MODEL FOR OPTIMAL BLOCK REPLACEMENT DECISION OF AIR CONDITIONERS USING FIRST ORDER MARKOV CHAINS WITH & WITHOUT CONSIDERING INFLATION

Transcription

1 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) MODEL FOR OTIMAL BLOCK RELACEMENT DECISION OF AIR CONDITIONERS USING FIRST ORDER MARKO CHAINS WITH & WITHOUT CONSIDERING INFLATION Y HARI RASADA REDDY * rofessor & Head Departmet of Mechacal Egeerg Sr ekatesa erumal College of Egeerg & Techology uttur 7 8, Chttoor (Dst), Adhra radesh, Ida profyhpr@gmalcom Moble: DRCNADHAMUNI REDDY rofessor & rcpal Sr ekatesa erumal College of Egeerg & Techology uttur 7 8, Chttoor (Dst), Adhra radesh, Ida prcpalsvpcet@gmalcom Moble: +9 9 DRKHEMACHANDRA REDDY Drector (Academc & lag), Jawaharlal Nehru Techologcal Uversty Aatapur, Aatapur, Adhra radesh, Ida koreddy@gmalcom Abstract: I ths paper, a mathematcal model has bee developed for group replacemet of a block of Ar Codtoers usg dscrete-tme Frst Order Markov Chas To make the model more realstc, three termedate states vz, Mor Repar State, Sem-Major Repar State ad Major Repar States have bee troduced betwee Fuctog State & Complete Falure States of the system The Trasto robabltes for future perods for Frst Order Markov Cha (FOMC) are estmated by Spectral Decomposto Method Usg these probabltes, the umber of systems each state ad accordgly the correspodg average mateace cost s computed The forecasted flato for Ar Codtoers ad the real value of moey usg Fsherma s relato are employed to study ad develop the real tme mathematcal model for block replacemet decso makg Key words: Replacemet, Markov Chas, Trasto robablty, Spectral Decomposto Method, Iflato, Forecastg Itroducto The replacemet decsos most of the multatoal compaes, Software Developmet Ceters, Star Hotels ad other major food processg dustres are mostly wth the ar codtoers The prmary ISSN : 97-6 ol No May 6

2 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) decso s geerally whether to replace the exstg block of ar codtog system cotag a large umber of ar codtoers or use for some more perod of tme The actvty of mateace volve repars & servce ragg from mor to major whch caot be defed ad computed exactly specfc, varous costs, ad fluece of varous ecoomc varables such as Iflato, value of moey etc Several researchers vestgated the optmal age replacemet models wth repars to reduce the cost Nuthall et al (98) studed the mpact of flato o replacemet costs alog wth the mpact of some other parameters vz facg method ad creased or decreased hours of use Che et al (7) preseted a agereplacemet model wth mmal repar based o cumulatve repar cost lmt I ths they cosdered the complete repar cost data order to decde whether to repar the ut or to replace Baga et al (99) dscussed optmal replacemet tme uder the age replacemet polcy for a system wth mmal repar that volves the replacemet of oly a very small part of the system Rupe et al () explored the mateace models for fte tme mssos by cosderg et preset value of costs There are some studes o the replacemet decsos for warrated products Zuo et al () dscussed replacemet polcy for mult state Markov deterorato of maches that are uder warraty a et al () exteded the work of Zuo et al () by cosderg more geeral state space wth tme parameters at each state Archbald et al (996) studed ad compared the optmal age-replacemet, Stadard Block Replacemet ad Modfed Block Replacemet (MBR) olces wth a ferece of MBR polcy s apprecably better tha the remag two However there s o much lterature o block or group replacemet model wth Markov cha trasto probabltes that s beg used may applcatos Markov chas cocept was used by Stelos et al (98), Jamg et al (), Shamsad et al (), Yg-Z L et al (9), Avk Gosh et al (), Carpoe et al (), for forecastg of dfferet parameters vz ower geerato, mosoo rafall, mapower supplyg, wd speed Wth the creasg popularty of use of Markov chas, some studes by Bruce Crag et al (998) ad Laa Cazacoc et al () are made o estmato ad evaluato of trasto probabltes usg Markov chas Markov Cha forecasts as observed by Yg-Z L et al (9) have some practcal value that yelded relatvely satsfed results Shamsad et al () ad Carpoe et al () observed that Secod order Markov chas resulted better forecast performace tha frst order Markov forecasts As the estmato of trasto probabltes for bgger state space S =,,,,m s much tme cosumg oe, Bruce A Crag et al (998) ad Sutawar et al (8) studed the spectral represetato of trasto probabltes ad Zheqg L et al () tred computer aded program to estmate the hgh-order Trasto robablty Matrx of the Markov Cha Ths paper dscusses a mathematcal model for group replacemet of block of ar codtoers wth three termedary states vz, mor repar state, sem-major repar state ad major repar states betwee workg ad falure states usg frst Markov Chas Though t s dffcult to detfy the specfc reparable termedate states, to make the model smplstc the varous repars pertag to ar codtoers are grouped as show Table- Table : Idetfcato of possble repars ad ther category Ar Codtoers Mor repars Sem - Major repars Major repars Rug capactor problem Fa capactor problem Temperature sesor problem Codeser fa motor problem Codeser blade problem Blower problem etc Capllary problem Gas leak problem Relay board problem Dsplay board problem Compressor roblem Evaporator problem Codeser col problem etc The trasto probabltes are estmated usg Frst Order Markov Chas Trasto probabltes for future perods are estmated by Spectral Decomposto Also the fluece of macroecoomc varables such as flato ad tme value of moey are cosdered to make the model yeld better results Methodology of Markov cha Markov process s a stochastc or radom process, whch has property that the probablty of trasto from a gve state to ay future state depeds o the preset state ad ot o the maer whch t was reached ISSN : 97-6 ol No May 6

