OPTIMAL BUS DISPATCHING POLICY UNDER VARIABLE DEMAND OVER TIME AND ROUTE LENGTH

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1 OPTIMAL BUS DISPATCHING POLICY UNDE VAIABLE DEMAND OVE TIME AND OUTE LENGTH Prof. Aml S. Kumrge Professor of Cvl Egeerg Uvers of Moruw, Sr L H.A.C. Perer Cvl Egeer Cerl Egeerg Cosulc Bureu, Sr L M.D..P. Jre Msers Sude, Fcul of Busess Sudes Uvers of Wollogog, Ausrl ABSTACT The problems of schedulg d schedule co-ordo bus operos hve coflcg objecves reled o user s cos d operor s cos. Pssegers would le o hve publc bus servces where here s less wg me. Operors o he oher hd would le o er prof wh lesser vehcle operg cos d mmum umber of buses. I developg coures where overlodg of buses hs log bee cosdered ecessr o esure bus rvel rems ffordble o mos socoecoomc groups, bus operors would ddo o lrger hedws, le o hve hgher lod fcors o crese reveue eve hough pssegers would prefer less lod fcors s provdes more comforble joure. All hese fcors re furher cosred b he fre levels, whch m o me he reveue deque o opere he mos ecoomcll opml frequec d lod fcor. Ths pper cosders mehod h s eeso o Newell s Opml Dspchg Polc, o deerme flee sze d dspchg re bsed o boh operor s cos d user s cos cludg he dsul of sdg, order o rrve globl cos opmum. I furher vesges he fcl vbl of provdg such servce d ses ou fcl vbl dom wh whch opmzo c occur prcce. If he resulg dspchg re s lower d does o fll wh he dom of fcl vbl, he operg subsdes re cosdered ecessr o m he ecoomcll opmum dspchg re. Ths mehod o compue opmzed dspchg res s bsed o scree-le cous cross gve locos log bus roues used cojuco wh lmed smple of o-bord bordg d lghg surves. Psseger reveues hve bee compued b process of mulplco of he rolzed org-deso mr b he fre for dsce rvelled bewee he respecve orgs d desos. Idcors hve lso bee developed o deerme verge rp leghs for ech roue d verge reveue per psseger ogeher wh he pos of mmum cpc log he roue. These dcors descrbe he ure of he demd h he bus roue serves.

2 The scree le cous provde he hourl vro demd over bus roue hroughou he d, whch hs bee epressed erms of poloml equo o deerme he vro of demd over dffere me perods. B combg boh fucos, compose fuco hs bee developed o deerme; he dl psseger demd o gve roue; he ol reveue for operors, he verge lod fcor d locos o he roue where mmum lodg occurs. INTODUCTION Dspchg re o bus roue s pere o boh s prmr seholders, he operors d commuers le. For he susece of he ssem, he pror d mos obvous requreme s h he operor hs o m fcl vbl of hs opero. If he dspch re ( defed erms of buses dspched over roue per hour s hgh, hs operg cos creses d cosequel fcl vbl s ffeced sgfcl uless here re deque pssegers eldg reveue, whch covers he cos of dspchg re. I hs suo hs fcl vbl depeds o wo oher fcors; ( he lod fcor of buses d (b reveue or fre level. I s herefore he objecve fuco for operors o deerme dspchg re wh he prmeers of he roue mel, he psseger demd, llowed lodg fcor d fre level. The operors objecve s herefore o dspch buses re lower h he fcll fesble dspchg re f. Therefore, we hve cosr: f I regulor regme, where bus fres re full dereguled, he operor hs he opo of cresg hs fre level so h, reveue opmzg commercl decso s rrved b rdg off lower dspchg res wh hgher fres. However, operg regme where fres re full reguled hs freedom s o wh he operor. Therefore, fcl vbl s usull cheved b he operor deermg dspchg re deermed o he mmum lod fcor llowed b lw or he bsece of such, b he phscl cpc of he bus. Uder such codos, over lodg creses, dspchg res reduce d cosequel he qul of servce for he oher seholder-he psseger deerores. Ths brgs us o he fcors h ffec he cos of he pssegers usull recogzed erms of fre, wg me, rvel me d lod fcor. As oe or more of hese prmeers creses, so does he cos o he psseger. Thus hgher wg mes whe dspchg re creses wll dmsh he ul (beef he obed from he servce, hereb cresg hs geerlzed cos, whch ur could reduce he pssegers o he roue. If oe were o rele o hpohecl codo where he bsolue epeco of he commuer s ssfed, where bus s vlble wheever he wshes o rvel, he wg me becomes zero. Ths del suo s he comprble o dvdul prve rspor. I s evde he h hese wo prmr seholders bus servces hve coflcg objecves owrds dspchg mes. Newell, 97 roduced he cocep of socl cos b g he opmum of he ggreged coss for boh seholders. Ths pproch however ssumes h he: Operor s ble o er deque reveue o cover cos of operg he servce he opmll deermed dspchg re. Operor s ble o eher vr hs fre wh vros coss, psseger demd or qulve fcors such s lower wg me or lod levels s m be

