A Condition-based Preventive Maintenance Policy for Markov Deteriorating Systems

Transcription

1 nternatonal Journal of erformablty Engneerng, Vol. 2, No. 2, Aprl 2006, pp RAMS Consultants rnte n na A Conton-base reventve Mantenance olcy for Marov Deteroratng Systems. NAGA SRNVASA RAO an V.N. ACHUTHA NAKAN Relablty Engneerng Centre, nan nsttute of Technology, Kharagpur , West Bengal, na. Receve on July 26, 2005) Abstract: A conton base preventve mantenance CBM) polcy s propose for a contnuously operatng evce whose conton eterorates wth tme n servce. The moel ncorporates both eteroraton an ranom falures. Deteroraton s moele as screte states process. The system unergoes ranom nspectons to now the conton, mean tme between nspectons are exponentally strbute. f the observe conton at an nspecton excees the threshol eteroraton level, reventve mantenance M) s performe. Else no acton taes place, contnue to run the system. Each M maes the system τstages ) younger. The propose moels conser an accumulate eteroraton base ncreasng ntensty for the ranom falures. A contnuously ncreasng falure rate for example Webull) s converte nto a stepwse ncreasng falure rate usng star-step approxmaton. An exact recursve algorthm computes the steay-state probabltes of the system. An operatng unt tme hour, ay, wee, etc.) base cost functon s efne usng fferent cost rates for the fferent types of outages. Base on maxmum avalablty or mnmum costs, optmal soluton of the moel s erve. Key Wors: conton base preventve mantenance CBM), eteroraton falure, ranom falure, mnmal repar, star-step falure rate.. ntroucton Optmal preventve mantenance polces are etermne for reparable evces by mnmzng total cost, maxmzng avalablty, or optmzng some other objectve. Deer [], ham an Wang [2], Deer [3], Hongzhou Wang [4], are the recent research reports on mantenance. reventve mantenance on eteroratng evces extens the useful lfe n two ways. The useful lfe s extene ether by reucng the accumulate eteroraton level,.e. reucng the prevously occurre eteroraton or by reucng the eteroraton rate of the evce after performng the M,.e., reucng the future eteroraton. Suprasa an Lelan [5], Sm an Enreny [6] an [7], Dongyan an Trve [8] moele the former case, Canfel [9], ar et al [0] consere the later. Ths paper proposes moels n the former case. Overhauls, replacements of the components etc, are the M actvtes n the former case. Later case nclues tghtenng an ajustments, balancng, lubrcaton, etc. Corresponng author s e-mal: naan@hjl.tgp.ernet.n 75