3 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) Frst Order Markov Cha (FOMC) The Frst Order Markov Cha (FOMC) assumes the probablty of ext state depeds oly o the mmedately precedg state Thus f t < t < < t represets the pots o tme scale the the famly of radom varables {X(t ) } whose state space S =,,,m s sad to be a Markov process provded t holds the Markova property: {X(t ) = X X(t - ) = X -,, X(t ) = X } = {X(t ) = X X(t - ) = X - } for all X(t ), X(t ),,X(t ) If the radom process at tme t s the state x, the future state of the radom process X + at tme depeds oly o the preset state x ad ot o the past states x, x,, x The smplest of the Markov rocess s dscrete ad costat over tme A system s sad to be dscrete tme f t s examed at regular tervals, eg daly, mothly or yearly Trasto robablty: The probablty of movg from oe state to aother future state or retag the same state durg a partcular tme perod s called trasto probablty Mathematcally the trasto probablty ca be expressed as: x(-), x() = {X(t ) = X X(t - ) = X - } ad s called FOMC trasto probablty that represets the probablty of movg from oe state to aother future state The trasto probabltes ca be arraged a matrx of sze mxm ad such a matrx ca be called as oe step Trasto robablty Matrx (TM), represeted as below: = m m m m mm Where m represets the umber of states The matrx s a square matrx of whch each elemet s o-egatve ad sum of the elemets each m row s uty e = ; = to m ad j j= j N j The tal estmates of j ca be computed as, j =, (, j = to m) where N j s the raw data N sample that refer the umber of tems or uts or observatos trastoed from the state to state j N s the raw data sample state Model Developmet: Notatos: N = Total Number of tems the System C = Idvdual Replacemet Cost er Ut C = Mor Repar Cost C = Sem-Major Repar Cost C = Major Repar Cost C = Group Replacemet Cost X I = roporto of uts fuctog state tally X = roporto of uts mor repar state tally X I = roporto of uts sem-major repar state tally X I = roporto of uts major repar state tally X = roporto of uts complete falure state tally X I = proporto of uts fuctog state at the ed of th tme perod X = proporto of uts mor repar state at the ed of th tme perod X I = proporto of uts sem major repar state at the ed of th tme perod, X I = proporto of uts major repar state at the ed of th tme perod, X = proporto of uts complete falure state at the ed of th tme perod j = robablty of tems swtchg from th state to j th state a perod TM = Trasto robablty Matrx φ t = Rate of Iflato durg tme t ISSN : 97-6 ol No May 6

