MODELLING PUBLIC TRANSPORT CORRIDORS WITH AGGREGATE AND DISAGGREGATE DEMAND

Transcription

1 MODELLIG PUBLIC TRASPORT CORRIDORS WITH AGGREGATE AD DISAGGREGATE DEMAD Sergio Jara-Díaz Alejandro Tirachini and Cristián Cortés Universidad de Chile ITRODUCTIO Traditional microeconomic modeling o public transport (usually a single line) has been based on aggregated demand description that is total ridership along a line and average journey length. This way aggregation is spatial rather than temporal as most public transport systems are designed or the period o largest demand. With this approach various authors have ormulated optimization models whose goal is to maximize social beneit or minimize total costs (considering both users and operators) inding optimal levels or the decision variables such as requency vehicle size number o bus stops and spacing between lines. Presently it is easible to capture more precise demand patterns in cases where either the payment device or the specialized inrastructure in buses and stations can provide not only passenger counts but also exact identiication o the origin and destination o the passenger journey (RFID cards along with wide-range devices on bus doors cameras etc.). In this paper we examine the advantages o having dierent levels o aggregation regarding demand inormation in order to explore improvements on the recommendations and conclusions obtained rom classical microeconomic models. Thus we establish the optimal conditions or the relevant decision variables (requency and vehicle size) on a public transport corridor with inelastic demand in cases where the demand data is only available at an aggregated level (at the level o an entire line or ridership per direction o movement) as well as cases in which it is easible to obtain more detailed inormation on the demand structure lie origin-destination matrices at the level o bus stops or number o passenger who board and alight each bus at each stop. In the ollowing section analytical expressions or the optimal requency and vehicle size are developed or each case. A theoretical comparison is presented in section 3 regarding optimal requencies and numerical experiments are conducted in section 4. We close with some relevant comments conclusions and urther research in the inal section. SIGLE LIE MODELS WITH AGGREGATE AD DISAGGREGATE DEMAD Demand description In what ollows a linear corridor is used as a representation o a generic public transport system which can correspond to either a single isolated bus line or a line inserted within an existing networ o ixed topology (number and position o bus routes). We assume that passengers arrive randomly to the stations a reasonable assumption or high requency corridors with headways o less than 0 minutes (Seddon and May 974; Danas 980). The demand is treated parametrically in the proposed ormulations. Our purpose is to ind the

2 optimal value or the design variables (in these developments optimal requency and vehicle size K) in order to maximize the social welare o the system which in case o inelastic demand is equivalent to minimizing the total cost taing into consideration both users and operators. Users costs will include both waiting and travel time o passengers both very dependent o the line requency the latter because o the eect o boarding and alighting at stations. Access time to bus stops is not included in the system optimization since the number and location o bus stops along the corridor are assumed to be ixed. As mentioned earlier the major objective o this paper is to compare the analytical expressions obtained or the optimal design variables under dierent demand aggregation levels. First a model based upon aggregated demand is ormulated in two versions one whereby only the total cycle demand y is now (denoted as Model or M) and a second model relying on aggregated demand inormation per direction o circulation say y and y on direction and respectively (denoted as Model or M). In both cases the average journey length is assumed to be nown. Then a model relying on disaggregated demand is presented (denoted as Model 3 or M3). Unlie M and M in this case we assume that a stop-to-stop OD matrix is available. From the more detailed demand data the cost unctions can be stated in a more comprehensive way mainly the components that depend on the in-vehicle time which is modeled just as an average value by the traditional aggregated demand models in the transit microeconomic literature. Besides or the M3 speciication the line contains stations in one direction (- stretches) as shown in Figure. The operation directions are denoted direction (rom station to ) and direction (rom station to ). Figure : Generic public transport corridor In the models we describe next the ollowing parameters are assumed nown and ixed: L: Length o the corridor [m] R : Bus movement travel time under normal service between stations and + including acceleration and deceleration times at bus stops [min] : Marginal passenger boarding time [seg/pax] l : Trip rate between stations and l [pax/hour] (used or M3). This demand is assumed ixed over the studied period deining a trip matrix o the orm: 0 0