2 76. Naga Srnvasa Rao an V.N. Achutha Naan Suprasa an Lelan [5] moele the eteroraton process as up or operatng states an one own or fale state. Mean tme between nspectons of the evce s exponentally strbute. Replacement or complete overhaul of the evce taes place f the entfe eteroraton stage of the evce at an nspecton excees a threshol n n s a preventve replacement threshol, <n<=), or n eteroraton fale state. The moel maxmzes the avalablty of the evce by smultaneously optmzng n an nspecton frequency. Sm an Enreny [6] moele the eteroraton process wth operatng states an one eteroraton fale state. They have also consere ranom falures wth constant ntensty nepenent of the eteroraton stage of the evce. eroc preventve mantenances are performe whch maes the evce one eteroraton stage younger. Sm an Enreny [7], Chen an Trve [8], Hossen et al [], Chen et al [2], an Moustafa et al [3] conser multple mantenance actons vz., mnmal an major mantenance. Sm an Enreny [7] s a statstc-base polcy by puttng a lmt on number of mnmal mantenance actvtes performe before a major mantenance or overhaul. [8], [], [2], an [3] are conton base mantenance polcy. No mantenance, mnmal mantenance, an major mantenance are the mantenance alternatves; any one of these may taes place base upon the observe conton of the evce. Mnmal mantenance restore the evce to prevous eteroraton state whereas major mantenance restore to as goo as new state. [8], an [] conser exponentally strbute mean tme between nspectons, [2] conser generally strbute mean tme between nspecton, an [3] conser contnuous nspectons. Borgonovo et al [4] consere stepwse ncreasng falure rate wth M, whch mae the evce one-eteroraton stage younger usng Monte Carlo smulaton methos. All the above stues propose optmal mantenance polces assume that the effectveness of M s ether mnmal one stage younger) or complete replacement/overhaul as goo as new) or both. n general, mantenance mnmal/major) s mperfect an that may not brng the evce to as goo as new state. Therefore, t s reasonable to moel the mantenance as mperfect an that maes the evce to τ eteroraton states younger τ> ). nspre by above observatons, n ths paper we propose conton base preventve mantenance moel for a eteroratng evce conserng: ) non-zero nspecton tmes ) ncrease ntensty for ranom falures, an ) effectveness of M actvty leang to recovery of τeteroraton stages ). For moelng the ncreasng ntensty e.g., Webull hazar rate) of ranom falures we use a star-step approxmaton algorthm [5]. The accuracy of the algorthm epens on the number of steps n our case number of stages n eteroraton process). The accuracy of the star-step approxmaton algorthm ncreases wth the number of states. But ncreasng states of the eteroraton process maes the effect of the M nsgnfcant f the recovery s only one stage. Therefore, by allowng more than one stage recovery an usng star-step approxmaton algorthm our moel gves better results wth esre accuracy by ncreasng the number of states n the eteroraton process). Rest of the paper s organze as follows: Assumptons of the propose moels are gven n secton 2. Steay state balancng equatons an recursve solutons for that s presente n secton 3, an secton 4 respectvely. Secton 5, scusses the optmzaton proceure. Example an scusson are gven n secton 6, an conclusons of the research wor are presente n secton 7.

3 A Conton-base reventve Mantenance olcy for Marov Deteroratng Systems 77 Notaton number of stages of eteroraton before eteroraton falure threshol eteroraton stage for performng preventve mantenances state of evce ) D nex for eteroraton states RF nex for ranom falure states M nex for preventve mantenance states nex for nspecton states DF nex for eteroraton falure state f steay-state probablty that the evce s beng replace followng a eteroraton falure pm steay-state probablty that the evce s out of servce ue to preventve mantenance rf steay-state probablty that the evce s out of servce ue to mnmal repar after a ranom falure /λ f mean tme between successve eteroraton falures, f no mantenance s ntate. l/λ mean tme to ranom falure n th eteroraton state l/λ mean tme between successve nspectons /μ f mean uraton of replacng the evce followng a eteroraton falure l/μ pm mean uraton of preventve mantenance /μ mean uraton of repar after ranom falure n th state /μ mean uraton of nspecton A steay-state avalablty of the evce TC total cost per unt operatng tme C f cost of eteroraton falure C rf cost of ranom falure C pm cost of performng a M actvty C cost of performng an nspecton C t cost of own tme per unt tme CT cycle tme tme between successve eteroraton replacements) TC f total cost of eteroraton falures n a CT TC rf total cost of ranom falures n a CT TC pm total cost for Ms n a CT TC total cost for nspectons n a CT total cost of own tme n a CT TC t Nomenclature The categores n each type are mutually exclusve an exhaustve. a. Falure type. Deteroraton: A process where the mportant parameters of a evce graually worsen. f left unattene, the process leas to eteroraton falure.. Ranom falure: These falures occur nstantaneously n any eteroraton stage an generally unprectable, so that such a falure cannot be prevente an nevtablty shoul be replace by a correctve acton.