4 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) r = Nomal Rate of Iterest r t = Real Rate of Iterest =, from Fsherma s Relato reset alue Factor (F) = j = robablty of tems swtchg over from th state to j th state a perod TM = Trasto robablty Matrx W(t) = Weghted average cost per perod group replacemet polcy, AC(t) = average cost per perod group replacemet polcy I ths paper, a group replacemet model for (N) tems that fal completely o usage, cosderg three termedate states e mor repar, sem-major ad major repar, s developed by usg frst order Markov cha So as t cossts m = states, the FOMC Trasto robablty Matrx(TM) ca be wrtte as: TM = = I I I I I I Where I,, I, I ad represets Workg, Mor repar, Sem-Major repar, Major repar ad Complete falure States respectvely Sum of the trasto probabltes each row s uty e j = ; = to The tal estmates of j ca be computed as, j = j= N j, (, j = to ) where N N j s the umber of Ar Codtoers trastoed from the state to state j N s the raw data sample state FOMC Trasto probablty, = D -, where = to for the future tme perods are computed usg Spectral Decomposto Method [Sutawar et al (8)] Spectral Decomposto Method: As the estmato of hgh order Markov cha trasto probabltes for bgger state space S =,,,,m s much tme cosumg oe, Bruce A Crag et al (998) ad Sutawar et al (8) studed the advatage of spectral represetato of trasto probabltes for mult state process ad Zheqg L et al () tred computer aded program to estmate the hgh-order Trasto robablty Matrx of the Markov Cha As several software for spectral decomposto are wdely avalable [Sutawar et al (8)], ths method provdes flexblty for the computato of trasto probabltes for mult state process Spectral Decomposto s based o ege values It s applcable to square matrx that wll be decomposed to a product of three matrces, oly oe of whch s dagoal As a result, the decomposto of a matrx to matrces composed of ts ege values ad ege vectors s called Ege or Spectral decomposto A x matrx always has ege values, whch ca be ordered ( more tha oe way) to form a x dagoal matrx D formed from the ege values ad a correspodg matrx,, of o zero colums (ege vectors) that satsfes the ege value equato: = D Ths gves the amazg decomposto of to a smlarty trasformato volvg ad D as = D - Furthermore, squarg both sdes of above equato gves = ( D - ) ( D - ) = D ( - ) D - = D - Mathematcally, Spectral Decomposto ca be represeted as = D - where = to ISSN : 97-6 ol No May 6

5 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) Therefore hgher order Trasto robablty Matrx (TM) of four state Markov cha ca be computed usg the equato, = D - where = to Calculato of umber of tems each state: The proporto (X ) of uts durg th perod varous states e, the state probabltes of tems dfferet states ca be computed as follows I I [ X X X X I I I I X ] = [ X X X X I X ] *, where = to X = (robablty of tems dfferet states durg tal perod) * ( TM) I geeral, X = X,,, of future perods for FOMC ca be calculated by spectral decomposto method The TMs, At the ed of the frst perod, the state probabltes ca be calculated from X ( X X X = ) I I I X X X X X = I I I [ X X X X X ] = [ ] Therefore, robablty of tems fuctoal state, at the ed of the frst perod I I I I X = X + X + X + X + X robablty of tems mor repar state, at the ed of the frst perod I I I X = X + X + X + X + X robablty of tems sem-major repar state, at the ed of the frst perod I I I I X = X + X + X + X + X robablty of tems major repar state, at the ed of the frst perod I I I I X = X + X + X + X + X robablty of tems rreparable state, at the ed of the frst perod I I I X = X + X + X + X + X Smlarly, the probabltes of tems fallg dfferet states future tme perods (= to ) are to be calculated by usg the equato, X = X The TMs ( ) for future tme perods =,,, are calculated by usg Spectral Decomposto method Usg these state probabltes the umber of dvdual replacemets ( ), mor repars (β ), sem-major repars (γ ) ad major repars (δ ) future tme perods ca be calculated as show the followg tables & : Table : No of Idvdual replacemets ad mor repars durg dfferet tme perods Tme perod No of dvdual replacemets st d X rd X X th X X X th X X X X S o o S o o No of mor repars β = β + β + β + β + NX βx + βx β X + βx + β X β X + βx + β X + β X β X S o o ISSN : 97-6 ol No May 6

