Deterioration-based Maintenance Management Algorithm

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1 Aca Polyechca Hugarca Vol. 4 No Deerorao-baed Maeace Maageme Algorhm Koréla Ambru-Somogy Iue of Meda Techology Budape Tech Doberdó ú 6 H-034 Budape Hugary a_omogy.korela@rkk.bmf.hu Abrac: The Road Maageme Syem (ad he PMS) uually do o ake o coderao he fuure raffc chage. The maeace ad rehablao aco ad he developme of he road ework rucure ad he chagg raffc rucure modfy he amou of he raffc o he road eco. The deerorao proce deped o moly he volume of he raffc. Tha why mpora o ake o coderao he chage of he raffc volume durg he plag me horzo. I he lecure ome echque are how whch hadle h problem: mulperod log me model a each plag perod he raffc volume chage ake o coderao. I rakg model he problem could be hadled ad olved. I he cae of oe perod Markov able model here ohg o do. I he mulperod model he problem could be olved alo. PMS Model. The Ma Queo of Deco Maker of he PMS The ma queo of deco maker of he Road Maageme Syem ad he PMS: How much budge requred each year for maeace ad rehablao he eleme for her whole lfeme o hold hem a cera codo level? Whch AM eleme codo drbuo would be expeced f he um meo above were o be avalable? Wha coequece would be expeced f h um were o avalable? Wha coequece would be expeced f he maeace expedure were gfcaly creaed? Wha are he opmum me for maeace aco? 9

2 K. Ambru-Somogy Deerorao-baed Maeace Maageme Algorhm Wha coequece would be expeced from he delay of eceary aco? Who beef ad who loe o wha exe ec?.2 The Model A deco model for paveme maageme ha bee developed here baed o lear programmg formulao. The egeerg ad deerorao model wa creaed by Gápár [5] he mahemacal oe by Bakó [2] Markov rao probably marce are roduce o model he deerorao proce of he road eco deermed by h marce. To every ype of road urface ad cla of raffc amou belog a cera Markov marx. The preeed model ad mehodology ued o deerme he opmal rehablao ad maeace polcy ework level. Depedg o he obecve fuco wo ype of problem could be olved by he model: he eceary fud calculao ad he opmal budge allocao for he ere ework. Several ype of oluo algorhm ca be ued depedg o he gve ak he avalable daa he budge cora ec. Two ma ype are he heurc ad he opmao algorhm. The heurc echque uually ued proec level bu could be ued ework level oo. The opmao model are olved by he radoal opmao algorhm. Depedg o he problem o be olved we ue eger a lear or a dyamc programmg algorhm. There are everal cora o be fulflled. We wll deoe he ukow varable by X k whch belog o he paveme ype o he raffc volume ad o he maeace polc. The oluo have o be Markov able. The Markova ably cora f = = k= ( E ) X = 0 Q () k k where E a ue marx. Becaue he equaly uually o fulflled or o derable we ue or relao ead of equaly (). There are everal furher cora whch are coeced wh oher uppoo. We uppoe ha he raffc volume wll o chage durg he plag perod: k= X k = b = 2... = 2K f (2) where b belog o he paveme ype ad he raffc volume. 20

3 Aca Polyechca Hugarca Vol. 4 No The oal area of he road urface ype wll rema he ame a he ed of he plag perod f = k= X k = d = 2 K (3) where d belog o he paveme ype ad d =. We have o apply oe of he maeace polc o every road eco f = = k= = X = (4) k We dvde he egme o 3 group: accepable (good) uaccepable (bad) ad he re. Le u deoe he ree e by J (good) by R (bad) ad by E (re of he egme) ad by H he whole e of egme. The relao for hee e are gve by J R= R E= J E=0 J R E=H The followg codo are relaed o hee e X k v J k J X k v R k R (5) v E X k v E where J R E are gve above ad v he oal legh of he good road afer he plag perod v R he oal legh of he bad road afer he plag perod v E he lower boud of he oher road v E he upper boud of he oher road. The meag of he fr codo ha he amou of good egme ha o be greaer ha or equal o a gve value. The ecod relao doe o allow more bad road ha fxed advaced. The hrd relao gve a upper ad lover lm o he amou of he re road. 2

