Optimal replacement and overhaul decisions with imperfect maintenance and warranty contracts

Transcription

1 Optmal replacemet ad overhaul decsos wth mperfect mateace ad warraty cotracts R. Pascual Departmet of Mechacal Egeerg, Uversdad de Chle, Caslla 2777, Satago, Chle Phoe: Fax: Abstract I ths artcle we develop a model to help a mateace decso-makg stuato of a gve equpmet. We propose a ovel model to determe optmal lfe-cycle durato ad tervals betwee overhauls by mmzg global mateace costs. We cosder a stuato where the customer, whch ows the equpmet, may egotate a better warraty cotract by offerg a mproved prevetve mateace program for the equpmet. The equpmet receves three kd of actos: repars, overhauls, ad replacemet. A overhaul represets a mperfect mateace acto, that s, the falure rate s mproved but ot a pot that the equpmet s as good as ew. Correctve mateace actos are mmal, the sese that the falure rate after each repar s the same as before the falure. The proposed strategy surpasses others see the lterature sce t cosders at the same tme the warraty egotato stuato ad the optmal lfe-cycle durato uder mperfect prevetve actos. We also propose a smplfed approach that facltates the task of mplemetg the method stadard spreadsheet solvers. Keywords: mperfect mateace, warraty, maagemet, lfe-cycle cost. 1. Itroducto May systems are sold wth a warraty that offers protecto to the buyers agast early falures durg the facy of the equpmet ad as a medum of promoto to the vedor. Whe the warraty perod teds to be large, degradato also appears durg t ad ths case prevetve mateace plays a mportat role order to reduce the falure rate, ad ts evoluto tme. Offerg perods of warraty mples extra costs for the vedor. There are repar costs correctve mateace) ad possbly pealty costs to be pad, whch are assocated to dowtme. A prevetve mateace program may reduce such correctve costs. Gve that the buyer does ot pay repars durg the warraty perod, there are o cetves for hm to sped prevetve actos the case that the warraty also covers for the dowtme costs). For the vedor, t s coveet to perform prevetve actos oly f they cost less tha the expected correctve costs. From the buyers pot of vew, vestg prevetve mateace durg ad after the warraty perod may have sgfcat effects o the lfe-cycle costs. Cosequetly, t may be coveet to defe prevetve polces all alog the lfe-cycle. The am of ths work s to preset a geeral framework for ths stuato from the pot of vew of the buyer. We cosder three kd of mateace actos: mmal repar as good as before the falure), mperfect overhaul betwee as good as after the prevous overhaul ad as good as before the overhaul) or replaced as good as ew). Each acto has ts ow costs ad may deped o varables such as age ad/or qualty. We wll cosder a sgle compoet aalyss, eglectg scale ecoomes that may appear from egotatg for mult-compoet systems. A cost aalyss for systems seres, parallel ad a combato of both may be foud Ba ad Pham[2]. Modellg of the system mprovemet due to mperfect mateace s crucal to establsh the cost model to be optmzed. Malk[9] troduced the cocept of vrtual age, whch essetally says that the system s youger tha before the acto by some terval T y. A lmtato of ths model s that t does ot alter the referece falure rate fucto. Nakagawa[11] assumed that the acto s mmal wth probablty a ad perfect wth probablty 1 a). As t s referred to mmal ad perfect actos, ths model mples that as tme passes the qualty of the overhauls must be mproved order to keep a costat. Ths has some cosequeces: f the orgal falure rate wthout overhauls of the system s a power fucto of tme, the falure rate s always bouded. Zhag ad Jarde [7] propose a falure rate model where after a overhaul, t s betwee as good as before ad as good as after prevous overhaul. Ths model does ot boud the falure rate as Nakagawa s but due to the dscotutes the falure rate fucto, t may be dffcult to evaluate the umber of expected falures durg the warraty perod, whch s varable our model. Djamalud et al.[4] propose to use a cotuous falure rate fucto whose growth parameter s depedet o the qualty cost) of the mateace polcy. For maxmum qualty the falure rate s costat, for mmum qualty, the falure rate correspods to the orgal falure rate whe o overhauls are performed. A serous lmtato of ths approach s how to model the relatoshp betwee the qualty of the prevetve polcy ad the falure rate. A summary of research o mperfect mateace s offered Pham ad Wag[5]. Cosderg warraty, Djamalud et al.[4] develop a framework to study prevetve polces whe the vedor offers a gve perod of warraty where he pays labor, materals ad dowtme costs f a falure occurs. Uder ths premse, the buyer s ot ecessarly commtted to prevetve mateace. Jack ad Dagpuar[6] use a vrtual age model to determe the qualty ad the perod betwee overhauls. I ther model, the optmal soluto s complete reewal at each overhaul age ). Jug ad Park[8] study the optmal perodc prevetve polces followg the exprato of warraty. They