6 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) Table : No of Sem-Major repars ad major repars durg dfferet tme perods Tme perod γ = No of sem-major repars st I NX d I I γ X rd I I I γ X X th I I I I γ X X X th I I I I I γ X X X X δ = No of major repars I NX I I δx I I I + δx δ X δ + δ + δ δ δ + δ I I I I + X + X X δ δ δ δ + δ I I I I I + X + X + X X S o o S o o S o o Iflato predcto: Iflato s predcted usg regresso model wth trgoometrc fucto ad the fluece of Iflato o the ar codtoers Ida from the year owards s studed over a perod of tme, forecasted ad compared wth actual values for the kow perods by employg varous forecastg techques to detfy the uderlyg model that best fts the tme seres data Subsequetly the flato s predcted for Ar Codtoers for the future tme perods by the developed Regresso model wth trgoometrc fucto, whch yelded relatvely mmal errors A susodal trgoometrc fucto s used the regresso model to accommodate cyclcal fluctuatos of flato For ths the followg mathematcal equato s cosdered φ = a + bt + c s (tπ + π/) () To fd the costats a, b & c the followg set of equatos are used φ = a + bt + c s (tπ + π/) --- () (φt) = at + bt + c [t s (tπ + π/)] --- () (φt ) = at + bt + c [ t s (tπ + π/)] --- () where φ s the flato, t s tme perod, s the umber of tme perods ad a, b & c are the coeffcets Regresso model wth trgoometrc fucto for predctg flato for a tme perod t s F= -+67T+799 S (Tπ + π/) Ifluece of Iflato ad tme value of moey Covetoal models are avalable to make the replacemet decsos cosderg the value of moey Here the Net reset Worth crtero based o the omal terest rate (r ) does ot reflect the real value of moey Real terest rates (r r ) are computed usg Fsherma s relato Whe the preset worth factors are computed ad multpled wth future moey, t gves purchasg power of moey reset worth factor= pwf= ν = + r t r φt Real rate of terest = r t =, from Fsherma s relato + φt To get more realstc results, the forecasted values of Iflato to get the real terest rates, usg Fsherma s relato, are used The Iflato values based o WI for ar codtoers are predcted by usg a regresso forecastg model usg a susodal trgoometrc fucto Total cost (TC) for tme perods, wthout the fluece of Iflato: TC = Group replacemet cost + Idvdual replacemet cost + Mor repar cost + Sem - Major repar cost + Major repar cost TC = NC Average cost per perod = + C + C β + C γ + C = = = = Total cost for ' ' perods umber of perods δ ISSN : 97-6 ol No May 6

