H. and Rahim, M.A., An inegraed economic design model for qualiy conrol, replacemen, and mainenance. Qualiy Engineering, 00,, -.

32 Inernaional Journal of Producion Research Page 0 of Linderman, K., McKone-Swee, K.E. and Anderson, J.C., An inegraed sysems approach o process conrol and mainenance. European Journal of Operaional Research, 00,, -0. Makis, V. and Fung, J., Opimal prevenive replacemen, lo sizing and inspecion policy for a deerioraing producion sysem. Journal of Qualiy in Mainenance Engineering,,, -. Makis, V. and Fung, J., An EMQ model wih inspecions and random machine failures. Journal of he Operaional Research Sociey,,, -. Mousafa, M. S., Abdel Maksoud, E. Y. and Sadek, S., Opimal major and minimal mainenance policies for deerioraing sysems. Reliabiliy Engineering and Sysem Safey, 00,, -. Özekici, S. and Günlük, N. O., Mainenance of a device wih age-dependen exponenial failures. Naval Research Logisics,,, -. Pham, H. and Wang, H., Imperfec mainenance. European Journal of Operaional Research,,, -. Pierskalla, W. P. and Voelker, J. A., A survey of mainenance models: The conrol and surveillance of deerioraing sysems. Naval Research Logisics Quarerly,,, -. Rahim, M.A. and Banerjee, P.K., A generalized model for he economic design of X - conrol chars for producion sysems wih increasing failure rae and early replacemen. Naval Research Logisics,, 0, -0. Tagaras, G., An inegraed cos model for he join opimizaion of process conrol and mainenance. Journal of he Operaional Research Sociey,,, -.

33 Page of Inernaional Journal of Producion Research Valdez-Flores, C. and Feldman, R.M., A survey of Prevenive Mainenance models for sochasically deerioraing single-uni sysems. Naval Research Logisics Quarerly,,, -.

34 Inernaional Journal of Producion Research Page of Table. Densiy funcions for qualiy shif and failure mechanisms Disribuion Qualiy shif mechanism Failure mechanism in sae i (i = 0,) Weibull Gamma c c λ f = λc e, >0, c>0, λ>0 >0, c i >0, λ i >0 c c λ f = λ e Γ c, >0, c>0, λ>0 c c i i λi i i i φ = λ c e, c i c i λ i φ = λ e Γ c, i i i >0, c i >0, λ i >0 0

35 Page of Inernaional Journal of Producion Research Table. Range of process parameer values Process parameer Range of values R R 0 - R 0 W W P W M 0.W - W 0.W P - W P Z = Z P Z M φ i : Weibull (mean µ i ) f: Weibull 0.Z - Z c i. - (0.) µ 0-0 µ 0 - µ 0 c. - (0.) (mean µ) µ - 0 φ i : Gamma (mean µ i ) f: Gamma c i - - µ 0-0 µ 0 - µ 0 c (mean µ) µ - 0

36 Inernaional Journal of Producion Research Page of Table. Parameer values for he numerical invesigaion λ λ R W P

37 Page of Inernaional Journal of Producion Research Se Table. Opimal soluions and evaluaion of he AQM and PQM policies Parameers λ λ R W P MM Opimum AQM Loss PQM Loss Policy m m0 EPT m0 (%) ( m =0) m = m0 (%) a b c a b.. 0 c a b c a b c a b c a b.. 0 c a b c a.. 0 b c a b c a b c a b c a b c a b c a b.0. 0 c a b c a b.. 0 c