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

Transcription

1 OPTIMAL BUS DISPATCHING POLICY UNDER VARIABLE DEMAND OVER TIME AND ROUTE LENGTH Amal S. Kumarage, Professor of Cvl Egeerg, Uversty of Moratuwa, Sr Laka H.A.C. Perera, Cetral Egeerg Cosultacy Bureau, Sr Laka M.D.R.P. Jayarate, Masters Studet, Faculty of Busess Studes, Uversty of Wollogog, Australa Treadbo 0, Hamlto Islad, Australa 2007 August

2 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Dspatchg Rate- Socal Perspectve Dspatchg rate o a bus route s pertet to both ts prmary stakeholders, the operators ad commuters alke. The optmum dspatchg cost accordg to Newell (97) should be whe the combed socal cost for both operator ad passegers a mmum. However o the part of the operator, he has to mata facal vablty to provde a cotug servce. h o

3 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Shortcomg Newell s polcy does ot take to accout the facal vablty of the optmum dspatchg rate. It also dd ot cosder varato of demad over space ad tme o a route. These are practcal lmtatos applyg the polcy.

4 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Newell s theory of Dspatchg Rate Newell (97) troduced the cocept of Socal Cost determg the optmum dspatchg rate assumg: Operator s able to ear adequate reveue to cover cost of operatg the servce at the optmally determed dspatchg rate Operator s able to ether vary hs fare wth varatos costs, passeger demad or qualtatve factors such as lower watg tme or load levels as may be demaded by the passegers or that he would be subsdzed by the govermet to compesate ay loss reveue Demad would ot vary alog the legth of the route durg the traffc day

5 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth ECO-Ecoomc Cost of Operato FRO -Facal Reveue to Operator ECP -Ecoomc Cost to Passegers Cost/Reveue (Rs/ Day) FCO h o -Optmum headway FRO ' h f -Facally feasble headway to the ' rght of (> ) h o h f " h f -Facally ufeasble headway to the left of h " (< ) o h f h f h o h f Headway (Mutes)

6 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Dspatchg Rate-Operators Perspectve A operator of a route wth a fxed demad wll operate buses at a partcular dspatchg rate Ǿ based o passeger demad, capacty of a bus ad reveue per trp. The operators objectve order to ear a proft s to dspatch buses at a rate lower tha the facally feasble dspatchg rate Ǿ f. Therefore, we have a costrat: Ǿ Ǿ f A chage operatg cost (e.g. crease fuel prces) leads to hgher dspatchg costs ad thus the eed for a hgher reveue per trp.

7 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Market Place- Prce Fxg I deregulated markets, the operator wll respod By creasg fares or By varyg the supply levels utl facal vablty s reached. I regulated markets where fare creases are cotrolled, ad where full effect of cost crease ot recoverable, facal equlbrum may be mataed by: (a) reducg dspatchg rate or (b) subsdzg operatos.

8 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Cocluso The ecoomcally optmum headway based o Newell s polcy, should also be h 0. The the reveue to operator should be based o the cost of provdg a headway of h o. FCO The FRO s based o the total reveue to operator for a gve operatg cycle. If FRO s less the the FRO, the a operatg subsdy ( f) s justfable to mmze the total socal cost. FRO f However f FRO s more tha FRO, the also t meas that fare levels are hgher tha what s requred to provde a socally optmum servce ad t s lkely that the operator wll make surplus profts.

9 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Estmato of FRO (Operatg Reveue) The estmate of the total reveue o a bus route s easly obtaed whe there s oly oe operator o the route. However, bus trasport systems where owershp s the hads of dvdual ower-operators, formato o reveue may ot be readly avalable. It therefore, becomes ecessary to have estmate-based crtera, whch could be adopted to determe the average daly passeger demad o a route as well as ts varatos agast tme ad space. Ths requres estmates of Passeger Demad alog the route Average Passeger Fare pad

10 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Ths form of most geeralzed boardg ad alghtg patter s used to estmate the demad fucto of a route over ts legth. Y f (t) () The demad for a route alog ts etre legth therefore has to be studed as a demad fucto represeted terms of a parabolc curve s the k th degree. Thus, the demad for passegers o a partcular route wth respect to the tme of the day (represeted as x the equato 2) would be a polyomal curve where the geeralzato form of the equato of the demad curve would be represeted as; 2 y a + a x + a x k a k x (2)