7 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) Weghted average cost, W(t) = TC / Total cost (TC) for tme perods, wth the fluece of Iflato: TC() = C ν ν + ν + C β + β ν + β + C [ ] [ ν + + β ν ] [ γ ν ν + ν ] + C [ δ + δ ν + δ ν + + δ ν ] + NC ν TC()=C ( ν ) + C (βν ) + C ( γ ν ) + C ( δ ν ) + NC ν Weghted average cost, W(t) = TC / ν olcy: s optmal whe the weghted average cost per perod s mmum e average cost per perod should be mmum th perod, to block replace th perod Case Study: I the preset study, a block of ar codtoers a software developmet cetre have bee studed ad the cost data for varous types of repars based o the formato gve by the ar codtoers servce egeers s assumed as gve below N = Total umber of Ar Codtoers the system = C = Idvdual replacemet cost per ut = Rs C = Mor repar cost = Rs C = Sem-major repar cost = Rs8 C = Major repar cost = Rs C = Group replacemet cost = Rs r = Nomal rate of Iterest = % X I I I [ X X X X X ] [ 8 9 ] = = = TM = The calculatos pertag to optmal block replacemet decsos of Ar Codtoers wthout cosderg the fluece of flato are show the table- ad cosderg the fluece flato are show table- Results ad Dscussos The average cost per year usg Frst Order Markov Cha are show the Table-6 Whe the fluece of flato s ot cosdered, Frst Order Markov Cha (FOMC) model resulted the replacemet age as 7 years Whe the fluece of predcted flato ad et worth of the moey s cosdered, FOMC resulted the early replacemet of block of Ar Codtoers at the age of years Therefore for the block of ar codtoers cosdered for study ths work, the optmal replacemet tme s at the ed of th year Table - 6: Average Cost lakhs per year Tme perod() FOMC Wthout flato FOMC wth flato * * 8 8 * Mmum Cost gvg Optmum perod for replacemet ISSN : 97-6 ol No May 66