3 demded b he pssegers or h he would be subsdzed b he goverme o compese loss reveue. Demd would o vr log he legh of he roue durg he rffc d ece wor o ( d ( b Pds d Kumrge ( hs cocluded h he fcll opmum dspchg re hs o be lws lower h he ecoomcll opmum dspchg re d pos ou h wo sreges could be emploed o rech uversl opmum where boh fcl d ecoomc opmum dspchg res re equl uque po. Ths pper herefore vesges more specfcll em ( of he bove mers bu wll use he heor developed for deermg such globl opmum. Fcl Cosrs o Dspchg Buses o Opmzed Ecoomc Crer Pds d Kumrge, ( developed he rgume h whe g he reveue of ere roue, he overll reveue does o chge wh respec o chges dspchg re d correspodgl he hedws bewee buses dspched. Ths s jus becuse suppl creses demd does o crese d s such s resoble o ssume h reveue or fcl reur o he bus operor rems cos FO. Ths s show Fgure log wh he vro of he fcl coss of opero (FCO o he operor, whch cosders h s dspchg re creses, he umber of buses requred d he umber of rps mde wll crese hereb cresg he cos o he operor s well s o soce erms of resource use. The curve Fgure follows dmshg cos curve (Newell, 97 d here ess po hf hedws bewee buses, fer whch cresg he dspchg re becomes ufesble. Fgure : Opmum Hedw of Bus Dspchg Bsed o Fcl eveue d Cos o he Operor Le us ow develop he ecoomc cos curve h cludes coss o pssegers. Ths s show Fgure, where ECO - Ecoomc Cos of Opero FO - Fcl eveue o Operor

4 ECP - Ecoomc Cos o Pssegers h o - Opmum hedw ' h f - Fcll fesble hedw o he rgh of h o (> h ' f " h f - Fcll fesble hedw o he lef of h o (< h " f Followg he sme role, f oe were ssumed h he reloshp bewee fcl coss d ecoomc coss re ler, (.e. ECO FCO, he we hve ecoomcll opmum hedw, ho, whch s he hedw correspodg o he desred dspchg re for soce. Fgure : Cosrs of Fcl eveue o Opmum Dspchg e I s resoble o ssume h operor would prefer o hve dspchg re h would " led o operg hedw o he rgh of h h o (e.g. f whch wll effecvel lower hs coss so h wh fed reveues, profs wll he crese. Therefore, operor s preferred dspchg re s represeed b correspodg hedws o he rgh of ho. I suo where cross-subsd s vlble bewee dffere mes of he d, he verge ho m dffer ' h f whe buses re beg dspched hedws whch s less h wh s opml (.e. o he lef of ho durg mes of hev demd d hedws hgher h ho durg le perods of demd such s durg off-pe perods, erl morg or le ghs. Ths forms erl cross subsd wh roue mged b he operor wh hs overll fcl vbl. I developg coures, regulors m w o specf such perods of mmum hedw s polc o fulfl cer mmum ffordbl codos s socl requreme. The esme of he ol reveue o bus roue s esl obed whe here s ol oe operor o he roue. However, bus rspor ssems where owershp s he hds of dvdul ower-operors, formo o reveue m o be redl vlble. I herefore,

