Proportional Intensity Model considering Imperfect Repair for Repairable Systems

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1 Iteratoal Joural of Performablty Egeerg Vol. 9, No. 2, March 203, pp RAMS Cosultats Prted Ida Proportoal Itesty Model cosderg Imperfect Repar for Reparable Systems YUAN FUQING * ad UDAY KUMAR Dvso of Operato ad Mateace Luleå Uversty of Techology, SE-97 87Lulea, SWEDEN (Receved o February 25, 202, revsed o Jauary 07, 203) Abstract: The Proportoal Itesty Model (PIM) exteds the classcal Proportoal Hazard Model (PHM) order to deal wth reparable systems. Ths paper develops a more geeral PIM model whch uses the mperfect model as basele fucto. By usg the mperfect model, the effectveess of repar has bee take to accout, wthout assumg a as-bad-as-old or a as-good-as-ew scheme. Moreover, the effectveess of other factors, such as the evrometal codtos ad the repar hstory, s cosdered as covarat ths PIM. I order to solve the large umber parameters estmato problem, a Bayesa ferece method s proposed. The Markov Cha Mote Carlo (MCMC) method s used to compute the posteror dstrbuto for the Bayesa method. The Bayesa Iformato Crtero (BIC) s employed to perform model selecto, amely, selectg the basele fucto ad remove the usace factors ths paper. I the fal, a umercal example s provded to demostrate the proposed model ad method. Keywords: Proportoal testy model (PIM), mperfect repar model, testy fucto, Markov Cha Mote Carlo (MCMC) method, model selecto.. Itroducto A reparable system ca be defed as a system whch s cotuous operato, ad whch s repared, but ot replaced, after each falure []. Itesve research has bee performed to address the problem of determg the relablty characterstcs of reparable systems, usg models based o the Homogeeous Posso Process (HPP), the No-Homogeeous Posso Process (NHPP), ad the mperfect repar model etc. Such models have bee prove successful egeerg. However, the weakess of such models s also crtczed by some researchers. As Ascher ad Fegold have metoed, most probablstc models the feld of relablty have bee smplfed hghly ad are based o urealstc assumptos. Most models cosder oly oe varable, amely the operatg tme. However, some stuatos, the operatg tme s ot the oly effectve factor fluecg relablty, some factors such as the evromet, repar hstory, load etc., wll also affect relablty greatly [2]. The Proportoal Hazard Model (PHM) s oe mportat model whch ca take the above-metoed factors to accout. The PHM was tally developed for the medcal dustry [3]. Ths model comprses two parts. The frst part s called the parametrc model or basele fucto, where the expoetal dstrbuto or the Webull dstrbuto ca be used. The secod part s called the covarat part. Ths part accommodates all the covarates, such as the evrometal factors, the repar hstory factors, ad so o. Kumar has carred out a thorough survey o the PHM [4, 5]. However, the PHM s sutable for o-reparable systems. Whe oe replace the basele fucto wth a testy fucto, the PHM s exteded to the Proportoal Itesty Model (PIM), by whch the problems of reparable systems ca be addressed [6]. *Correspodg author s emal: yua.fuqg@ltu.se 63

2 64 Yua Fuqg ad Uday Kumar For reparable system, Guo ad Love have developed a seres of PIMs [7, 8]. These models assume that the system udergoes Prevetve Mateace (PM) ad Correctve Operato (CO). After each PM acto, the system s assumed to be restored to a asgood-as-ew state. Hece the system each PM perods ca be cosdered as a ew detcal system. Ths assumpto facltates parameter estmato for covarates. Based o ths assumpto, oe ca use the partal lkelhood method to estmate the covarat parameters, regardless of the basele fucto [9]. Ths s oe of the sgfcat characterstcs of the PHM. However, whe o detcal systems are observed, ths parameter estmate method s ot feasble. I state-of-the-art applcatos, some NHPP models are usually used as the testy fucto PIM [7, 8]. The NHPP assumes that, after repar, the system s restored to a same-as-old state. Guo et al. argue that ths assumpto s rarely satsfed practce [0]. I the preset paper, we employ the mperfect repar model stead of the NHPP model as the basele fucto. I our paper, we assume that the operatg codtos are varat betwee each repar ad that o PM s carred out o the system. As o detcal systems are observed, the partal lkelhood estmate method caot be appled to ths case. We propose a Bayesa method to perform parameter estmato. I the Bayesa method, a MCMC method based o slce samplg s used to approxmate the posteror dstrbuto. I the remag of ths paper, Secto 2 dscusses the model developmet corporatg mperfect model to PIM. Secto 3 dscusses the parameter estmate for the developed model. Secto 4 presets a umercal example. Secto 5 presets the cocluso ad dscusses the lmtato of ths proposed model. Notato V q x λ 0 λ z F R f t α L Vrtual age at falure Repar factor Iter-arrval tme betwee th ad - th falure Itesty fucto wthout covarate Itesty fucto of PIM model Covarat such as repar hstory, repar hstory Coeffcet of covarat Falure probablty Relablty Probablty desty fucto (PDF) The th Falure tme Scale parameter power law model Shape parameter power law model Lkelhood fucto D Observed falure data G, ) Gamma dstrbuto wth parameter k k ( 2 U (...) Uform dstrbuto k ad k 2