11 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth The Prcple of Least Squares method s used to fd out the costats a, a 2, a 3,, ad a k. To approxmate the gve set of data, (x,y ), (x 2,y 2 ),, (x,y ), where, the best fttg curve of predcted demad, y f(t), has the least square error, (.e. the resdual of equato (2)) s gve by R 2 2 k [ y ( a0 + ax + a2x ak x )] 2 Demad Per Hour D (3) Df(t) hour Tme

12 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Ths matrx equato ca be solved umercally ad the aswer gves the values for the ukow costats a, a 2, a 3,, ad a k. Cosequetly the best ftted curve of the predcted demad curve, y f(t), s avalable for further aalyss (0) + + k k k k k k k y x y x y q a a x x x x x x x x M M L M O M M L L

13 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth The average fare o a route I geeral, there are three dfferet evets volved the carrage of a passeger that s mplctly bult to the fare computato o most routes. These are: A passeger boards the bus A passeger alghts from the bus A passeger s carred past a stop or secto of the route

14 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth If the fare computato s gve terms of passeger charge for boardg (β), alghtg (α) ad for carryg past oe bus stop (γ), the reveue R of a sgle bus at a ay partcular locato o the route ca be represeted as: R ( A α ) + ( B β ) + ( X A ) + γ () Where A umber of passegers alghtg secto B umber of passegers boardg secto ad X umber of passegers carred wth secto Secto Secto 2 Secto 3 Secto -2 Secto - A, B A 2, B 2 A 3, B 3 A 4, B 4 A -2, B -2 A -, B - A,B

15 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Therefore, the Facal Reveue to all operators o the route per day s gve by: Rt k t f t D t (7) Where f t fare to travel total legth of the route ad D t total demad of passeger per trp. k t Average Reveue Rato of (,j)

16 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Valdato The 24 km log urba bus route, Paadura Nugegoda (Route # 83), s used as a case study. The aalyss uses loadg data at a md way locato. The demad fucto at ths md way pot over tme s gve as: D 0.006t 0.093t +.7t 2.26t t t

17 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth

18 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth R 96 α + 96 β γ (9)

19 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Reveue Ratos are calculated for dfferet trps durg the dfferet tmes of a traffc day for each of the locatos where roadsde loadg surveys have bee carred out. The calculato of Reveue Rato for the th trp for the three loadg survey locatos o ths partcular route s as follows; k t R ( B.& A, ) R ( B.& A, ) R ( B.& A, ), R LS- R LS-2 R + + LS-3 / (20)

20 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth The reveue of ay trp occurrg tme t o ths route ca be foud as R t k f [0.006t 0.093t +.7t 2.26t t t t t ] Summg the reveue of all trps occurrg durg the traffc day would gve the total reveue o the route. Therefore, the total route reveue to operators FRO ca be computed as: FRO k t 2 Where t f [0.006t 0.093t +.7t 2.26t t t ] t t t ad t 2 are the startg tme ad fshg tme of the traffc day.

21 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth 7 mts 0 mts

22 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth The ecoomc cost curves for bus operatos (ECO) ad passeger costs (ECP) ad the total ecoomc cost to socety (TEC) show that the optmum headway for ths route should be 7 mutes. Ths meas that the actual average headway h a o the route a s to the rght of h o. O examato of the reveue for ths route as dscussed above, the FRO curve s see to tercept the ECO curve at h f whch s the headway at whch the route operatos become facally feasble. Thus we have a codto that h a > h f > h o. Ths meas two thgs amely; (a) that fare levels are suffcet ad (b) that operators are stll makg excess profts.

23 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth Coclusos The method preseted ths paper s a exteso to Newell s optmzato method of determg optmal headways by computg reveue varatos alog a route both tme ad space order to determe the facal vablty of such a optmal headway computed as a socal cost to the ecoomy. The paper provdes a method for computg fares routes where there are multple operators ad as such, ths method may be used for ay type of bus route. A case study s used to compute these values ad the paper shows how the exstg headways s hgher tha the facally vable headway ad much hgher tha the ecoomcally optmum headway. The paper cocludes by showg the extet to whch operatoal mprovemets could be effected order to reduce watg tmes for passegers ad what reveue based chages are requred order to esure that the actual operatg headway s equal to the ecoomcally optmum headway. `

24 Thredbo 0, Australa, August 2007 Optmal Bus Dspatchg Polcy Uder Varable Demad over Tme ad Route Legth