5 becomes ecessr o hve esme-bsed crer, whch could be doped o deerme he verge dl psseger demd o roue s well s s vros gs me d spce. Accordg o Srhm e l (999, chges hedw vro d ru mes were used o esme he l beefs of hs d of ssem wh respec o opero coss, psseger wg d psseger rvel me. Followg Housell d McLeod (998, hedw vro ws lso used o derve mesure of ecess wg me h pssegers hd o eperece due o urelble servce. Demd Esmo o oue Served b Mulple Operors Abowz d Egelse (984 hve developed ler regresso model o esme he me rug me of bus roue. I hs model, psseger bordg d psseger lghg behvour re cosdered s depede vrbles. Ths model hs show h he me rug me s hghl flueced b fcors such s bordg d lghg, rp dsce, me of he d d dreco of rvel. Abowz d Tozz (986 hve lso developed oher mhemcl model o vesge he mpc of fve bordg d lghg profles o he effecveess of hedw bsed corol. These profles specfed h; Pssegers bord he begg d lgh he ed of he roue (oe o oe Pssegers bord he begg d lgh he mddle d he ed of he roue (oe o specfed sops. Pssegers bord he begg d lgh he mddle of he roue (oe o oe. Pssegers bord d lgh uforml log he roue (specfed o specfed. Pssegers bord he mddle d lgh he ed of he roue (oe o oe. All he bove refer o suos where he locos for bordg d lghg log roue re specfed. However, locl bus servces, such sops co be specfed d operos re mosl m o m.e; pssegers m bord po d lgh oher po log he roue. Ths form of mos geerlzed bordg d lghg per s used o esme he demd fuco of roue over s legh. Y f ( ( The demd for roue log s ere legh herefore hs o be suded s demd fuco represeed erms of prbolc curve s he h degree. Thus, he demd for pssegers o prculr roue wh respec o he me of he d (represeed s he equo would be poloml curve where he geerlzo form of he equo of he demd curve would be represeed s;... ( The Prcple of Les Squres mehod s used o fd ou he coss,, 3,, d. To pprome he gve se of d, (,, (,,, (,, where, he bes fg curve of predced demd, f(, hs he les squre error, (.e. he resdul of equo ( s gve b

6 [ ]... ( (3 I s oed h,, 3,, d re uow coeffces whle d re gve s me vlue d demd d respecvel. To ob he les squre error, he uow coeffces,, 3,, d mus eld zero frs prl dervves. [ ]... ( ( (4 [ ]... ( ( (5 [ ]... ( ( (6 These led o he equos... (7... ( (9 Equo 7, 8 d 9 c lso be represeed mr form, s show equo ( q (

7 Ths mr equo c be solved umercll d he swer gves he vlues for he uow coss,, 3,, d. Cosequel he bes fed curve of he predced demd curve, f(, s vlble for furher lss. The equo of he predced demd curve llusred s show Fgure 3, whch gves he umber of pssegers crred o ll buses cross gve po log he roue, over me perod. Fgure 3: Demd of Pssegers o oue over Trffc D Compuo of eveue o oue Served b M operors I geerl, here re hree dffere eves volved he crrge of psseger h s mplcl bul o he fre compuo o mos roues. These re: A psseger bords he bus A psseger lghs from he bus A psseger s crred ps sop or seco of he roue. If here re umbers of sops o prculr roue, he here would be - umber of secos, where seco would be defed s he seco of roue bewee wo djce sops. Fgure 4 llusres hs log wh A, B (where,,, represeg he respecve umber of people cull bordg d lghg h sop. If he fre compuo s gve erms of psseger chrge for bordg (, lghg ( d for crrg ps oe bus sop (, he reveue of sgle bus prculr loco o he roue c be represeed s: ( A ( B β ( X A α γ ( Where A umber of pssegers lghg seco, B umber of pssegers bordg seco d X umber of pssegers crred wh seco