3 Proportoal Itesty Model cosderg Imperfect Repar for Reparable Systems 65 N (...) Normal dstrbuto (...) k π Dstrbuto Bayesa ferece PIM PHM PM CO HPP NHPP MCMC BIC OPSK HOILQ SCSK STEMP ENDUS Number of parameters BIC Proportoal Itesty Model Proportoal Hazard Model Prevetve Mateace Correctve Operato Homogeeous Posso Process No-Homogeeous Posso Process Markov Cha Mote Carlo Bayesa Iformato Crtero operator skll hydraulc ol qualty mateace crew skll hydraulc system temperature evrometal codtos 2. Imperfect Repar Model wth Covarates 2. Vrtual Age Model It s assumed that the system s rejuveated after each repar. The effectveess of each repar s represeted by a reducto of the expereced age. Such mperfect repar models are called vrtual age models state-of-the-art research. The Kjma I ad Kjma II models are two mportat represetatves of such models []. The Kjma I model s descrbed as: V = V + qx () where x deotes the ter-arrval tme betwee th ad - th falure, V ad V deotes vrtual age at the respectve th ad -th repar. q s the mportat mperfect repar factor, whch accommodates the degree of repar effectveess. Usually, q s bouded wth a rage [0,]. The correspodg Kjma II model s V ( ) = q V + x (2) I both Kjma models, q = 0 mples the system has bee restored completely. The vrtual age model s hece degeerated to a reewal process model. Whe q =, the vrtual age model s degeerated to a NHPP model. For the sake of smplcty, we descrbe the Kjma models as a fucto: V = f V, x ) (3) ( 2.2 Proportoal Itesty Model wth Covarates Assume the system s experecg mperfect repar. Let the testy fucto of ths system uder mperfect repar represeted by ( t; ), the correspodg PIM model λ 0 q

4 66 Yua Fuqg ad Uday Kumar λ represeted by ( t, z;. Smlar to the classcal Proportoal Hazard Model[3], we propose our PIM as follows: λ t, z; = λ ( t; exp( z + z + z +... ) (4) For smplcty, we rewrte Formula (4) as: t z q t q where f z) = z + z + z... b. ( b λ (, ; ) = λ0 ( ; )exp{ f ( z)} (5) ( I ths paper, we assume the covarat betwee each falure are costat ad the covarat vary from each falure ter-arrval tme. Based o ths assumpto, the orgal testy fucto, wthout mperfect repar ad wth costat operato codto, s shfted horzotally due to mperfect repar. Furthermore, due to the varablty of operatg codtos, the orgal testy fucto would be shfted vertcally. Ths scearo s llustrated Fgure, where the λ ( t, z; s show ot cotuous. λ( t, z; t t+ t j Fgure : Effect of Imperfect Repar ad Operatg Codtos The covarates ca be some evrometal factors such as temperature, humdty, dust etc. Moreover, as Ascher has clamed [2], the repar hstory ca also fluece the repar rate. Thus the covarates ca be some factors regardg repar hstory. Percy et al. has developed a PIM that cosders the repar hstory factor [2]. I ther model, they cosder repar hstory data such as: the tme sce the last PM, CO, the total umber of PM actos ad Cos, as covarates. I ther paper, the factors regardg repar hstory are sgfcat. Therefore, cosderg the fluece of the repar hstory s ecessary some stuato. Moreover, the covarates ca corporate some codto motorg data. 2.3 Cumulatve Dstrbuto Fucto of PIM Based o the prevous falure occurrece, the codtoal probablty of the ext falure ca be obtaed from [3]: It s equvalet to F( x + V ) F( V ) F ( x / V ) = (6) R( V ) R( V ) R( x + V ) R( x + V ) F ( x / V ) = = (7) R( V ) R( V ) Substtutg Formula (5) to (7), the ) = t F x V e t 0 } ( / λ ( t; exp{ f ( z dt The correspodg Probablty Desty Fucto (PDF) s: (8)

