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M. Medina-Tapia, R. Giesen and J.C. Muoz 1

1

A MODEL FOR THE OPTIMAL LOCATION OF BUS STOPS AND ITS2

APPLICATION TO A PUBLIC TRANSPORT CORRIDOR IN SANTIAGO345

6Marcos Medina-Tapia7

Departamento de Ingeniera Geogrfica, Universidad de Santiago de Chile8Enrique Kirberg Baltiansky 03, Estacin Central, Chile; Tel: (+562) 718 2206718 22309

[email protected]

Ricardo Giesen *13Departamento de Ingeniera de Transporte y Logstica, Pontificia Universidad Catlica de Chile14

Av. Vicua Mackenna 4860, Macul, Chile; Tel: (+56 2) 354 [email protected]

* Corresponding author171819

Juan Carlos Muoz20Departamento de Ingeniera de Transporte y Logstica, Pontificia Universidad Catlica de Chile21

Av. Vicua Mackenna 4860, Macul, Chile; Tel: (+56 2) 354 [email protected]

24252627

Word Count: 6,185 plus 1 Table and 4 Figures = 7,4352829303132333435363738

July, 201239 revised November, 201240414243

Submitted for presentation at the 92ndAnnual Meeting of the Transportation Research Board44January 2013, Washington D.C., and publication in Transportation Research Record45

46

TRB 2013 Annual Meeting Paper revised from original submittal

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A MODEL FOR THE OPTIMAL LOCATION OF BUS STOPS AND ITS APPLICATION1TO A PUBLIC TRANSPORT CORRIDOR IN SANTIAGO2

34

Marcos Medina-Tapia, Ricardo Giesen and Juan Carlos Muoz5

678

ABSTRACT910

The location and number of bus stops are key to the operational efficiency of the services that use11them, affecting commercial speed, reliability, and passenger access times. In defining the12number of stops, a tradeoff arisesbetween reduced access time, which widens a routes coverage13area, and both the operational speed of the route and users in-vehicle travel time.14

15The objective of this paper is to present the development of a model for optimally locating stops,16and applying it to a public transport corridor in the city of Santiago, Chile. The proposed model17employs a continuous and multiperiod approximation of corridor demand, allowing for the18determination of the density of stops which minimizes the sum of operator costs and total costs to19passengers. The model simultaneously solves for the optimal stop density and the headway20between successive buses.21

22The proposed model was applied to the Grecia Avenue corridor (in Santiago, Chile). Finally, the23actual stop locations were compared with the optimal locations suggested by the model, and24many similarities were found.25

2627

KEYWORDS:Public Transport, Location Models, BRT, Continuous Approximation Method.2829


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1 INTRODUCTION AND OBJECTIVE12

The location and number of bus stops in a corridor have a large influence on the operational3efficiency of service (commercial speed, reliability, passenger walking time), with a tradeoff4existing between a lower access time (or equivalently, a larger coverage area) and the routes5

speed (Vuchic, 2007).67The stop location problem has been approached in the literature through various focuses that can8be classified along two dimensions.9

10One dimension is the methodology employed: integer programming models, simulation, or11analytic models. In the first category, the location is generally determined from a set of12predetermined candidate points (e.g. Drezner and Hamacher, 2002; Murray, 2003; Murray and13Wu, 2003; Bruno et al., 2002; Laporte et al., 2011). A second category includes simulation14methods that allow for the determination of the number and spacing of stops (e.g. Fitzpatrick et15al., 1997; Alterkawi, 2006). Third are analytical models based on the continuous approximation16approach (see Daganzo, 2005), which usually determine the optimal distance between stops17(Wirasinghe and Ghoneim, 1981; Vuchic, 2005), or which determine the minimum number of18stops in a transit network considering critical distances to residential areas (Ceder, 2007).19

