Multiple Periodic Preventive Maintenance for Used Equipment under Lease

Transcription

1 Mulle Perodc Prevenve Manenance or Used Equmen under ease Paarasaya Boonyaha, Jarumon Jauronnaee, Member, IAENG Absrac Ugradng acon revenve manenance are alernaves o reduce he used equmen alures rae whch have more dsruons han he new. The omal manenance olcy wll eecve o he mos alure decrease. Ths research s o deermne he omal PM acons ha mnmze he oal manenance cos or leased equmen when we consder he enales or equmen alures rears me over lm. The ormulaed models are combned rom he advanages o sequenal PM erodc PM olces. Assumon o alures orm s aled by Nonhomogeneous Posson rocess (NHPP) alures dsrbuon s arased Webull. The omal soluon s acheved rom he omal ugrade level wo merames aroach whch s obaned he omal o me nervals o carry ou PM he omal level o PM acons. Index Terms Perodc revenve manenance, Relably, ease, Used equmen. I. INTRODUTION Equmens under leasng s he one sraegy o busness managemen he mos moran o leasng busness s a relably. The used equmen should be ocused because o alure rae s hgher han he new one less relably. The ugradng acon suable PM are necessary o decrease alure rae, moreover hey can reduce he oal manenance cos. Mul erodc revenve manenance olcy or he new equmen s combnng he advanages o sequenal erodc PM olcy under lease condon whch s more lexble han mlemenaon [7. Ths olcy s aly rom sequenal PM olcy or leased equmen whch a un s revenvely mananed a unequal me nervals. Usually, he me nervals become shorer shorer as me asses, consderng ha mos uns need more requen manenance wh ncreased ages [5 erodc PM olcy or leased equmen, unle he sequenal PM olcy, whch a un s revenvely mananed a xed me nervals o T, =,,..., over he lease erod. Ths mles ha he mlemenaon o erodc PM olcy s more convenence han he sequenal PM olcy because he me nervals beween successve PM acons are consan [. The lease erod s dvded n o wo sages whch s he rs he second lease erod. Each erod have equal me nervals or PM acons, bu he requency o each PM acons wll be deren n he rs he second lease erod e.g., he rs Paarasaya Boonyaha s wh he Thammasa Unversy, Bango, Thal (corresondng auhor o rovde hone: ; e-mal: boonyaha@homal.com). Jarumon Jauronnaee, was wh Thammasa Unversy, Bango, Thal. She s now wh he Dearmen o Indusral Engneerng, Thammasa Unversy, Thal (e-mal: arumon@engr.u.ac.h). lease erod, he PM acons are carred ou a erodc mes o T, =,,..., n he second lease erod, he PM acons are carred ou a erodc mes T /, =,,...,. Frequency o PM acons n he second erod s hgher han he rs because o ncreasng alure rae corresondngly. The searaed lease mng wll be consdered aer he equmens have been used A years ugradng wll be carred ou beore lease. Any nervenen alures over he lease erod, we wll assume o be reced hrough mnmal rears. II. MODE FORMUATION We use he ollowng noaon: F ( ) = alure dsrbuon uncon ( ) ( ) = alure densy uncon assocaed wh F ( ) r = alure rae [hazard uncon assocaed wh F ( ) λ ( ) = alure nensy uncon wh no PM [= r ( ) λ ( ) = alure nensy uncon wh PM acons Λ ( ) = cumulave alure nensy uncon wh no PM = λ ( x) dx Λ ( ) = cumulave alure nensy uncon wh PM acons = λ ( x) dx N ( ) = number o alures over [, Y = me o rear G ( y) = rear-me dsrbuon uncon g ( y) = rear-me densy uncon [= dg ( y) / dy = lease erod = s lease erod = nd lease erod T = erod o me nsan o carry ou PM = number o PM acons over he s lease erod = number o PM acons over he nd lease erod = me nsan or erod h PM acon over he s lease