8 !!!!! " #$ " % #$ % " & #$ & " ' #$ ' " (% #$ (% " ( #$ ( " #$ Fgure 4: Bordg & Alghg Pers o Bus oue wh Sops I he bsece of d from o bord cous, esmes c be mde locos log he roue, where he reveue of bus roue c be deermed from smple of buses for whch bordg d lghg surves hve bee crred h loco. Ths we wll represe erms of vrble eveue o defed he ro of reveue bewee h whch s esmed for prculr rp,, (, d he mmum possble reveue erble jh loco ssumg h ll pssegers h loco were o rvel he ere dsce (j,m. The eveue o c hus be wre s: eveue o (,j, (, j ( j,m Where; s he eveue o for h rp j h loco. (, j Averge of ll (, j over roue d rffc d, s clled Averge eveue o, s gve equo 3. l m j (, j (3 (, j Where l s umber of rps d m s umber of locos If we ssume h he demd s uform over he legh of he roue, he he fcl reveue from bus operos o he roue durg he rp (, s gve b he epresso: f D (4 Where f fre o rvel ol legh of he roue d D ol demd of psseger per rp. Averge eveue o of (,j

9 Bu sce we hve lred ssumed h demd for pssegers over me, s fuco of me (, he: D f ( (5 Therefore, he reveue of rp gve equo 4, c be clculed usg he Averge eveue o gve equos 3 b mulplg he demd fuco prculr loco represeed b equo 5 so h ol fre of he roue s ( f f (6 Therefore, he Fcl eveue o ll operors o he roue per d s gve b: f D (7 Vldo The 4 m log urb bus roue, Pdur Nugegod (oue # 83, s used s cse sud. The lss uses lodg d md w loco ml o he roue. Tble shows he summr of lodg d hs loco oe-hour ervl hroughou he rffc d. Tble shows vro he demd per po o roue durg he ere rffc d from 6AM o 6PM. I order o f he demd d o prbolc curve, he order of he poloml fuco s ssumed o be 6 d 3 whch deoes he umber of d smples. The uow coss of he poloml equo c be foud whe he respecve d from Tble s subsued he equo, s me vlue d demd vlue, so h: D (8

10 Tme (srg Tble : Lodg Surve D No. of Buses Demd (per hour Suppl (ses per hour Lod Fcor 6: : : : : : : : : : : : : Fgure 5: Demd of Pssegers o oue over perod Fgure 5 supermposes he ormlzed demd curve o he observed hourl demd d he esg hourl suppl erms of ses dspched per hour, whch ur gves he dspchg re.

11 . / *,-- Fgure 6: Summr of Psseger Bordg & Lghg for Sgle Trp The bordg d lghg surve d of sgle bus rp o he sme roue s show below, where he fre seco s ppromel ms legh s gve Tble Fgure 6 gves he correspodg umber of pssegers ech seco compued from he followg ble. Tble : Psseger Bordg & Alghg D for Sgle Trp Sop Dels Tol Crred o Seco Tme Bordg Alghg e seco. Pdur Wl Gor Horeuduw Moruw Kubedd ml M.Lv Dehwl Klubowl Nugegod TOTAL I hs prculr roue ol of e seco smples re e (.e: (- d herefore b pplg equo (. 96 α 96 β 546 γ (9 The epresso bove revels h ( he ol umber of pssegers who borded he bus ws 96 d (b he ggreged umber of secos he rvelled cross ws 546. Wh mmum fre beg s 5. (roud US 5 ces he vlue for he coss α, β re ech ssumed o be ½ of hs vlue. The vlue for s clculed g he verge fre per seco for hs

12 roue d foud o be ppromel s..8 per seco. Therefore, he ol reveue for hs bus rp s s,6 compued s follows: ,96.8 (upees Summg he demd over he rffc d d mulplg wh he correspodg reveue per rp provdes he ol fcl reveue o ll operors o he roue. eveue os re clculed for dffere rps durg he dffere mes of rffc d for ech of he locos where rodsde lodg surves hve bee crred ou. The clculo of eveue o for he h rp for he hree lodg surve locos o hs prculr roue s s follows; ( B.& A, ( B.& A, ( B.& A,, / 3 LS- LS- LS-3,96.8,96.8,96.8, / ,. The clculed reveue ro for he h rp for he bus roue # 83, Pdur Nugegod s.. The ol verge reveue ro for he roue s verge of ll umbers of rps s dcg equo ; (,,..., ( The reveue of rp occurrg me o hs roue c be foud from he equo f [ ] -( Summg he reveue of ll rps occurrg durg he rffc d would gve he ol reveue o he roue. Therefore, he ol roue reveue o operors FO c be compued usg equo 7 d s gve s: FO Where f [ ] (3 d re he srg me d fshg me of he rffc d. Dspchg o esure Operor s Fcl Vbl As dscussed erler, lhough he demd vres from rp o rp would o chge he ol reveue o ll operors o he roue. Fgure 7 shows he curve FO wh correspodg ECO