2 use the expected mateace cost rate per ut tme from the buyer s perspectve. They also use a vrtual age model for the falure rate ad cosder that prevetve actvtes start just after the ed of the warraty. Ths lmts the optmalty of the prevetve mateace sce early prevetve actos may reduce eve further the lfe-cycle costs[4]. Research dealg o warraty cost aalyss has bee summarzed Blschke ad Murthy[3] ad Murthy ad Djamalud[1]. The model that s preseted here cosders that prevetve polces are take from the begg of the lfe-cycle. I ths way, the falure rate s reduced ad costs assocated to correctve mateace are reduced. We shall propose a falure rate model that takes the advatages of both the model of Zhag ad Jarde ad the oe by Djamalud et al.. Based o the proposed model we mmze the expected cost per ut-tme. It allows a optmal decso o lfe-cycle durato ad the umber of perodc overhauls to perform durg t. It also permts a egotato of the warraty perod wth the vedor. We show results for the case whe the referece falure rate fucto s growg expoetally wth tme. We llustrate the methodology through a umercal example ad we obta optmal values for the umber of overhauls ad the lfe-cycle durato. The mprovemets made over the referece models are: lfe-cycle durato s a decso varable; we obta a optmal value for the warraty perod to be egotated wth the vedor; we cosder explctly the dowtme costs whch are ot pad geeral by the vedors durg the warraty perod; we cosder the cost of a overhaul as a fucto of ts qualty; we propose a cotuous falure-rate model whch s equvalet to the model of Zhag ad Jarde but t s easer to evaluate at ay stat. 2. Geeral model 2.1. Statemet of the problem Equpmet breaks dow from tme to tme, requrg repars. Also, whle the equpmet s beg repared, there s a loss producto output. I order to reduce the umber of falures, we ca perodcally overhaul the equpmet ad perform prevetve actos. After some tme t may be ecoomcally coveet to replace the equpmet by aother ew oe. Our purpose s to determe the optmal lfe-cycle durato ad the terval betwee overhauls that mmze the global cost per ut tme Costructo of model Let us cosder the followg codtos: Equpmet s subjected to three types of actos: mmal repar, mperfect overhaul, ad replacemet; each acto has ts ow assocated costs; Equpmet s repared whe t fals; Equpmet s perodcally replaced; Equpmet receves 1 overhauls durg ts lfe; 1, teger,postve 1) the terval betwee overhauls T s s costat. The lfecycle durato T l s the T l = T s 2) a overhaul mproves the equpmet term of ts falure rate λ, all repars are mmal, that s, they oly retur the equpmet to producto but they do t mprove the falure rate. The qualty of a overhaul as well as ts cost) s depedet o the mprovemet factor p; < p < 1 3) Materal ad tradesme to perform a repar cost c m ; dowtme cost of a repar s c f m ; overall cost of a overhaul s c o t cludes: materal, tradesme, dowtme costs); overall cost of a replacemet s c r vestmet, labor, materal, dowtme costs); falure rate wth perodc overhauls s λt); falure rate f o overhauls are performed s λt); The vedor oly pays labour ad materal costs durg the warraty perod T w ; dowtme costs are assumed by the customer; The customer ad the vedor agree to egotate a exteded warraty perod f the customer performs overhauls durg the durato of the cotract; we have: T w < T l 4) repars beyod the warraty perod are pad by the customer; recovery value of the equpmet s eglgble; the qualty p s cosdered costat, as well as ts cost, the mea tme to repar s eglgble frot of the mea tme betwee falures; the vedor offers a basele warraty perod T w where the customer s ot oblged to perform overhauls; the customer may egotate a exteso to the warraty so, T w T w 5) Our am s to determe the umber of overhauls, ther terval T s or equvaletly the lfe-cycle durato T l ), ad the warraty terval T w, that mmze the total expected cost per ut tme c g.