20A second dimension discussed by Chien and Qin (2004) classifies past work by the complexity of21the problem approached: models that use simplified demand distributions; models that optimize22operation and stop distance simultaneously; and models that consider varying demand across23time. The first category of studies is focused on locating stops by using simple demand24distributions that do not vary in time (Wirasinghe y Ghoneim, 1981; Laporte et al., 2002). In the25second category are studies that focus on the joint optimization of service design and stop26spacing, such as in Kocur and Hendrickson (1982), Kuah and Perl (1988), Gibson and Fernndez27(1995), Chien and Schonfeld (1997), and Chien and Quin (2004). Third are studies that optimize28the public transport system while considering that demand varies in time. Work along these lines29has been presented in Hurdle (1973), Clarens and Hurdle (1975), and Chang and Schonfeld30(1991). However, only Hurdle (1973) and Chang and Schonfeld (1991) solve the problems of31bus stop location of bus stops and headways simultaneously.32

33The objective of this paper is to present the development of a simultaneous optimization model34for stop location and headways for a service with multiperiod demand based on a continuous35approximation model. For this, a demand profile of passengers who wish to board and alight is36defined at each point along the corridor. The model considers two variables: the density of stops37in the vicinity of each point of the corridor and the frequency of operation in each time period.38The objective is to minimize the sum of total passenger and operator costs. Increasing the39density of stops reduces walking time, but it increases travel times, fleet requirements, and40infrastructure investment and maintenance costs.41

42The proposed stop location model is presented in Section 2 below. In Section 3, the main43analytical results of the model are presented. In Section 4, the application of the model to the44Avenida Grecia corridor is discussed. Finally, Section 5 summarizes the primary conclusions of45this investigation.46

47


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2 PROPOSED OPTIMIZATION MODEL12

2.1 System Studied34

The system modeled consists of a corridor of length L in which buses travel in both directions5

and stop at a set of predetermined stops according to their schedules. It is assumed that the buses6 have a fixed capacity, belong to only one service, and are dispatched from terminals at the ends7of the corridor according to the frequency of the service, which varies according to the period (i8= 1m periods) and is equal in both directions.9

102.2 Modeling Approach11

12The modeling approach adopted in this study involves a continuous analysis of the corridor, that13is, certain model variables and parameters are represented through continuous functions that vary14depending on the position, x, in the corridor. The model seeks to optimize the stop density15(

, where

is the direction), minimizing both the passenger costs as well as the operator and16

infrastructure costs, while also determining the optimal frequency for each period of analysis.1718

The main assumptions of the model are the following:1920

Only one route travels on the corridor, and it stops at all stops.21 The user costs are composed of walking access times, waiting time, and in-vehicle travel time.22

For this analysis, the weights of three components are assumed to have the same value for all23users, regardless of age, socioeconomic status, period of the day, etc.24

Passenger demand at each point of the corridor during a given period is described by a known,25deterministic function that varies according to position.26

We consider a high frequency service in which there are no timetables. Thus within each27

period passenger arrivals are random, i.e. follow a Poisson process with demand rate specified28 at each point.29 The buses have sufficient capacity to transport all users, and users always board the first bus30

that passes.31 It is assumed that no bus congestion occurs at stops, so each bus opens its doors as soon as it32

arrives at a stop.3334

2.2.1 Model Parameters and Variables3536

In all of the following definitions, the location, x, is defined between 0 and L, the direction, r, is37defined for each of the two directions, and iis defined for each one of the mperiods. The units in38

which each of these variables is measured are also included below.3940 : Number of passengers who would like to board at x, in direction r, in period i41 [passengers/Km-Hr]42 : Number of passengers who would like to alight at x, in direction r, in period i43[passengers/Km-Hr]44 : Cumulative number of passengers who have alighted from the bus between the start45of the route and a pointxin direction r [passengers/Hr]46


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M. Medina-Tapia, R. Giesen and J.C. Muoz 5 : Cumulative number of passengers who have alighted from the bus between the start1of the route and a pointxin direction r [passengers/Hr]2 : Total user cost in the corridor [$/day]3

: Total operator cost in the corridor [$/day]4

: Total access and egress cost for users walking to the nearest stop in the corridor5

[$/day]6 : Total cost of in-vehicle travel time in the corridor [$/day]7 : Total cost of waiting time in the corridor [$/day]8 : Total vehicle operating cost in the corridor [$/day]9 : Total installation and maintenance cost for stops in the corridor [$/day]10 : Passenger load for buses at pointx in direction r of the corridor [passengers/Hr]11 : Average walking speed for bus stop access and egress [Km/Hr]12 : Cruising speed of a bus in period i[Km/Hr]13 : Duration of period i [Hr], where , the span of service of the route operating14on the corridor15