3 Additionally or M3 also the ollowing unctions are deined: Passenger boarding rate at station whose destination is among stations l and l + inclusive [pax/hour]: ( l l ) l l l l Passenger alighting rate at station whose destination is among stations l and l inclusive [pax/hour]: ( l l ) l l l Thus rom these unctions we can deine the ollowing quantities: + + Passenger boarding rate at station direction : ( ) l + l l + Passenger alighting rate at station direction : ( ) l l + + Passenger boarding rate at station direction : ( ) l l Passenger alighting rate at station direction : ( ) + l l + We assume that the boarding process dominates over the alighting process and thereore in the model only the irst phenomenon (quantiied through the parameter ) is considered. From these deinitions we can relate the demand deined in the context o the three models as ollows ( + + ) y y + y + Besides the vehicle arrival distribution to the stations is crucial or a correct computation o the passenger waiting time which decreases as the headways become more regular. Following Delle Site and Fillipi (998) two possible bus arrival patterns to stations are considered namely scheduled service (regular headways) and random bus arrivals (Poisson process). The ormer is associated with systems with low variability in both running and passenger transer times at bus stops or instance special bus segregated corridors with eicient transer operations at stops. In this case the average waiting time turns out to be hal o the headway. Random arrivals are characteristic o networs with high variability in running times (poorly controlled bus systems); in this case the expected waiting time is equal to the average headway. Recalling that the pursued objective is to minimize the total system cost expression with respect to the relevant design variables (requency and vehicle size) next we deine analytically the dierent cost components considered or the three models rom both the users and operator standpoints. The ormer comprises the waiting and in-vehicle time costs while the latter comprises two operational cost expressions associated with distance and time respectively. ()

4 Users costs The waiting time cost (C w ) is considered as the product between the total expected waiting time experienced by all customers and the subjective value o waiting time (P w ). I x is an auxiliary binary variable (equals to i buses arrive Poisson 0 i buses arrive at constant headway as discussed earlier) we can compute the waiting time cost component as ollows: + x y Cw Pw () where is the operational requency computed as the inverse o the headway. Expression () applies to all models regardless o the demand aggregation level applying expression (). The in-vehicle time cost (C v ) is modelled as the product between the total expected in-vehicle time and the subjective value o the in-vehicle time (P v ).Unlie the waiting time component C w C v adopts a dierent orm depending on the demand aggregation level. For the M model in-vehicle travel time is expressed as a raction o the total cycle time i.e. the ratio between the average journey length l and the total route length L (Mohring 97; Jansson 980; Jara Díaz and Gschwender 003). On the other hand or the M model it is possible to split this component per direction as the average journey length on direction i (namely l i ) o journey over the route length (L). Moreover the cycle time t c is computed as the sum o the running time by direction (denoted as R i ) and the total stopping time at stations (computed as the total expected passenger boarding time). The latter component is computed as the product between the average number o passengers boarding a vehicle yi / and the marginal boarding time. Analytically or M: l y Cv Pv R R y L β + + (3) and or M l y l y Cv Pv β + Ry + β + Ry (4) L L For M3 no approximation is needed; travel time t l or each OD pair (l) is given by l + i Ri + β si < l i tl (5) + i Ri + β sil < i l+ Then multiplying (5) times l adding over all OD pairs and multiplying by P v. the total invehicle travel time cost is obtained in monetary units. Analytically l + + i i Cv Pv Ri + β l + Ri + β l (6) l + i l i l+

5 Operator cost denote c( K ) as the cost per vehicle-hour ($/veh-h) and ( ) In this computation we tae into account two components or the operator cost as some items are better represented on a temporal basis (labor) and others over a spatial basis (running cost maintenance etc.). Following Jansson (980) and Oldield and Bly (988) a linear dependency on the vehicle capacity K is assumed or the operator cost unctions. Let us c K as the cost per vehicleilometer ($/veh-m). Analytically ( ) ( ) c K c + c K c K c + c K (7) 0 0 Thereore the operator cost can be expressed as: o ( ) '( ) C c K F + c K v F where v is the commercial speed and F is the leet size given by requency times cycle time tc discussed in section.. Thus (8) can be rewritten as a unction o t c and as ollows Co c( K ) tc + c' ( K ) L (9) and inally y Co c( K ) R R c ( K ) L β (0) which applies or the three models. Optimal Value o the Frequency and the Vehicle Capacity. The total cost minimization problem comprises the joint minimization o both users and operators costs encompassing equations (); (3) (4) or (6); and (0). In order to ind the optimal value o the variables and K irst order conditions (FOC) are applied. The vehicle capacity is adjusted in order to accommodate the demand o the most loaded segment along the corridor q max which can be easily obtained rom the OD matrix in M3. For models M and M this value must be assumed to be nown (or accurately estimated). Then by deining a saety actor η ( 0] and computing K qmax η the FOC yield the ollowing values or the optimal requency or M M and M3 respectively: * * + x l qmax Pe y + Pv β y + c β y L η c0 ( R + R ) + c0l + x l l qmax Pe y + Pv β y + y+ c β y L L η c0 ( R + R ) + c0l (8) () ()