8 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) Cocludg remarks A Markov cha based mathematcal model for group replacemet model has bee developed for a block of computers system usg Frst Order Markov Chas Ths paper cosders fve dscrete states- workg codto, mor repar, sem-major repar, major repar ad break dow state of a ar codtog system to make the mateace cost more realstc FOMC has resulted the optmal replacemet age of 7 years wthout cosderg flato ad moey value, where as cosderg flato ad tme value of moey t was years Also the model was more realstc as the fluece of macroecoomc varables vz flato ad tme value of moey o replacemet model are cosdered The authors of ths paper have checked ths wth the replacemet decso whe the fluece of flato s ot cosdered Refereces [] Avk Ghosh Dastdar, Deepawta Ghosh, S Dasgupta ad UK De (): Hgher order Markov Cha Models for mosoo rafall over West Begal, Ida, Ida Joural of Rado ad Space hyscs, ol 9, pp 9-, February [] Berchtold Adre ad Adra E Raftery (): The Mxture trasto Dstrbuto Model for Hgh-Order Markov Chas ad No- Gaussa Tme Seres,Statstcal Scece, ol 7, No,8-6, [] Bruce A Crag ad eter Sed(998): Estmatg the Trasto Matrx of A homogeeous Markov Cha, Techcal Report #98-, Departmet of Statstcs, urude Uversty, Jue 998 [] Carpoe A, Lagella R, Testa A (), ery Short-term robablstc Wd ower Forecastg based o Markov Cha Models, IEEE th Iteratoal Coferece o robablstc Methods Appled to ower Systems (MAS), pp7-, [] Che Y H ad Che J A (7): Optmal Age-Replacemet Model wth Mmal Repar Based o Cumulatve repar Cost Lmt ad Radom Lead Tme, proceedgs of the 7 IEEE IEEM, pp 66-69, 7 [6] Har rasada Reddy et al (): Block Replacemet Modelg for a Block of Ar Codtoers wth Dscrete-Tme Markov Cha Approach, Iteratoal Joural of Appled Egeerg Research, ol 7, Number (), pp79-9 [7] Isha Baga ad Kacha ja (99): Improvemet, deterorato ad Optmal replacemet uder age- replacemet wth mmal repar, IEEE trasactos o relablty, ol, No, :6-6, March 99 [8] Jaso W rupe, U S West (): Optmal Mateace Modelg o fte Tme wth Techology Replacemet ad Chagg Repar Costs, IEEE proceedgs of Aual Relablty ad Mataablty Symposum, pp 69-7, [9] Jamg HU, Jgya SONG, Guoqag YU & Y Zhag (): A Novel Networked Traffc parameter Forecastg Method based o Markov Cha model, IEEE Trasactos, 9-6, [] Laa Cazacoc ad Elea Cora Cpu (): Evaluato of the Trasto robabltes for Daly precptato Tme seres usg a Markov cha Model, roceedgs of rd Iteratoal Colloquum Mathematcs Egeerg ad Numercal hyscs, October 7-9,, pp:8 to 9 [] Navee Klar, Reddy C N, Balu Nak (9): Forecast of rce Iflato for Electroc Sytems, roceedgs of the Iteratoal Coferece o Global Iterdepedece ad Decso Sceces, Dec 9, MacMlla advaced research Seres, -8 [] Nuthall L, Woodford KB ad Beck AC (98): Tractor replacemet olces ad cost Mmsato, Dscusso paper o7, Agrcultural Ecoomcs research ut, Lcol College, New Zealad, Nov 98, ISSN -77 [] Raftery A E (98a): A model for hgh-order Markov Chas, Joural of Royal Statstcal Socety, Seres B 7, 8-9, 98 [] Ruey Hue Yeh, Gaug-Cheg Che & Mg-Yuh Che (): Optmal age-replacemet polcy for No-reparable products uder reewg free-replacemet warraty, IEEE trasactos o relablty, ol, No, :9-97, March [] Shamsad A, M A Bawad, W M A wa hussa, T A Majd ad S A M Sasu (), Frst ad Secod order Markov Cha models for Sythetc Geerato of wd speed tme seres, Scece Drect, eergy, volume, ssue, pp 69-78, Aprl [6] Stelos H Zaaks ad Mart W Maret (98): A Markov Cha Applcato to Mapower Supply lag, The Joural of the Operatoal Research Socety, ol, No, pp9-, Dec 98 [7] Sutawar Darws ad Kulsa(8): Spectral Decomposto of Trasto Matrx, Joural Matematka Da Sas, September 8, ol, No, pp 97- [8] Thomas W Archbald & Rommert Dekkar (996): Modfed Block replacemet for Multple-compoet systems, IEEE trasactos o relablty, ol, No, :7-8, 996 [9] Yg-Z L ad J-cag Nu (9), Forecast of ower geerato for Grd-Coected hotovoltac System Based o Markov Cha, rd IEEE Coferece Idustral Electrocs ad Applcatos, 8 ICIEA 8 [] Yue a ad Marl U Thomas (): Repar ad Replacemet Decsos for warrated products uder Markov Deterorato, IEEE trasactos o relablty, ol 9, No, :68-7, Jue [] Zheqg L ad ad Wemg Wag (): Computer aded solvg the hgh-order trasto probablty matrx of the fte Markov Cha, Elsever joural of Appled Mathematcs ad Computato (Artcle ress) [] Zuo M J, Lu B ad Murthy DN (): replacemet-repar olcy for Mult-state Deterorato roducts Uder warraty, Europea joural of Operatoal Research,, ol, pp 9-, Iterpretato of colums the followg tables s gve below: A=Tme erod t B= t = umber of dvdual replacemets C=Idvdual replacemet Cost D=β t = umber of mor replacemets E=Mor replacemet cost F= γ t = umber of sem-major replacemets G=Sem - Major replacemet Cost H=δ t = umber of major replacemets I=Major replacemet Cost J= Total Mateace Cost = C+E+G+I K= Cumulatve Mateace Cost ISSN : 97-6 ol No May 67

9 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) L=Group Replacemet Cost M= Total Cost = K+L N =Average Mateace Cost/perod = Iflato Q = Real Iterest rate R = reset Worth Factor S = Dscoutg Factor T = Cumulatve Dscoutg Factor The calculatos pertag to optmal block replacemet decsos of Ar Codtoers wthout cosderg the fluece of flato are show the table- ad cosderg the fluece flato are show table- ISSN : 97-6 ol No May 68

10 Y Har rasada Reddy et al / Iteratoal Joural of Egeerg Scece ad Techology (IJEST) Table - : Calculatos of optmal block replacemet decso usg Frst Order Markov Cha wthout cosderg flato A B C D E F G H I J K L M N * ** Table - : Calculatos optmal block replacemet decso usg Frst Order Markov Cha cosderg the fluece of flato A Q R S T B C D E F G H I J K L M N * ** ISSN : 97-6 ol No May 69