3 2.3. Cost model The expected umber of falures durg the warraty perod s gve by: λt)dt ad durg the rest of the lfe-cycle by T w λt)dt The total lfe-cycle cost for the customer s gve by C g,t l, p,t w ) = c r + c o p) 1) + c f m λt)dt+ 6) ad for the vedor by c m T w λt)dt as a cosequece, T w s ot a actve decso varable the sese that t may be obtaed oce λt) s modelled. Of course, the vedor wll agree to exted the warraty oly f at least oe overhaul s performed durg [,T w ]: T s T w 2.5. Dscotuous falure rate model The falure rate s a crucal dcator of the equpmet codto sce t permts falure forecastg ad establsh approprate prevetve measures lke overhauls. We wll cosder that the falure rate after a overhaul fals betwee as bad as just before ad as well as just after the prevous overhaul wth some mprovemet factor p, 1). Let λ k t) be the falure rate after the k th overhaul. We wll express the falure rate as: λ k t) = pλ k 1 t T s ) + 1 p)λ k 1 t) 11) c m λt)dt the lfe-cycle cost per ut tme for the customer s the c g,t l, p,t w ) = C g,t l, p,t w ) T l 7) 2.4. Costrats If o egotato occurs, the vedor agrees to pay the labor ad materal requred to mmally repar the equpmet durg the warraty perod: 1. whch produce dscotutes λt) as observed fgure λ 1/days) c m λt)dt O the other had, the customer performs overhauls durg the lfe of the equpmet order to reduce the umber of falures durg the lfe of the equpmet. Ths would reduce the expected umber of falures durg the warraty perod ad the expected cost for the vedor. Performg overhauls mply costs to be pad by the customer so the vedor could compesate hs efforts by extedg the warraty perod. The vedor does ot wat to crease hs expected cost so he would agree to exted the warraty f the expected cost durg the exteded warraty does ot surpass the expected cost for the basele warraty, that s: c m λt)dt c m λt)dt 8) or, terms of umber of falures, λt)dt λt)dt 9) Gve that λt) s o egatve, we observe that the last term 6): c m T w λt)dt s decreasg as T w creases. The mmzato of the total cost of the customer mples forcg the costrat 9) to the equalty: λt)dt = λt)dt 1) Tme days) Fgure 1: Model from Zhag ad Jarde vs proposed from the example, p =.7 I a geeral stuato the agg process shows the patter observed fgure 1. If the mprovemet factor p s, the λ k t) = λ k 1 t) other words, the falure rate s the same as before the overhaul so t may be cosdered as a mmal repar see fgure 2). λ 1/days) 1.6 x Tme days) Fgure 2: Perfect p = 1) ad mmal p = ) overhaul p= p=1

4 If the mprovemet factor p s 1, λ k t) = λ k 1 t T s ) the overhaul returs the falure rate to the level obtaed just after the prevous overhaul. Sce all overhauls have the same level ad ther perodcty s costat, overhaul may be cosdered as a replacemet fgure 2). It may be prove that 1 λt)dt = = 2.6. Expoetal growth model If the falure rate wth o overhaul follows from 12) y 13) we have ) Ts p 1 p) 1 λt)dt 12) λt) = e α +α 1 t, α 1 > 13) ) Ts λt)dt = p 1 p) 1 λt)dt = ) = p 1 p) 1 eα e α 1T s 1 ) = α 1 e α ) = 1 p)α 1 p [ 1 p) e α )] 1T s = e α ) 1 p)α 1 p 1 p) 1 = [ ] = e α p 1 p)e α 1 T s 1 1 p)α Cost of a overhaul It s logcal to mpose a depedecy betwee the qualty of a overhaul ad ts cost, we propose where c o p) = c o,m e βp 14) β = log c o,m c o,max ad c o,m y c o,max are mmum ad maxmum complete reewal) costs for a overhaul see fgure 3). Cmax 3. Proposed model The model preseted 2.5 show dscotutes every tme a overhaul s performed. Gve that the replacemet problem cosders a log term, stataeous falure rate values are ot mportat to the optmzato problem ad a log term approxmato s useful to easly determe the umber of falures durg the lfe-cycle ad durg the warraty perod Equvalet parameters By observato of fgure 1), let us cosder a log term falure rate model of the type ˆλt) = e ˆα + ˆα 1 t 15) where ˆα ad ˆα 1 are determed by p. To estmate these parameters we explot 11), λ 1 T s ) = pλ ) + 1 p)λ T s ) If we cosder the model preseted 2.6, λ 1 T s ) = pe α + 1 p)e α +α 1 T s 16) We may estmate ˆα 1 by usg the pots,α ) ad T s,λ 1 T s )) whch gve us a lower boud for the falure rate at ay stat, ˆλ f t) = e α + ˆα 1 t 17) wth substtutg 16) to 17), ad we obta, t [,T l ] e α + ˆα 1 T s = pe α + 1 p)e α +α 1 T s 18) α 1 = log pe α + 1 p)e α +α 1 T s ) α T s 19) I order to estmate ˆα, we cosder that the cotuous model should produce the same umber of falures for each terval T s ad that the slope a loglear dagram s ˆα 1, Cp) Ts ˆλt, ˆα 1 )dt = Ts λt)dt e ˆα ) e ˆα 1T s 1 = eα e α 1 T s 1 ) ˆα 1 α 1 ad we obta, Cm 1 p Fgure 3: Overall cost of a overhaul as a fucto of p 1 see ref. [7]. α = log e α ˆα 1 e α 1 T s 1 ) ) α 1 e ˆα 1 T s 1 ) 2) The use of 17) greatly smplfes the evaluato of costrat 1) ad facltates the optmzato process.