5 Proportoal Itesty Model cosderg Imperfect Repar for Reparable Systems 67 f ( t / t t 0 t 0 ) = λ ( t )exp( f ( z ))exp( λ ( t)exp( f ( z )) dt) (9) 2.4 PIM uder Power Law Process I ths paper, we assume the orgal testy fucto as NHPP. As metoed before, the NHPP s wdely used modellg reparable systems. NHPP assumes that the successve falures of a system are depedet of each other. I other words, the accumulatve umber of falures has depedet cremets. The most popular of NHPP models are Power Law Process Models ad Cox-Lews model [4]. The Power Law Process Model defes the testy fucto as a power fucto ad the Cox-Lews model defes t as a log-lear fucto. Ths paper uses the Power Law Process model as orgal testy fucto. The defto of the Power Law Process model s as follows [5, 6]: t λ( α α where α s scale parameter, s shape parameter. t ) = ( ) (0) Based o Power Law Process, whe the Kjma I model s used to accommodate the effectveess of repar, the testy fucto λ ( t, z; s descrbed as: q xl + x l = 0 λ( x, z; = ( ) exp( z + 2z2 + 3z b) α α where we assume x 0. 0 = Whe the Kjma II model s used to accommodate the effectveess of repar, the testy fucto λ ( t, z; s: where V + x (, ; ) ( λ x z q ) exp( z 2z2 3z3... b = ) α α V s defed Equato (2) ad we assume V Parameter Estmato ad Iferece Whe the umber of covarat cosdered s large, estmatg ther correspodg parameter s also dffcult. As there are o detcal systems observed, the popular partal lkelhood method caot be appled [7]. I ths secto, we propose a Bayesa method to estmate parameter. 3. Lkelhood Fucto I ths paper, the parameters of terest are tme s α, q, 0 =, t 2 t, the correspodg covarates are z, z2,..., z t,..., testy fucto Formula () s the: () (2), (=,2, ). Assume the falure. The PDF based o the

6 68 Yua Fuqg ad Uday Kumar f α x + qt α ( qt ) ( x + qt )).exp ( α α ( t / t ) = ( ) exp( f ( z )exp( f ( z The correspodg lkelhood fucto s the rewrtte to: = ) (3) L( α,, q,,...) = f ( t / t ) (4) Whe the Kjma I model s used, the correspodg l s: l L = l( ) + α = = l lα + f f ( z ) + ( ) = ( z) = z + 2z2 + f ( z ) + ( ) x + qt l α L + ( = = x + qt qt l( x + qt ) + ( α) = = e e f ( z ) f ( z ) u ( ) α {( qt )... (5) The correspodg Kjma II model s smlar to above Formula (5), we omt the expresso here. 3.2 Bayesa Estmator Whe the umber of parameters s larger, oe ca employ Bayesa ferece to estmate parameters [9]. The Bayesa estmator cosders the parameter as a varable stead of a costat. The Bayesa lkelhood fucto s: L t) = where f / ) s defed the Formula (3). t t ( Usg the observed tme to falure data D: fucto s f ( t / t ) ( (6) L( D,, q, ) = t t,..., t, 2, the jot pror lkelhood α (7) = f ( t / t ) Oe mport step to perform Bayesa ferece s the selecto of the pror dstrbuto for each parameter. We use o-formatve prors to estmate parameters ths paper. Smlarly to the pror used by Hamada et al. [20], we employ a Gamma dstrbuto as the pror dstrbuto for the scale ad shape parameters. For the mperfect repar factor q, we use the stadard uform dstrbuto. The pror for the coeffcet of the covarate s selected as ormal dstrbuto. I summary, the pror dstrbutos are assumed to be: (, ) (, ~ U (0, α ~ G k a k α 2, ~ G k k 2), q ), ~ N ( µ, ). k, k ) k a, k for scale parameter α. G( a 2 s the Gamma dstrbuto wth parameters 2 α α G ( k, k 2) s that for shape parameter wth parameters k, k 2. N ( µ, ) deotes multomal Normal dstrbuto of covarate coeffcet wth mea vector µ ad covarate matrx. du) )) ( qt + x ) }