2 = me nsan or erod h PM acon over he nd lease = reducon n nensy uncon due o acon over he s lease erod = reducon n nensy uncon due o h PM h PM acon over he nd lease erod A = age o equmen beore lease x = ugrade level beore lease (ercenage o A ) = ugrade cos u ( ) = cos o PM acon resulng n a reducon nensy uncon over he s lease erod ( ) n = cos o PM acon resulng n a reducon nensy uncon over he nd lease erod = oal cos o PM acons τ n = average cos o M acon o recy alure = oal cos o M acons = rear me lm [arameer o lease conrac n = enaly cos er un me rear no comleed whn τ [Penaly- = enaly cos er alure () > φ φ = oal cos due o Penaly- = oal cos due o Penaly- A. ease onac N [Penaly- For lease erod, aer A years, ha comose o +, when have a requency o PM acon more han, ha bene o relably ncreasng. The equmen s leased or a erod wh wo yes o enaly resulng rom alures. Penaly-: The lessor ncurs a enaly he me o rear alure exceeds τ. e Y denoe he me o rear, hen here s no enaly τ ( Y τ ) Y >τ [. Y a enaly Penaly-: The lessor ncurs a enaly cos alure ha occurs over he lease erod [. B. Modelng Falures PM Acons n or each Equmen alures are reced hrough mnmal rears he rear mes are small relave o he mean me beween alures. We assume ha equmen had been used beore new lease. Furhermore, a he rmal lease no PM needed a early hase o equmen because o low alure. As a resuls, s no worh o PM, however he lessor has rovded PM when ever equmen aled. The rs alures mng s a dsrbuon uncon F () alures rae r ( ) s an ncreasng uncon o me. For he rs erod, he lessor carres ou erodc PM wh a erod o T T / over he second lease erod o comensae ncreasng o alure rae. The me nsans o PM acons are gven by = T, =,,..., or he rs lease erod = + T /, =,,..., or he second lease erod. Each PM acon resuls n a reducon n he nensy uncon. The reducons resulng rom he h h PM n he rs erod s gven by or PM n he second erod [7. The rs erod, he alures over he lease erod or used equmen, no PM acon beore lease ha occurs accordng o he nensy uncon gven by () = λ ( A + x) λ or + () Where = = x hs reer o no ugrade case = A x. = = ( A ) s consraned as ollows = λ x ( ) A + x = λ or () The second erod, () = λ ( A + x) λ (3) or + = = = A + where x s consraned as ollows ( A + x) = = = λ (4) or ; Fg. Plo o alure nensy uncon o boh erod o. os o essor () os o M Acons e N ( ) s a number o alures over [, s a mean cos o rear. The cos o rearng alures s gven by = N( (5) () os o PM Acons The cos o PM acon deends on he reducon resul n he nensy PM cos uncon. We model hs hrough a xed cos he varable cos s gven by

3 ( ) = a + b, =,,..., (6) or he rs lease erod ( ) = a + b, =,,..., (7) For he second erod wh a > b. Hence, he oal cos o PM acons s gven by (, ) = ( a + b ) + ( a + b ) (8) = = where =,,..., =,,..., () Ugrade oss The ugrade cos u (x) s an ncreasng uncon o x gven by ϕ ( x) ( ) /( A u x = ω x e ) (9) Where > ϕ > whenω s a scale arameer ω ϕ s a shae arameer o u (x) (x) Penaly oss Penaly- resuls rom alure o comlee a rear whn a seced me τ. e Y (a rom varable rom a dsrbuon G ( y) ) denoe he me o recy he h alure, N(. Then, he oal Penaly- cos ncurred s gven by N ( φ, Y, τ ) = max[, Y τ () = Penaly- resuls whenever here s any alure over he lease erod, he lessor ncurs he enaly- coss hs s gven by φ ) = max[, N( ) () { } n III. MODE ANAYSIS A. Execed number o alurwes The equmen alures wh no PM acons occur accordng o a Non homogeneous Posson rocess (NHPP) wh nensy uncon λ ( ) = r( ) where r() s alure rae [hazard uncon assocaed wh he dsrbuon F() we dene [3 Λ( = λ ( ) d () The execed number o alures over he lease erod, wh no PM acon, s gven by E[ N( = Λ( A + x) Λ( A x) (3) The execed number o alures over he lease erod, wh PM acon, s gven by E[ N( ( A + x) Λ ( A x) = Λ ( ) ( ) = = (4) where =,,..., =,,..., B. Execed os () Execed M cos From (5) (4), he oal execed cos o M acons s gven by E( ) = E[ N( (5) he used equmen consderaon n (5) resul n E( [ Λ ( A + x) Λ ( A x) x A + x ) = ( A + ) ( ) (6) = = () Execed Penaly- cos From () (4) he oal execed Penaly- cos s gven by E[ φ, Y, τ ) = E[ N( ( y τ ) g( y τ ) Usng negrang by ars on (6) resuls n dy E[ φ, Y, τ ) = E[ N( ( G( y)) dy τ (7) (8) () Execed Penaly- cos From () (4) he oal execed Penaly- cos s gven by E[ φ ) = ne[ N( (9) ombnng all o coss, execed M cos, oal PM cos, ugrade cos, execed Penaly- cos execed Penaly- cos, yelds he oal execed cos o he lessor gven by ) = N( + ( a + b ) + ( a + b ) = = ϕ ( A x) + ωx /( e ) + N( ( G( y)) dy τ + n N( () =,,..., where =,,..., We can dene = + τ ( G( y)) dy + Then, () can be rewren as ) = Λ ( A+ x) Λ ( A x) ( ) ( ) A+ x A+ x = = + ωx /( e ϕ ( A x) ) + = n ( a + b ) + =,,..., = where =,,..., ( a + b ) ()