6 * l + x + + qmax Pe y + Pv β l i l i + + c β y l + i l i l+ η c 0 R + c0 L (3) AALYTICAL COMPARISO By looing at expressions () () y (3) we can observe that the only dierence in the optimal requency values appears in the term associated with the in-vehicle travel time within the square root because in-vehicle time cost is quadratic with the demand. Thereore the relevant comparison involves l y (a) L l l y + y (a) L L l + + l i + l i l + i l i l+ (3a) ote irst that in systems where the stop time at stations is ixed the three ormulae (equations and 3) provide the same result. On the other hand by writing the average journey time length as a unction o the disaggregated quantities we obtain L l l l y ( ) l + ( ) l ( ) L l l l y ( y y )( ) ( ) L l l ( l ) + l ( l) + l + l First we ocus our analysis in models M and M. Let us dl i ote D i the total traveled distance along direction i that is D i l i y i that yields L D l ( ) l l + ( ) l l ( ) L D l ( ) Then (a) and (a) can be rewritten as ollows l D + D y ( y + y ) L L l l D D y + y y + y L L L L Comparing these two expressions is equivalent to analyzing the sign o the expression y y D D. Then expression () will be larger than () i either ( )( ) ( y > y ) ( D > D ) or ( y y ) ( D D ) < < which is equivalent to say that the traveled

7 distance is the largest on the smallest demand direction o movement. On the other hand i the largest demand direction matches the largest traveled distance (which is a very reasonable intuitive assumption) the most aggregated model (expression ) underestimates the optimal requency compared with that obtained rom the model that dierentiates both directions behind expression (). ow let us compare ormulae (a) and (3a) i.e. M and M3. By writing (a) in a disaggregated way we have: l l D D + l + l y + y y + y l l L L L L l + + l The previous expression must be compared against (3a). A priori it seems not possible to perorm any comparison between both expressions under a generic situation. In order to approach to the solution let us to examine some interesting particular cases. First let us examine the case o equal trip rates in each direction that is l i < l i > l In such a case (a) and (3a) yield the same result given by ( ) ( ) + (4) Moreover expression (a) becomes ( ) ( ) 4 + (5) ote that (5) is always lower or equal than (4). Besides in this case the average length o trip is the same in both directions. Let us now see the case in which the number o stations equals three (3). In this case (a) and (3a) become respectively (by simplicity we analyze only direction ): 3 ( ) (b) (3b) ( ) ( ) The comparison then is reduced to + (c) (3c).

8 The relative value o the trip rates will determine the value o the optimal requencies. Two particular cases: a) I 3 (c) is equivalent to (3c) and both optimal requencies turn out to be the same. b) I 3 3 and 3 then i > (c) is larger than (3c) and consequently () is larger than (3); the other case is analogous. Thereore we can not establish a priori a raning among optimal requencies obtained rom the dierent models. That raning mostly depends upon the value o the matrix cells. Apparently the more heterogeneous the matrix cells become the higher the probability o getting dierent values or the optimal requencies rom the various proposed models. Thereore it seems clear that the concentration o trips is a relevant issue when comparing the analytical recommendations obtained rom the dierent aggregation level models. In the next section we complement these analytical insights with the conclusions rom some numerical examples. UMERICAL COMPARISO In this section we conduct some numerical computations o the optimal design variables (requency vehicle and leet size) obtained by applying the dierent demand aggregation models (M M and M3). We concentrate our analysis on two numerical cases. One is the study o a public transport corridor in Santiago Chile called Los Pajaritos rom where we have origin-destination demand matrix or the most demanded morning pea hour on a typical day o operation (MTT 998). Los Pajaritos is a corridor o 7 m. with 9 segments and 0 stations along each direction. The second example is the experiment proposed by Delle Site and Phillipi (998) where also a detailed station to station origin destination demand matrix is available. The matrices used in both examples were properly generated rom real data o aluence at the level o stations. In both examples the assumed parameters are those shown in Table next. Table : Summary o the assumed model parameters Parameter Value c0 c β Saety actor η!"#$%&"'$()*+ "'$