7 Proportoal Itesty Model cosderg Imperfect Repar for Reparable Systems 69 The correspodg jot posteror dstrbutos are π ( α,, q, D) L( D α,, q, ) π ( α k π ( k, k 2 a, k α 2 ) π ( q 0,) π ( µ, The posteror dstrbuto s a complex compoud. We employ a MCMC method wth a slce sampler to compute the posteror dstrbuto. Oe ca refer [9] for the theoretcal dscusso of the MCMC method. The result of the MCMC estmato wll be seres of traces. The estmated parameter values of terest are derved from these seres of traces. Oe ca refer [20] for the procedure of obtag the mea, varace ad cofdece terval from the seres of traces. 3.3 Model selecto I Bayesa ferece, model selecto s a broad term. I ths paper, the model selecto covers the selecto of a sutable basele fucto ad the removal of usace factors. Before we dscuss the detaled procedure of model selecto, we troduce a method for assessg the performace of the model. The Bayesa Factor s usually used to measure the goodess of models [9]. Yet the Bayesa Factor (BF) s dffcult to be obtaed practcally. Whe the umber of parameters s kow, usually a approxmate of the BF s used, amely the Bayesa Iformato Crtero (BIC). The defto of the BIC s as follows: α (9) BIC = 2 l L( D,,..) + k l where k s the umber of parameters, s the sze of data sets. The models wth a lower BIC are more preferable. I ths paper, we frstly employ the Bayesa MCMC method to estmate the parameters cosderg all covarates ad the compute the cofdece terval of the estmated parameters. All the covarates whose cofdece terval of ts cover zero wll be removed. After that, we re-estmate the parameters for the reduced covarates aga. Ths procedure s performed teratvely utl o covarate cover zero. 4. A Numercal Example I order to demostrate the methodology proposed above, we troduce a example whch has bee dscussed by Ghodrat ad Kumar [2]. The hydraulc brake pump s a crtcal part of the hydraulc loader. It s kow that the followg factors ca fluece ts relablty: the operator skll (OPSK), mateace crew skll (SCSK), hydraulc ol qualty (HOILQ), hydraulc system temperature (STEMP), ad evrometal codtos (ENDUS). Our paper uses the data Table AI from ther paper [2]. Several PIM models were appled to model the testy fucto cosderg covarates. They dffer at ther basele fucto: models based o the NHPP ad the Webull dstrbuto, ad the Kjma I ad Kjma II models, whch are mperfect repar models. The Webull dstrbuto-based model essetally assumed that q = 0 ad hece assumed that the repar effectveess restored the system to a as-good-as-ew state. The NHPP-based model assumed that the repar effectveess restored the system to a sameas-old state. ) ) (8)

8 70 Yua Fuqg ad Uday Kumar The repar hstory was also take to accout the model. The umber of repars expereced was used as a covarate for the PIM, whch s abbrevated as CO. I summary, 5 covarates were cosdered tally. It was assumed that these covarates were depedet of each other. We developed our model based o Equato (4). Thereafter we employ Bayesa method to estmate parameter,,. The pror dstrbuto was assumed to be: ~ G(3,2) ~ N(0,00). α q, α, ~ G(2,2), ~ U (0,) q, Usg MCMC method, after 2000 bur- teratos, 3000 teratos were a statoary state. The mea, stadard devato, ad cofdece terval for the parameters of terest were derved from the traces whch are statoary state. The parameters wth a cofdece terval coverg zero were cosdered sgfcat. After removg all the sgfcat factors, we re-evaluated the parameter. All the cofdece tervals of the remag covarates excluded zero. The results usg dfferet models are tabulated Table. Table : Results usg Several Models Basele Fucto Model Covarates BIC No covarate / 04 NHPP Full covarates Reduced covarates No covarate / 20 Kjma I Full covarates Reduced covarates No covarate / 95.6 Kjma II Full covarates Webull Dstrbuto 2 4 Reduced covarates 54 No covarate / 292 Full covarates Reduced covarates From Table, t shows that the NHPP-based seres of models ad models wthout covarates are ot sutable for the data, as the BICs are all hgh. Therefore, the assumpto of restorato to a same-as-old state s ot reasoable ad some covarates should be corporated to the model. The Kjma I ad the Webull dstrbuto-based models exhbt a smlar performace ths applcato. The reaso behd s that, whe the Kjma I model s used, q s ear zero. Therefore, the effectveess of q ca be cosdered as restorg the system to a as-good as-ew state. The best model s the Kjma II based PIM model wth the covarates 2, 4 correspodg to BIC=54. The detaled results for ths model are tabulated Table 2 where SD s stadard devato.