4 . Omzaon The omal arameers o he PM olcy are arameer values ha yeld a mnmum or ). We oban he omal values usng a wo rocess. In Sae one we aly. In Sae we derenal calculus mehod o oban ( T ) oban T by usng one-dmensonal mnmzaon mehod wh he erave rocedure. () Sae Fx, oban rom = T, < < <... < < b / (3) < < <... < < b (4) where, / < < < b / (5) As a resul, ) s a lnear uncon o consraned as ndcaed n (),(4),(3)-(5). Thereore, he omal values are he end ons o he consran nervals. Ths yelds + λ ( A x) < b / = or = b / > (6) where = A x = λ( A x) λ ( A + x) < b / = = = b / or > where = A + x = Ths mles ha he omal PM acon a (7) = T, =,,..., or = + T /, =,,..., s o reduce alure nensy by he maxmum amoun when < b / or < b / no o carry ou any PM when () Sae We oban b / or T b /., he omal T, by mnmzng T ) usng ( ) obaned rom Sage. One can oban T by usng one-dmensonal mnmzaon mehod wh he erave rocedure accordng o he algorhm gven by. Se : Gven =. Se : Evaluae sde consrans o T rom / + < T. / Se 3: =,,..., = + T /, =,,...,. Fnd T over he nerval / + < T / wh As a resul, ) s only a uncon o, rom () one dmensonal mehod se sze hen comue (4), he s consraned. Deermne he exreme on rom = / T = ( ) / T o ) by deermnng he rs aral dervaves o Se 4: omue rom ) corresondng o as below = T, =,,..., J ( T, ) / =... = J ( T, ) / = J ( T, ) /... = ( J, ) / = + T /, =,,...,. () Se 5: hen we have he consrans o as ollow Evaluae by lacng n (6) (7) resecvely. Se 6: Evaluae ) rom () Se 7: Se new, reea Se onwards unl + = max where max = Λ ( ) / a, hen go o Se 8. Se 8: Search or T whch yelds he smalles values or ). Usng hs, he omal PM acons are gven by T = ( ) he mnmum execed cos o he lessor gven by J ( T, ( T )). IV. NUMERIA EXAMPE We assume ha he alure dsrbuon or he equmen s gven by he wo-arameer Webull dsrbuon [. As a resul, β λ ( )( ) = β / α / α (8) wh scale arameer α > shae arameer β > (mlyng an ncreasng alure rae). Accordng o [3 we can assume α = because o he scale arameer α has no nluence o he model hey are wo or hree arameer Webul model. The rear me, Y, s a rom varable wh dsrbuon uncon G (y). We assume ha G (y) s also a wo-arameer Webull dsrbuon uncon gven by G( y) = ex[ ( y / ϕ ) m ; y (9) wh he scale arameer ϕ < he shae arameer m < (mlyng a decreasng rear rae). We consder he ollowng nomnal values or he model arameers

5 = 5 (years), = (years), = $, = 3$, = $, ( ω) =, u ( ϕ) =. n, a = $, b = 5$, τ = (days), β = 3, α = m =. 5, ϕ =. 5 A. The Omal Parameers or he PM Polcy As a resul, T =. 755 years ha means me nerval o PM acon s 3.3 monhs or he rs erod.66 monhs or he second erod we have, 7, u = =, = 7 ha means under leasng me, PM acons me s 7 he omal o ugrade level s 87% ha gve he mnmum oal manenance cos $9,88.. The omal arameers o mul erodc PM olcy wh ugrade cos are gven n Table. Table. The omal arameers or he PM olcy wh ugrade cos ( A = 5) A =5 x % T J ($) , , , , , ,696.7 Table. The omal arameers or he PM olcy wh ugrade cos (comare A wh he several age) Ugrade A x % T J , , , , , , ,4.98 Toal cos 3,. 5,.,. 5,.,. 5,. Toal cos age o equmen beore lease comarson Age o equmen beore lease Fg. Toal cos (J ) age o equmen beore lease (A) comarson Toal cos 4,.,. Toal cos ugrade level comars on We can comare he urher age o used equmen beween usng mul PM olcy wh ugrade mul PM olcy whou ugrade. The resuls are shown n he Table.3, ugradng acon have eeced o decrease oal manenance cos. Moreover he older used equmen, we have been releved oo much manenance cos wh mul PM olcy wh ugrade as shown n he Table.3,. 8,. 6,. 4,., Ugrade level Fg. Toal cos (J ) ugrade level (x%) comarson As Fg. we can noce ha he hghes ugrade level have no gven he mnmum manenance cos bu he ugrade level wll drecly varable wh age o he equmen as he resul n Table. Table.3 The omal arameers or he PM olcy wh ugrade cos whou ugrade cos Ugrade No ugrade A x % J J Δ J ($) 63 7, ,7.9 5, , ,77.7 4, , ,7. 56, , ,79.75, , , , , ,986. 3, , , ,369.5