13 d ECP curves d he ol ecoomc cos gve b curve TEC. For he roue queso, he observed verge dl dspchg re s 8 buses over he rffc d of hours gvg verge hedw of mues. The observed vro s show Tble s bewee low of 3 buses o hgh of buses per hour durg he pe perod. The ecoomc cos curves for bus operos (ECO d psseger coss (ECP d he ol ecoomc cos o soce (TEC show h he opmum hedw for hs roue should be 7 mues. Ths mes h he cul verge hedw h o he roue s o he rgh of ho. Ths mes h eher fre levels re deque or he operors re mg super orml profs he epese of cresed ecoomc cos o pssegers. O emo of he reveue for hs roue s dscussed bove, he FO curve s see o ercep he ECO curve hf whch s he hedw whch he roue operos become fcll fesble. Thus we hve codo h h > hf > ho. Ths mes wo hgs mel; ( h fre levels re suffce d (b h operors re sll mg ecess profs. I s herefore possble o reschedule mebles o reduce hedws from h o hf whou crese fre. However order o esure h he roue operes ecoomc opmum hedw of ho, fre levels hve o be cresed b percege of (FO FO/FO or lerel he operors should be gve operg subsd o h vlue of f whch s FO - FO. Fgure 7: Effec o he Hedw wh eveue Icreme CONCLUSIONS The epeco of pssegers d operors ehbs fudmel coflc rspor plg. I order o solve hs coflc he rspor pler eeds o rrve ccepble compromse b equll reg he wo compble plg objecves. The mehod preseed hs pper s eeso o Newell s opmzo mehod of deermg opml hedws b compug reveue vros log roue boh me d spce order o deerme he fcl vbl of such opml hedw compued

14 s socl cos o he ecoom. The pper provdes mehod for compug fres roues where here re mulple operors d s such, hs mehod m be used for pe of bus roue. A cse sud s used o compue hese vlues d he pper shows how he esg hedws s hgher h he fcll vble hedw d much hgher h he ecoomcll opmum hedw. The pper cocludes b showg he eed o whch operol mprovemes could be effeced order o reduce wg mes for pssegers d wh reveue bsed chges re requred order o esure h he cul operg hedw s equl o he ecoomcll opmum hedw. ACKNOWLEDGEMENTS The uhors grefull cowledge he fcl sssce provded b he Nol scece Foudo hrough s Gr No 999/E/ for he l reserch ledg o he d colleco for hs pper s well s Trs Ssems Ausrl for he sposorshp o ed he Thredbo coferece order o prese hs pper. EFEENCES Abowz, M. d Egelse, I., Mehod for Mg Trs Servce egulr, Trsporo eserch Bord, Trsporo eserch ecord 96, 984, pp. l-8. Abowz, M d Tozz, J, Trs oue Chrcerscs d Hedw-Bsed elbl Corol", Trsporo eserch ecord N78, Trs Prcg d Performce. Housell, N. d F. McLeod AVL mplemeo pplco d beefs he U.K. Pper preseed he 77h ul meeg of he Trsporo eserch Bord, Wshgo, D.C. Jur -5. Lm T. Evlug Publc Trs Beefs d Coss, 6 M,. Pds J.D.A.I., Opmzo of Bus Dspchg From A Gve Terml, MSc Thess, Uvers of Moruw, 5. Kumrge A.S. d Pds J.D.A.I, Formulo of Bus Dspchg Polc o Acheve Smuleous Ecoomc d Fcl Opmzo of oue Newell G.F. Dspchg Polces for Trsporo oue, Trsporo Scece 5, 97, 9 5. Phlppe L. To. Trsporo modellg mehods d dvced rspor elemcs (ATT, epor 9/4 Mrch 3, 993. Srhm, J., K. Dueer, T. Kmpel,. Gerhr, K. Turer, P. Tlor, S. Clls, D. Grff, d J. Hopper Auomed bus dspchg, operos corol, d servce relbl: Bsele lss. Pper No. 9993, Trsporo eserch Bord, 78h Aul Meeg, Wshgo, D.C.