4 78. Naga Srnvasa Rao an V.N. Achutha Naan b. Restoraton type. reventve mantenance or smply Mantenance: A restoraton wheren an unfale evce, at pre-selecte but not necessarly entcal) ntervals, has ts accumulate eteroraton reuce or elmnate.. Repar: A restoraton wheren a fale evce s returne to a worng conton. c. Mantenance & repar types. Mnmal mantenance: Mantenance of lmte effort an effect. For eteroraton moele as screte stages, mnmal mantenance restores the evce to the prevous eteroraton stage or to eteroraton stage, whchever s worse.. Major mantenance: Mantenance wheren a evce s restore to as goo as new.. General mantenance: t restores the evce to the prevous τeteroraton v. stages or to eteroraton stage, whchever s worse. Mnmal repar: Repar wheren the evce s returne to the operable state t was n just before falure. v. Replacement/ overhaul: Repar wheren the evce s returne to as goo as new. 2. Assumptons an the Moel 2. Devce. The evce s contnuously operatng except when remove for mantenance or repar. 2. Falure types: eteroraton an ranom, an t assumes that the falure states of the evce can be scovere wthout any nspecton. 3. The evce has a eteroraton falure mmeately followng the completon of stages of eteroraton. 4. Ranom falures have the ncreasng rate or ntensty wth ncreasng the accumulaton of eteroraton. 5. The evce s reparable. The falure type s self-announcng. The mean tme to repar epens on the falure type. 2.2 System System conssts of the evce an mantenance actvtes.. The uraton of each eteroraton stage has an exponental strbuton wth rate λ. 2. Followng eteroraton falure, the evce s overhaule or replace,.e., restore to as goo as new. The overhaul uraton s exponentally strbute. 3. Ranom falure occurs at an ncreasng rate or ntensty, epenng on the eteroraton stage of the evce. Mnmal repar s performe. The mnmal repar uraton s exponentally strbute. 4. erocally the evce s remove from operaton for preventve mantenance, whch maes the evce τeteroraton states younger.

5 A Conton-base reventve Mantenance olcy for Marov Deteroratng Systems The tme between successve mantenances s exponentally strbute. 6. reventve mantenance uratons are exponentally strbute. 7. There are no transtons between the falure & mantenance states. 2.3 The Moel Contnuous tme Marov chan CTMC) moel for general conton preventve mantenance that maes the evce to τstages younger s shown n Fg.. States D to D are the operatng states n ncreasng orer of eteroraton. D s the best operatng state an D s the worst. D represents the state n whch the evce s n th eteroraton stage an s n worng conton. represents the state n whch the evce s n th eteroraton stage an uner nspecton. M s the state where the evce s n th eteroraton stage an s uner preventve mantenance. RF s the state where the evce s n th eteroraton stage an s uner repar after a ranom falure. State DF s the eteroraton falure state. The evce s nspecte after a ranom pero that s exponentally strbute wth mean /λ. Fg. Marov State Transton Dagram for the ropose CBM olcy By ths moel, the evce experences no mantenance when the eteroraton stage s etermne by an nspecton to be <, experences a preventve mantenance f the evce s foun to be n eteroraton stage such that, by whch the evce s restore to -τ eteroraton stage wth a mean uraton of /μ pm. When the evce s n ranom falure state n eteroraton stage, mnmal repar taes place that brngs the evce nto operatng conton, an just before falure eteroraton stage. When the evce s n eteroraton falure state DF, complete replacement s carre out to brng the evce to state D, wth a mean uraton /μ f. 3. Steay-state Balancng Equatons Usng the Marov approach assocate wth Fg., we obtan the steay state balancng equaton [6] to represent the steay-state contons of the process n fgure. Ther soluton yels steay state probabltes.

6 80. Naga Srnvasa Rao an V.N. Achutha Naan Depenng upon the value of τ,, an the followng fve cases are exst: Case : <=τ+) Case 2: >τ+), <=τ+) an -tow< ) Case 3: >τ+), <=τ+) an -tow>= ) Case 4: >τ+), >τ+) an -tow< ) Case 5: >τ+), >τ+) an -tow>= ) Steay state balancng equatons for the above cases are gven below. Case : ) rf f f.... ) 0 )... 0, 2,..., 2) rf.. 0,,..., 3) rf Case 2: ), ) an ) rf.. f. f. ) 0 rf.... 0, 2,..., 5)... 0,,..., 6) rf.. 0,..., 7) rf Case 3: ), ) an ) rf.. f. f. ) , 2,..., 9) rf. rf.. 0,,..., 0).. 0,..., ) rf Case 4: ), ) an ) rf f f )... 0, 2,..., 3) rf.... 0,,..., 4) rf. rf.. 0. rf. 0,,..., 5),,..., 6) 4) 8)