4 K. Ambru-Somogy Deerorao-baed Maeace Maageme Algorhm Le u deoe by c k he u co of he maeace polc k o he paveme ype ad raffc volume. Our obecve o chooe uch a X whch fulfl he codo gve above wh mmal rehablao co. The obecve = f = k= X C m! (6) k k Le u deoe h value by C. The budge C* whch avalable for he maeace purpoe uually le ha C o C*<C. I h cae we modfy our model: he above meoed rehablao co fuco become coraed: X k Ck C * (7) ad we ue a oher obecve. Le u deoe he beef by h k where h he beef of he ocee whe we apply o he paveme ype ad wh he raffc volume he maeace polc k. Our am o deerme uch a oluo X whch fulfl he cora ()-(5) ad (7) ad maxmze he oal beef of he ocey. The obecve h cae X max! (8) k h k Gápár [6 7] ummarze he dealed of he ug algorhm. The Road Maageme Syem (ad he PMS) uually do o ake o coderao he fuure raffc chage. The deerorao proce deped o moly he volume of he raffc.tha why mpora o ake o coderao he chage of he raffc volume durg he plag me horzo. 2 Traffc Plag ad Forecag A lo of problem are kow o be coeced wh raffc plag forecag drbuo ad agme: deermao of he pah wh mmal ravellg me bewee every par of po (SP-problem); foreca of he fuure raffc arg from he ource po (Forecag); deermao of he raffc o he road egme (Agme problem). 22

5 Aca Polyechca Hugarca Vol. 4 No SP Problem A he SP problem le N be a fe e of po E he e of edge ad x y x y. The ravellg me o pah d ( ) > 0 he ravellg me o he edge ( ) P = ( x x K ) 2 x r r = ( P ) = d ( ) L x x (9) +. The mulermal hore pah problem ca be formulaed a follow: deerme he pah P wh mmal ravellg me for every par of po. There are everal algorhm for olvg h problem. The be mehod wa gve by Warhall [ 9 ]. Th algorhm co of he followg ep: d d ( 0) = d( x x ) ( k ) ( ( r) ( r) ( r = m d d + d )) r r where he umber of po. Th algorhm eed he lea umber of compuaoal ep; oly 3 addo ad comparo are eeded. The marx ( ( ( d ) D = gve he legh of he mmal pah. I order o oba he pah hemelve we compue aoher o called labellg marx (0) ( k) = (k ) S : (0) ( r) ( r) ( r) ( r) f d dr + dr () = ( r) r Koherve 2.2 Traffc Forecag ad Agme Fraar problem: Gve a marx A = ( a ). The eleme a mea he amou of raffc from po o po. Le u deoe by k repecvely by l he amou of raffc leavg repecvely eerg o po. The Fraar problem (F-problem) o deerme he fuure raffc marx from he marx A ug he gve k ad l value. I he cae of Gravy problem Iead of marx A marx D gve where he eleme d mea he mmal ravellg me from po o po. Thee wo problem eem o be dffere bu boh problem ca be olved by he forecag of he pu-oupu (I/O) able. 23

6 K. Ambru-Somogy Deerorao-baed Maeace Maageme Algorhm Le K K2 KK m be ource ad le m marx A mea he raffc from l he oal raffc leavg k l = = = = a ( = 2K a ( = 2K. K o K repecvely L L2 KL be k. The eleme a 0 of L. Le u deoe by k repecvely by L : The marx A called I/O able ad eleme value of I/O able A. (2) k l are called he margal Le k k2 K k ad λ λ2 K λ be ew margal value ( k 0 λ 0 = 2 K. The forecag of he I/O able o deerme he ew I/O able X from A ad gve margal value k λ o ha he able A ad X are mlar. The wo able are mlar whe x ca be deermed he followg form x = ξ a η (3) where ξ η are ukow. Klafzk [8] Bakó [] ha how by he help of geomerc programmg ha h problem equvale o he mmalzao of formao dvergece. The able X ha o afy he equao (2). = = x x = k = λ ( = 2 K ( = 2 K Ug equao (3) ad (4) we oba ξ ηa = k η ξa = λ (5) D Eopo [4] uggeed he followg mehod for olvg he equao (6): Sep 0 η ( 0) = λ ( = 2 K (4) 24