9 Proportoal Itesty Model cosderg Imperfect Repar for Reparable Systems 7 Table 2: Evaluated parameter values Lower boud Upper boud Parameter Mea SD (0.025) (0.975) Scale Parameterα Shape Parameter Imperfect Repar Factor q OPSK SCSK STEMP I the optmal PIM model, the mea for q s I order to demostrate the effectveess of repar, we plot ts testy fucto agast tme wth q = 0.099, as show Fgure 2, where the testy fucto s for the 4th to the 7th falure. We ca see from ths fgure that repar s sgfcatly effectve as the system s testy fucto has almost bee restored to zero after each repar. Fgure 2: Effectveess of Repar Moreover, the optmal PIM model, the remag sgfcat factors are: OPSK, SCSK ad STEMP, whch are all sgfcat at level To llustrate the effectveess of these factors, we take the factor STEMP as a example. Fgure 3 plots the 95% cofdece terval of the testy fucto, whe STEMP s ad the other covarates are zero. Both the upper boud ad the lower boud are uder the No Covarate curve. It mples that uder the hgher STEMP, the system has hgher relablty tha the ormal STEMP. Addtoally, order to compare the effectveess of all 3 factors o the system relablty, Fgure 4 s plotted. I Fgure 4, testy fucto OPSK supposes the OPSK to be ad the other factors are 0. The other curves for factor STEMP, SCSK follow ths. Ths fgure shows the effectveess of the 3 factors o relablty follows the order: OPSK> STEMP > SCSK. The factor OPSK flueces the system relablty most sgfcatly.

10 72 Yua Fuqg ad Uday Kumar Fgure 3: 95% Cofdece Iterval for Itesty Fucto Fgure 4: Sestvty of Sgfcat Factors CO, whch represets the repar hstory ths example, s sgfcat ths optmal PIM model. Ths does ot mea that the repar hstory ca be eglected. I some other stuatos, cosderg the repar hstory s stll ecessary. Fally, t s ecessary to meto that Lao has developed a mperfect repar model that cosders the cumulatve umber of falures as a covarate [22]. He argues that hs mperfect repar model ca outperform ay other mperfect repar model. Essetally, the mperfect repar PIM model proposed our paper has geeralzed that preseted Lao s paper. I our model, ot oly the cumulatve umber of falures, but also ay factors relevat to falure ca be accommodated as covarates. 5. Coclusos Ths paper has proposed a model that combes the mperfect repar model ad the proportoal testy model. Usg ths proposed model, the effectveess of repar ad covarates are corporated to the model. The paper essetally provdes a framework for accommodatg all possble factors to a model to aalyze ther effectveess. These factors could be the operatg codtos, the evrometal fluctuato, ad the repar or mateace hstory etc. The troducto of a large umber of factors to the model