7 A Conton-base reventve Mantenance olcy for Marov Deteroratng Systems 8 Case 5: ), ) an ) rf f f )... 0, 2,..., 8) rf. rf... 0,,..., 9)... 0,,..., rf. rf. 20) 0,,..., 2) The followng equatons are common to all fve cases:.. 0,,2,..., 22).. 0,,2,..., 23) rf pm pm.. 0,2,..., 24) ) f f 26) 4. Recursve Soluton pm rf f Followng recursve soluton s evelope for solvng the balancng equatons erve n secton 3, whch gve steay state probabltes of fferent eteroraton an mantenance states. Case : ) f 27) f,,..., 28),,..., 2 29) Case 2: ), ) an ) f 30) f,,..., 3),,..., 32)

8 82. Naga Srnvasa Rao an V.N. Achutha Naan Case 3: ), ) an ),,..., 2 33) f 34) f,,..., 35),,..., 36) Case 4: ), ) an ),,..., 2 37) f 38) f,,..., 39),,..., 40),,..., 4),,..., 2 42) Case 5: ), ) an ) f 43) f,,..., 44),,..., 45),,..., 46)

9 A Conton-base reventve Mantenance olcy for Marov Deteroratng Systems 83,,..., 2 47) An the common equatons for all the cases are 5. Optmzaton. / ),,2,..., 48) pm pm / ),,2,..., 49) rf / ),2,..., 50) The optmal soluton can be obtane by etermnng the values of an λ, whch maxmzes the avalablty of the evce, or the combne costs, assocate wth the falure an mantenance outages. For example, the optmal value of λ for any preventve mantenance threshol, can be etermne such that A s maxmze. A n most practcal cases, the mnmzaton of the total costs nvolve rather than maxmzaton of the avalablty s of nterest. The costs nclue the mantenance, repar, an out-of-servce costs. A smple an more approprate moel for the cost analyss s propose. Assgn costs C, C pm, C rf, C f, C t ) for the unt tmes of the varous outages nspectons, preventve mantenances, repars after ranom an eteroraton falures, an own tme), an respectve cost are calculate for a eteroraton replacement cycle CT). CT s the expecte tme between successve eteroraton replacements. From these costs, cost per operatng hour or cost per unt operatng tme s calculate for ecson-mang purpose. f CT s the expecte tme between successve eteroraton replacements, then for a CT tme there s one eteroraton replacement. Therefore, 5. Cost Moel CT * f f f f 5) / CT / * ) 52) Expecte costs per CT for eteroraton falures, preventve mantenance, ranom falures, an own tme are calculate as follows:. Expecte cost of eteroraton falure urng CT For a eteroraton falure cycle CT) there exsts only one expecte replacement ue to eteroraton falure. Therefore, E TC f ) C f 53). Expecte cost for performng nspectons urng CT E TC ) ).. C CT 54).. Expecte cost for performng M actvtes urng CT

10 84. Naga Srnvasa Rao an V.N. Achutha Naan v. E TC ) ).. C CT 55) pm m pm pm. Expecte cost for ranom falures urng CT E TC ) * )* C CT 56) rf rf rf * v. Expecte cost of own tme urng CT E TCt ) pm rf f ) * Ct * CT Usng above to v total cost expressons, the expecte cost rate per unt operatng tme s expresse as: E TC ) E TC ) E TC ) E TC ) E TC ) / A * ) E TC) CT 58) 5.2 Optmzaton roceure f pm. Depenng upon the accuracy requre an/or eteroraton replacement rate choose an approprate unt for.. a. Number of eteroraton states ), b. Tme between nspectons /λ ). Set equal to one. Set /λ equal to one. v. Calculate fferent eteroraton state probabltes usng equatons 27) to 50) v. Calculate avalablty Aλ ) at tme between nspectons /λ ). v. Usng equatons 53) to 58) fn the expecte total costs for varous cost components, an cost per operatng hour. v. ncrease /λ ) by one, an repeat the calculatons from step v, untl /λ ) reachng maxmum lmt /λ f ). v. Select the tme between nspecton nterval /λ ) that maxmzes the avalablty or mnmzes the cost per operatng hour whchever be the crtera. x. ncrease the preventve mantenance threshol by one, an repeat the calculatons from step, untl reachng maxmum lmt ). x. Select an optmum mantenance polcy ), an nspecton nterval / λ ) from all selecte polces n step v. 6. Examples an Dscusson Let λ = 0.0/wee, μ pm = 0.0/wee, μ f = 0.5/wee, μ = 00/wee, μ rf = /wee, ranom falures of the evce has Webull ntensty wth scale parameter η= 00, shape parameter β= 3, the cost ata n thousans as C pm =20, C rf =0, C f = 00, an C t =. Tae = 50 an τ= 20. Avalablty an total cost of the system s calculate at fferent tme between rf t 57)