: Average fixed dead time per stop (to open and close doors, etc.) [Hr/stop]16

: Boarding time per passenger [Hr/passenger]17 : Alighting time per passenger [Hr/passenger]18 : Acceleration of a bus leaving stops [Km/Hr2]19 : Deceleration of a bus entering stops [Km/Hr2]20 : Value of walking access time [$/passenger-Hr]21 : Value of in-vehicle travel time [$/passenger-Hr]22 : Value of waiting time [$/passenger-Hr]23 : Vehicle cost per unit distance covered at cruising speed [$/veh-Km]24 : Fixed cost per bus [$/veh-day]25

: Hourly salary of drivers [$/driver-Hr]26

: Additional operating cost (beyond the cost of operating at cruise speed) for27

acceleration and deceleration per stop [$/stop]28 : Cost of idling at a stop [$/veh-Hr]29 : Fixed cost of installation (or construction) of a stop [$/stop-day]30 : Operation and maintenance cost of a stop [$/stop-Hr]3132

2.2.2 Decision Variable3334

There are two decision variables in the model: the density of stops per kilometer of the corridor in35direction r, , which is a continuous function of x(); and the headway, ,or time between36successive buses in each period of operation i.37

382.2.3 Demand Functions39

40In the continuous modeling approach used, it is assumed that a stop can be located at any point41along the corridor (Vuchic, 2005). Accordingly, boarding passengers are represented by 42and alighting ones are represented by for period i in direction r; these are continuous43density functions in units of passengers per unit length (Km) and time (Hr), as shown in Figure441(a). In Figure 1(b), the cumulative number of passengers who have boarded and who45have alightedbetween the start of the corridor and pointx; these can be calculated directly46


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from and . Finally, Figure 1(c) shows the passenger load profile (), which is1calculated as the difference between the total number of passengers who have boarded and2alighted between the start of the corridor and pointx (Vuchic, 2005).3

4

Figure 1. Passengers who board and alight, and the load profile of a corridor.5Source: Vuchic (2005).6

7

82.3 Cost Functions9

10The total cost () is the sum of daily costs incurred by users () and the operator (). The11former is detailed in Equation 2 and is the sum of stop access and egress costs (), waiting costs12(), and in-vehicle travel time costs () of passengers who use the route. The second cost13function, presented in Equation 3, corresponds to operator costs, composed of fleet operating14costs () and stop installation and operation costs ().15

16

( 1)

17

( 2)18 ( 3)19

bri(x)

ari (x)

Bri (x)

Ari (x)

x [Km]

x [Km]

x [Km]

Densidad

[pax/Km-Hr]

P.Acumulado

[pax/Hr]

Carga

[pax/Hr]

x

Pri (x)=Bri (x) - Ari (x)

(a)

(b)

(c)

Load

[pax/hr]

CumulativePassengers

[pax/hr]

Density

[pax/km-hr]


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2.3.1 User Costs12

a) Access Cost34

For the access cost (

), it is assumed that the walking access and egress stages are comprised of5

two components: one perpendicular to the corridor (corridor approach), which is independent of6 stop locations, and one parallel to the corridor (stop approach), which is part of the cost function7to be minimized. The latter walking distance is approximated as , where dr(x) is the8 stop spacing in direction r in the vicinity of point x, which is equivalent to the inverse of the9density between stops.10

11 ( 4)12

Accordingly, represents the expected walking distance for a passenger. By multiplying13by the inverse of walking speed (

), the expected walking time is obtained. If the expression is14

then multiplied by both the number of passengers who access and leave the corridor at point x15and the value of walking access time (), the estimated total access cost for the corridor is as16 follows:1718 ( 5)

19b) Waiting Cost20

21The total cost of waiting (

) represents the monetized value of passengers waiting time. For22

this we assume that there is always available capacity in the buses, which allows for the23

additional assumption that the waiting time of users during period i grows linearly with the24average headway between buses, . This linear relationship is governed by a parameter which25 describes the routes regularity. If the headways between buses are perfectly regular, the26parameter takes a value of 0.5, and if the bus arrivals occur according to a Poisson process, it27takes a value of 1. The following equation shows the total waiting cost.28