7 Aca Polyechca Hugarca Vol. 4 No Sep r ( r ξ ) = ( r η ) = k λ ξ ( ) ( = 2 K r a ( ) ( = 2 K r a (6) Bergma [3] proved he covergece of he algorhm. The algorhm ermaed whe he error π k a ξ max (7) k uffcely mall. Traffc agme he proce of allocag all he rp oe or more O/D marce o her roue he ework reulg he flow o lk. The raffc agme mehod employ hree bac ep: SP problem agme a par of he raffc o he egme ad check for covergece. 3 Combed Algorhm The maeace ad rehablao aco ad he developme of he road ework rucure ad he chagg raffc rucure modfy he amou of he raffc o he road eco. The deerorao proce deped o moly he volume of he raffc. Tha why mpora o ake o coderao he chage of he raffc volume durg he plag me horzo.i he fr par he PMS model are ummared. I he ecod par we llurae he raffc foreca drbuo ay agme algorhm (TFDA). We propoed a ew model wh:ew egme each me perod raffc forecag ad agme each plag perod.now he bac dea of combed algorhm demoraed. The algorhm are a log rage mulme perod oe we compue he maeace ad rehablao aco for each year. Afer deermao he aco a year a TFDA (raffc foreca drbuo ay agme algorhm) ep ake where he ew raffc o he egme are deermed. 25

8 K. Ambru-Somogy Deerorao-baed Maeace Maageme Algorhm Afer ha ew raffc caegore formed ad he ex PMS ep (for he ex me perod) ue h ew raffc caegore. The algorhm demorae boh cae he ep of he algorhm. The fr algorhm how he cae of heurc (rakg) mehod: Sep Th a PMS rakg for he fr me perod. Sep 2 A TFDA ep gve. Sep 3 Th compuao coue ul he la perod fhed. The umber of he me perod. The combed algorhm how he cae of Markov deerorao model he followg: Sep The fr ep o compue he Markov able oluo. Here raffc chage o ake o coderao. Sep 2 The yearly deerorao ad maeace proce deermed. Sep 3 Afer each year a TFDA algorhm ake ad he reul of ha ued o form he ew raffc caegore for he ex year. Sep 4 The algorhm fhed whe we compue he deerorao ad maeace aco for he la me perod. The procedure gve above erve beer reul becaue he raffc chage are ake o coderao each year. Referece [] Bakó A.. Mahemacal ad Compuaoal Evaluao ad Developme of Traffc Plag ad Forecag Caddae derao HAS 980 p. 220 [2] Bakó A.: Deco Supporg Model for Hghway Maeace Aca Polechca Hugarca 2004 pp [3] Bergma L.: Dokazayelzvo zodmozy G.B. Selekkozkov dla zadac z razporüm orgacamam Zural Vüczl. Ma. I. Ma. Fzk 967 pp [4] D Eopo D. A.: Lekkowz A Algorhm for Compug Ieroal Trafer Ug he Gravy Model OPNS Re. 963 pp [5] Gápár L.: Developme of he Fr Hugara PMS Traporao Reearch 992 pp ( Hugara [6] Gápár L.: E ezbezogee Maagemeyem für de Shaeerhalug Ugar Srae ud Auobah 992/8 pp [7] Gápár L.: Road Maageme Hugara Academy of Scece 2003 p. 36 ( Hugara [8] Klafzky E.: O he Predco of he Ipu-Oupu Table Repor of Compug ad Au. I pp. -3 ( Hugara [9] Warhall S.: A Theorem of Boolea Marce J. of ACM pp