11 A Conton-base reventve Mantenance olcy for Marov Deteroratng Systems 85 nspectons MTB e, /λ ), an fferent preventve mantenance threshols ). Varaton of avalablty, an total cost per operatng wee as a functon of MTB are plotte n Fg. 2 an Fg. 3 respectvely for = ). From these two fgures the maxmum avalablty an mnmum total costs are foun an shown n Table. Varaton of maxmum avalablty an mnmum total cost per operatng wee as a functon of preventve mantenance threshol ) are plotte n Fg. 4 an Fg. 5 respectvely. The optmal mean tme between nspectons OMTB) an preventve mantenance threshol ) are shown n Table. Fg. 6, Fg. 7 an Table 2 shows a comparson of conton base an tme base M polces. From these results t s clear that conton base M polces gve better results compare wth tme base M polces Avalablty MTB Fg. 2: Varaton of Avalablty wth ncreasng MTB for = ) 2.6 Total cost MTB Fg. 3: Varaton of Total Cost wth ncreasng MTB for = )

12 86. Naga Srnvasa Rao an V.N. Achutha Naan Avalablty maxmzaton Total costs mnmzaton Avalablty reventve mantenance thershol ) Fg. 4: Varaton of Optmum Avalablty wth ncreasng Table : Optmal CBM olcy when = Objectve MTB n wees) Avalablty Total cost/wee Avalablty maxmzaton Total cost mnmzaton Table 2: Optmal CBM an Tme base reventve Mantenance TBM) olces for the Example Data Mantenance polcy CBM polcy TBM polcy Optmzaton Crtera Avalablty Maxmzaton Total cost mnmzaton Avalablty Maxmzaton Total cost mnmzaton Avalablty Total costs Rs/wee) OMTB n wee)

13 A Conton-base reventve Mantenance olcy for Marov Deteroratng Systems 87.4 Avalablty maxmzaton Total costs mnmzaton.2 Total cost reventve mantenance thershol ) Fg. 5: Varaton of Mnmum Total Cost wth ncreasng Avalablty maxmzaton Total cost mnmzaton Avalablty reventve mantenance thershol ) Fg. 6: Comparson of TBM polcy wth CBM polcy whle Maxmzng Avalablty 0.9 Avalablty maxmzaton Total cost mnmzaton 0.85 Total cost reventve mantenance thershol ) Fg. 7: Comparson of TBM polcy wth CBM olcy whle Mnmzng the Total cost