6 V. ONUSION The roosed o hs aer s o research he mulle erodc revenve manenance olcy whch or used equmens under lease. Ugrade s he acon whch decrease alure rae oal manenance cos. Avalable mul erodc PM s more advanage han erodc PM cause o corresond wh degeneraon o he used equmen whle sequenal PM s more delcae o ado han mul erodc PM whch dvde me o wo nervals. Resuls o hs aer can suor leaser o carres ou he used equmen or lease wh he mnmum cos nvoe ros as long as he oal mnmum cos sll hgher han nves he new one. [3 S. Sheu,. Kuo, T. Naagawa, Exended omal age relacemen olcy wh mnmal rear, RAIRO: Recherche Oeraonalle, 7(3), 993, [4 H. Wang, A survey o manenance olces o deeroraon sysems, Euroean ournal o oeraonal research, 39,, REFERENES [ R. E. Barlow,.. Huner, Omum revenve manenance olces, Oeraons Research, 8, 96, 9-. [ R. E. Barlow, F. Preshan, Mahemacal Theory o Relably, New Yor: Wley, 965. [3 W. R. Blshche, D. N. P. Murhy, Relably, modelng, redcon omzaon, New Yor: John Wley & Sons,. [4. Y. heng, M.. hen, The erodc revenve manenance olcy or deerorang sysems by usng mrovemen acor model, Inernaonal ournal o aled scence engneerng, (), 3, 4-. [5 J. Jauronnaee, D. N. P. Murhy, R. Boondsulcho, Omal recenve manenance o leased equmen wh correcve mnmal rears, Euroean ournal o oeraonal research, 74, 6, -5. [6 D. n, M. J. Zuo, R.. M. Yam, General sequenal merec revenve manenance models, Inernaonal ournal o relably, qualy saey engneerng, 7(3),, [7 X. u, V. Mars, A. K. S. Jardne, A relacemen model wh overhauls rears, Naval research ogscs, 4, 995, [8 J. Medh, Socasc rocesses, New Delh: Wley Easern, 98. [9 H. Mormura, On some revenve manenance olces or IFR, Journal o he oeraonal research socey o aan, (3), [ H. Mormura, H. Maabe, A new olcy or revenve manenance, Journal o he oeraonal research socey o aan, 5, 963a, -4. [ H. Mormura, H. Maabe, On some revenve manenance olces, Journal o he oeraonal research socey o aan, 6, 963b, [ D. N. P. Murhy, D. G. Nguyen, Omal age olcy wh merec revenve manenance, IEEE Transacons Relably, 3, 98, 8-8. [3 T. Naagawa, Imerec revenve manenance, IEEE Transacons on Relably, 8(5), 979, 4. [4 T. Naagawa, Sequenal merec revenve manenance olces, IEEE Transacons on Relably, 37(3), 988, [5 T. Naagawa, S. Osa, The omum rear lm relacemen olces, Oeraonal research Quarerly, 5, 974, [6 P. D. T. O onner, N. Newon, R. Bromley, Praccal relably engneerng, Wes Sussex: John Wley & Sons,. [7 T. Nyamosoh, J. Pongech, Mulle-erodc revenve manenance olcy or leased equmen consderng wo yes o enaly The 37h Inernaonal onerence On omuers Idusral Engneerng, 7 [8 T. Nyamosoh, J. Pongech, Omal Mulle-Perodc Prevenve Manenance Polcy or eased Equmen Inernaonal DSI / Asa Pacc DSI7, 7 [onlne Avalable: h://as.nda.ac.h/resource/ascon_resource/ads7/aers/fnal_ 9.d [9 H. Pham, H. Wang, Imerec manenance, Euroean ournal o oeraon research, 94, 996, [ J. Pongech, D. N. P. Murhy, Omal erodc revenve manenance olcy or leased equmen, Relably engneerng sysem saey, 5, -6. [ J. Pongech, D. N. P. Murhy, R. Boondsulchoc, Manenance sraeges or used equmen under lease, Journal o qualy n manenance engneerng, (), 6, [ S. S. Rao, Omzaon heory alcaons, New Delh: Wley Easern, 978.