5 3.2. Optmal value for the warraty perod If the falure rate wth o overhauls follows 13), the expected umber of falures durg the referece warraty perod T w s: T w λt)dt = eα ) e α 1 T w 1 α 1 the, the o-lear costrat 9) takes the form e ˆα ) e ˆα 1T w 1 = eα ) e α 1 T w 1 ˆα 1 α 1 from where we obta T w explctly: T w = ) ) log e α ˆα 1 e ˆα α 1 e α 1 T w ) ˆα 1 Fgure 4: Study of the topology of the cost fucto for p = Numercal example Let us cosder the followg data, whch are very smlar to those used referece [7], cosderg the extra parameters eeded for our model: c g c r = 2 KUSD/reewal c o,m = 8 KUSD/overhaul c o,max = 32 KUSD/overhaul c f m = 1 KUSD/repar c m = 1 KUSD/repar T w = 73 days T l Fgure 5: Mma for varyg lfe-cycle duratos p =.7) 1 the referece falure rate wth o overhaul) follows λt) = e 15+.1t c g wth t days. We have ĉ g = c r + c o,m e βp e ˆα 1) + c f m ˆα 1 e ˆα 1 T l 1 ) e + c ˆα m ˆα 1 e ˆα 1 T l e ˆα ) 1T w T l T s the followg results are obtaed: = 7 T l = 1877 days T w = 16 days c g,t s ) =.155 USD/day So the system must be replaced every 5.15 years ad must be overhauled every 1877/7 1) = 313 days. The warraty may be egotated wth the vedor from 2 years to 16/365 = 2.76 years. Fgure 6: Mma for varyg tervals betwee overhauls p =.7) The example was solved usg the solver of EXCEL[1]. Table 1) gves the optmal solutos for varous mprovemet factors. For the gve cost structure the larger the mprovemet factor s, the more overhauls should be performed but also the lfe-cycle ad the exteded warraty are larger. A sestvty aalyss for p =.7 s show fgures 4-6) respectvely. We observe fgure 4) that the cost per ut tme s qute usestve to T l a the rage 15,25) days. No local mma appear for ths varable. If stead of T l, we use T s as decso varable as t s doe Zhag ad Jarde)

6 some local mma may perturb serously the fdg of the global mmum as t s observed fgure 6). Optmzato s the doe terms of T l. p T l ĉ g T w Table 1: Optmal solutos for dfferet p 5. Coclusos Refereces [1] Excel 2: User Maual. E Publshg Ltd. [2] Ba, J. et Pham, H. Warraty cost models of reewable rsk-free polcy for mult-compoet systems. IN- FORMS QSR Best Studet Paper Competto, 22. [3] Blschke, W. R. et Murthy, D. N. P. Product Warraty Hadbook. Marcel Dekker, Ic., [4] Djamalud, I., Murthy, D. N. P., et Km, C. S. Warraty ad prevetve mateace. Iteratoal Joural of Relablty, Qualty ad Safety Egeerg, vol. 8, o. 2, 21, p [5] H, H. P. et Wag, H. Imperfect mateace. Europea Joural of Operatoal research, o. 94, 1996, p [6] Jack, N. et Dagpuar, J. S. A optmal mperfect mateace polcy over a warraty perod. Mcroelectrocs Relablty, vol. 34, o. 3, 1994, p We have preseted a model to relate the log term falure rate wth the mprovemet factor of the overhauls ad ther terval. We establshed a cost optmzato model to determe optmal levels of prevetve mateace. The model cludes easly a egotato crtero for extedg the warraty perod. The formulato has bee smplfed to permt the use of stadard spreadsheet solvers to solve the mmzato problem. The exteso of the model to cosderate dscouted costs s straghtforward. I the ear future the author s cosderg the relaxato to the costrat that mposes equal tervals betwee overhauls. As falure rates crease, t would make sese to crease the prevetve strategy ad further reduce costs. [7] Jarde, A. K. S. et Zhag, F. Optmal mateace models wth mmal repar, perodc overhaul ad complete reewal. IIE Trasactos, vol. 3, 1998, p [8] Jug, G. et Park, D. Optmal mateace polces durg the post-warraty perod. Relablty Egeerg ad System Safety, o. 82, 23, p [9] Malk, M. Relable prevetve mateace schedulg. IEE Trasactos, vol. 11, 1985, p [1] Murthy, D. N. P. et Djamalud, I. New product warraty: A lterature revew. Iteratoal Joural of Producto Ecoomcs, o. 79, 22, p [11] Nakagawa, T. Optmal polces whe prevetve mateace s mperfect. IEEE Trasactos o Relablty, o. R-28, 1979, p