11 Proportoal Itesty Model cosderg Imperfect Repar for Reparable Systems 73 complcates the estmato of parameters. I cotrast to some PIMs cosderg several detcal systems exstg, where the partal lkelhood estmator ca be used, the paper employs a Bayesa ferece method based o MCMC estmato. Ths parameter estmator does ot requre detcal systems. Oe lmtato of the preset research s the fact that the factors are assumed to be mutually depedet. I practce, especally whe the repar hstory factor s cosdered, some factors ca be hghly correlated. The teracto betwee factors should therefore be cosdered. Aother lmtato of ths paper s the fact that the covarates are assumed to be tme-depedet. Further research wll exted ths proposed model to cosder tmedepedet factors. Refereces [] Crow, L. H. Relablty Aalyss for Complex, Reparable Systems, ARMY MATERIAL SYSTEMS ANALYSIS ACTIVITY A692020, 975. [2] Ascher, H. ad H. Fegold, Reparable Systems Relablty : Modelg, Iferece, Mscoceptos ad ther Causes. M. Dekker, New York, 984. [3] Cox, D. R. Regresso Models ad Lfe-tables, Joural of the Royal Statstcal Socety, 972; 34(2): [4] Kumar, D. ad B. Klefsjo, Proportoal Hazards Model-A Revew, Relablty Egeerg & System Safety, 994; 44(2): [5] Kumar, D., Proportoal Hazards Modelg of Reparable Systems, Qualty ad Relablty Egeerg Iteratoal, 995; (5): [6] Lugtghed, D. Bajevc ad A.K.S. Jard, Modelg Reparable System Relablty wth Explaatory Varables ad Repar ad Mateace Actos, IMA Joural of Maagemet Mathematcs, 2004; 5(2):89-0. [7] Love, C. E. ad R. Guo, Applcato of Webull Proportoal Hazards Modelg to Badas-Old Falure Data, Qualty ad Relablty Egeerg Iteratoal, 99; 7(3): [8] Love, C. E. ad R. Guo, Usg Proportoal Hazard Modelg Plat Mateace, Qualty ad Relablty Egeerg Iteratoal, 99; 7():7-7. [9] Cox, D. R. ad D. Oakes, Aalyss of Survval Data. Chapma ad Hall, New York, 984. [0] Guo, R., H. ASCHER ad E. Love et al., Geeralzed Models of Reparable Systems: A Survey va Stochastc Processes Formalsm, ORON, 2000; 6(2): [] Kjma, M. ad U. Sumta, A Useful Geeralzato of Reewal Theory: Coutg Processes Govered by No-Negatve Markova Icremets, Joural of Appled Probablty, 986; 23():7-88. [2] Percy, D. F. ad K. A. H. Kobbacy, Usg Proportoal-testes Models to Schedule Prevetve Mateace Itervals, Joural of Mathematcs Appled Busess & Idustry, 998; 9(3): [3] Kjma, M., Some Results for Reparable Systems wth Geeral Repar, Joural of Appled Probablty, 989; 26(): [4] Coetzee, J. L., The Role of NHPP Models the Practcal Aalyss of Mateace Falure Data, Relablty Egeerg & System Safety, 997; 56(2): [5] Klefsjo, B. ad U. Kumar, Goodess-of-Ft Tests for the Power-Law Process Based o the Ttt-Plot,IEEE Trasactos o Relablty, 992; 4(4): [6] Rausad, M. ad A. Høylad, System Relablty Theory : Models, Statstcal Methods, ad Applcatos, Wley-Iterscece,New York, [7] Cox, D. R., Partal Lkelhood,Bometrka, 975; 62(2): [8] Kalbflesch, J. D. ad R. L. Pretce, The Statstcal Aalyss of Falure Tme Data. Wley, New York, 980. [9] Gelma, A., Bayesa Data Aalyss, Chapma ad Hall, Lodo, 2004.

12 74 Yua Fuqg ad Uday Kumar [20] Hamada, M.S., A.G.Wlso, C.S.Reese ad H.F. Martz, Bayesa Relablty, Sprger, [2] Ghodrat, B. ad U. Kumar, Relablty ad Operatg Evromet-based Spare Parts Estmato Approach, Joural of Qualty Mateace Egeerg, 2005, (2): [22] Guo, H.R., H. Lao, W. Zhao ad A. Mettas, A New Stochastc Model for Systems uder Geeral Repars, IEEE Trasactos o Relablty, 2007; 56(): Yua Fuqg obtaed hs M.Tech. System Egeerg at Bejg Uversty of Aeroautcs ad Astroautcs, Cha, the year He joed the Dvso of Operato ad Mateace Egeerg, Luleå Uversty of Techology, Swede, September 2007 to work for the degree of Ph.D. Hs area of research deals wth relablty data aalyss ad statstcal learg theory. Uday Kumar obtaed hs B.Tech. Ida durg the year 979. After workg for 6 years Ida mg compaes, he joed the postgraduate programme of Luleå Uversty of Techology, Luleå, Swede, ad obtaed the degree of PhD the feld of Relablty ad Mateace durg 990. Presetly, he s Professor of Operato ad Mateace Egeerg at Luleå Uversty of Techology, Luleå, Swede. Hs research terests are equpmet mateace, equpmet selecto, relablty ad mataablty aalyss, system aalyss, etc. He has publshed more tha 70 papers teratoal jourals ad coferece proceedgs.