9 Example : Corridor Los Pajaritos The morning pea hour OD matrix and the associated load proile are shown in Table and Figure respectively. Load imbalance between both directions o movement is evident. Results are summarized in Table 3. Models M and M3 show similar results while M clearly underestimates not only the optimal requency but also the optimal leet size. Table : OD matrix Los Pajaritos corridor Load [pax/h] Load proile Direction Direction Station Figure : Load proile Los Pajaritos corridor Table 3: Optimal design variables Los Pajaritos corridor Model Regime Frequency [veh/h] [veh] Fleet size Cap veh [pax/veh] M M Scheduled M M M Poisson M

10 Example : Delle Site and Phillipi (998). The morning pea hour OD matrix and its load proile are shown in Table 4 and Figure 3. In this case demand imbalance between directions is concentrated in only a group o stations. Results or M M and M3 are summarized in Table 5 which shows that optimal requency increases with the level o demand o inormation available suggesting that lesser inormation will result in an underestimation o the optimal requency. The leet size remains unchanged. As the dierence among models only happens in travel time Table 6 shows the results o a sensitivity analysis increasing both travel time value Pv to 800 $/h and travel time between stations to 3 minutes which could be caused by traic congestion. The dierence in optimal requency becomes larger and the optimal leet size is now dierent or the three models. Table 4: OD matrix Delle Site and Phillipi (998) example Load [pax/h] Direction Direction Load proile Station Figure 3: Load proile Delle Site and Phillipi (998) example

11 Table 5: Optimal design variables Delle Site and Phillipi (998) example Frequency Fleet size Cap veh [pax/veh] [veh/h] [veh] M Sched M Sched M3 Sched M Poisson M Poisson M3 Poisson Table 6: Modiied optimal design variables Delle Site and Phillipi (998) example Frequency Fleet size Cap veh [pax/veh] [veh/h] [veh] M Sched M Sched M3 Sched M Poisson M Poisson M3 Poisson In both numerical examples the optimal requencies obtained rom M were systematically smaller than those obtained rom M and M3; and these latter yield dierent results in the second example only. As a general rule it seems that the better represented is transit demand the larger the optimal requency and the smaller vehicle size. This maes even more dramatic Jansson s (984) observation regarding the underestimation o optimal requency and overestimation o vehicle size when users cots are not taen into account considering that he was using an M type model. Both the analytical developments and the numerical examples suggest that the larger the cost associated with in-vehicle travel time the more liely is that the various aggregated models predict dierent (smaller) values or the optimal requencies. COCLUSIOS In this paper we examine the advantages o having detailed demand inormation when using classical public transport microeconomic models. We have establish the optimal conditions or requency on a public transport corridor with inelastic demand in cases where the demand data is only available at an aggregated level (at the level o an entire line or ridership per direction o movement) as well as cases in which it is easible to obtain more detailed inormation on the demand structure lie origin-destination matrices at the level o bus stops or number o passenger who board and alight each bus at each stop. We have developed two analyses one which is purely analytical and another based on the result o applying the dierent aggregation level models to two examples in which real public transport data or pea periods are available. From the analytical part we clearly identiied those terms in the optimal requency expression that remars the dierences among the models. However some conclusions with respect to the raning among requencies obtained in each model could only be obtained rom the analysis o the empirical results. This

12 comment motivates a deeper analysis o the analytical expressions and also and more importantly more numerical examples to validate our conclusions by discarding some possible cases that could be imposed numerically in the analytical developments but not very liely to occur in reality. Overall however our results suggest that the underestimation o optimal requency and overestimation o vehicle size when not accounting or users costs is even more important than predicted by Jansson (984). ACKOWLEDGEMETS This research was partially inanced by Fondecyt Chile grants 066 and and the Millennium Institute "Complex Engineering Systems. REFERECES Danas A. (980) Arrival o Passengers and Buses at Two London Bus-stops. Traic Engineering and Control (0) Delle Site P.D. and F. Filippi (998).Service Optimization or Bus Corridors with Short-Turn Strategies and Variable Vehicle Size. Transportation Research A 3() 9-8. Jansson J. O. (980) A Simple Bus Line Model or Optimization o Service Frequency and Bus Size. Journal o Transport Economics and Policy Jara-Díaz S.R. and A. Gschwender (003) Towards a General Microeconomic Model or the Operation o Public Transport. Transport Reviews 3(4) Mohring H. (97) Optimization and Scale Economies in Urban Bus Transportation. American Economic Review Oldield R.H. and P.H. Bly (988) An Analytic Investigation o Optimal Bus Size. Transportation Research B Seddon P.A. and M.P. Day (974) Bus Passenger Waiting times in Greater Manchester. Traic Engineering and Control 5(9)