14 88. Naga Srnvasa Rao an V.N. Achutha Naan 7. Conclusons A moel s propose for the applcaton of conton base preventve mantenance polcy for Marov eteroratng system. The propose moel ncorporate tme varyng falure rate for ranom falures. The propose moel allows more then one-eteroraton stage recovery by performng M. For a set of gven ata, the propose algorthms smultaneously optmze the mean tme between nspectons an preventve mantenance threshol ). The propose moel gves a mantenance polcy ether by maxmzng the avalablty or mnmzng the total cost whchever be the objectve. Relatons between fferent parameters are stue for maxmzng avalablty. References []. Deer, R., Applcatons of Mantenance Optmzaton Moels: A Revew an Analyss, Relablty Engneerng an Systems Safety, Vol. 5, No. 3, pp , 996. [2]. ham, H., an H. Wang, mperfect Mantenance, European Journal of Operatons Research, Vol. 94, pp , 996. [3]. Deer, R., R. E. Wleman, an F. A. Van Der Duyn Schouten, A Revew of Multcomponent Mantenance Moels wth Economc Depenence, Mathematcal Methos n Operatons Research, Vol. 45, No. 3, pp , 997. [4]. Wang, H., A Survey of Mantenance olces of Deteroratng Systems, European Journal of Operatons Research, Vol. 39, pp , [5]. Suprasa, A. V., an M. Lelan, Optmal Desgn of a Conton-base Mantenance Moel, Relablty, Avalablty an Mantanablty Symposum RAMS), pp , [6]. Sm, S. H., an J. Enreny, Optmal reventve Mantenance wth Repar, EEE Transacton on Relablty, Vol. 37, No., pp , 988. [7]. Sm, S. H., an J. Enreny, A Falure-repar Moel wth Mnmal an Major Mantenance, EEE Transactons on Relablty, Vol. 42, No., pp , 993. [8]. Chen, D., an K. S. Trve, Close-form Analytcal Results for Conton Base Mantenance, Relablty Engneerng an Systems Safety, Vol. 76, pp. 43 5, [9]. Canfel, R. V., Cost Optmzaton of eroc reventve Mantenance, EEE Transacton on Relablty, Vol. 35, No., pp. 78 8, 986. [0]. ar, D. H., G. M. Jung, an J. K. Yum, Cost Mnmzaton for eroc Mantenance olcy of a System Subjecte to Slow Degraaton, Relablty Engneerng an Systems Safety, Vol. 68, pp. 05 2, []. Hossen, M. M., R. M. Kerr, an R. B. Ranall, An nspecton Moel wth Mnmal an Major Mantenance for a System wth Deteroraton an osson Falures, EEE Transactons on Relablty, Vol. 49, No., pp , [2]. Chen, D., Y. Cao, an K. S. Trve, reventve Mantenance of Mult-state System wth hase-type Falure Tme Dstrbuton an Non-zero nspecton Tme, nternatonal Journal of Relablty, Qualty an Safety Engneerng, Vol. 0, No. 3, pp , [3]. Moustafa, M. S., E. Y. A Masou an S. Sae, Optmal Major an Mnmal Mantenance olces for Deteroratng Systems, Relablty Engneerng an System Safety, Vol. 83, pp , 2004.

15 A Conton-base reventve Mantenance olcy for Marov Deteroratng Systems 89 [4]. Borgonovo, E., M. Marseguerra, an E. Zo, A Monte Carlo Methoologcal Approach to lant Avalablty Moelng wth Mantenance, Agng, an Obsolescence, Relablty Engneerng an Systems Safety, Vol. 67, pp. 6-73, [5]. Sun, H., an J. J. Han, nstantaneous Avalablty an nterval Avalablty for Systems wth Tme-varyng Falure Rate: Star-step Approxmaton, acfc Rm nternatonal Symposum on Depenable Computng 200, Seoul, Korea), pp , 200. [6]. Msra, K. B., Relablty Analyss an recton: A Methoology Orente Treatment, Elsever Scence ublshers, Amsteram, N. Srnvasa Rao s a senor research scholar of the Relablty Engneerng Center, nan nsttute of Technology Kharagpur, an s worng towars hs h.d. egree. He s also a faculty wth the Department of Mechancal Engneerng of the Nagarjuna Unversty, A.., na. He has taen hs B. Tech. egree n Mechancal Engneerng, an Master s egree n nustral Engneerng. Hs areas of nterest nclue Relablty Engneerng, Mantenance Management an nustral Engneerng. V.N.A. Naan s an Assocate rofessor n the Relablty Engneerng Center of the nan nsttute of Technology, Kharagpur, na. He too hs Masters an Doctorate egrees n nustral Engneerng & Management wth specalzaton n Qualty & Relablty Engneerng from nan nsttute of Technology, Kharagpur, na. He has wore wth Unon Carbe na Lmte at Calcutta, nan nsttute of Management, Ahmeaba, an nan Space Research Organzaton, Ahmeaba. He has vste the Chnese Unversty of Hong Kong for ong postoctoral research. He has publshe hs research papers n many nternatonal journals, an presente papers at several conferences. He s also actvely nvolve n nustral consultancy an sponsore projects. Hs areas of nterest are, Qualty an Relablty Engneerng, Mantenance Engneerng an Management, Spare parts an Supply Chan Management, an Tolerance esgn an Analyss.