29

( 6)30

c) In-Vehicle Cost3132 The total in-vehicle cost () represents the in-vehicle costs experienced by all of the corridors33 passengers. This cost is composed of three costs, as can be seen in Equation 7.3435 ( 7)36


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Where:1 : Cost associated with the travel time when the vehicle is at cruising speed2[$/day]3

: Cost associated with the travel time when the vehicle is accelerating from or4

decelerating for a stop [$/day]5

: Cost associated with the time stopped for the boarding and alighting of6 passengers at each stop [$/day]78 In this analysis we assume that the stops are sufficiently spaced for the bus to reach its cruising9speed in each inter-stop interval. The cost associated with time aboard the bus while the bus is10traveling at a constant cruising speed can be expressed as the sum along the corridor of the time11experienced by people onboard the bus at each point. This is formulated as the product of the12load of the bus in pointx integrated over the length of the segment, the duration of the period of13time, the inverse of the cruising speed of the bus, and the value of time of the users:14

15

( 8)

16The additional cost for time lost accelerating and decelerating can be approximated as the time17lost at each stop () multiplied by the quantity of people who experience it, which can be18expressed in the following form:19

20

( 9)21

The extra time lost at each stop has components for bus deceleration (

) and acceleration22

( ) :2324 ( 10)25

Third, once the bus is stopped, all passengers aboard experience a delay equal to the time it takes26for other users to board and alight ():27

28

( 11)

29This dwell time, , is related to the number of passengers who board and alight, which can30 be modeled in two ways: parallel or sequential boardings and alightings (Fernndez y Planzer,312002). The parallel approach is normally used to consider an onboard fare payment system in32which one door is used for boardings and one or more doors are used for alighting and the dwell33time is governed by the whichever process takes longer to be completed (Equation 12). The34sequential approach is used for corridors with pre-payment on the bus platforms, so all of the35buses doors are shared by boarding and alighting passengers. Accordingly, the dwell time36


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consists of the sum of boarding and alighting times (Equation 13). Clearly, in both cases the1parameters bri and ari will vary depending on available fare payment technologies and the2physical design of the bus boarding points.3

4

(

) ( 12)

5 ( 13)62.3.2 Operator Costs7

8The operator cost function is composed of the vehicle operating costs () and the costs of9installing and maintaining costs ().10

11a) Vehicle Operating Cost12

13The formulation of the cost associated with vehicles (

) is based on the multiperiod model14

presented by Fernndez et al. (2005), who decompose it into fleet requirement costs, driver15salaries, and vehicle operation costs according to the following expression:16

17 ( 14)18

Where:19 : Cost associated with the fleet [$/day]20 : Cost associated with driver pay [$/day]21

: Cost associated with the distance traveled by buses at cruising speed [$/day]22

: Cost associated with the distanced traveled by buses accelerating from or23

braking for bus stops [$/day]24 : Cost associated with time spent idling at bus stops [$/day]2526The cost of the fleet () is the product of the fixed unit cost per bus () and the number of27buses required for peak period service, which determines the fleet size required.28

29 ( 15)30

Where:31

: Required fleet size to maintain an average headway, h, between buses [veh]32

: Peak-period headway [Hr], such that 3334 The fleet size () required for operation corresponds to the quotient of the cycle time ( ) and35the peak period headway (), that is, . The cycle time of period i () is36related to the sum of the time spent traveling the corridor (in both directions) at cruising speed37( ), accelerating and decelerating at each stop ( ), opening and closing doors, and38allowing for the boarding and alighting of passengers in each stop ( ). That is:39


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1

( 16)2

To estimate driver costs (

), we assume that a fixed wage,

,is paid for each on-duty hour of3

work, independent of the structure of shifts required to cover the required service.45 ( 17)67

On the other hand, the cost of operating the vehicle at cruising speed in each period equals the8product of the number of cycles ( ), the distance covered per cycle at this speed ( ) and9the unit cost per kilometer traveled (

):10

11

( 18)12

The additional cost for acceleration from and deceleration for stops can be expressed as:1314

( 19)

15Finally, the cost for idling while passengers board and alight is expressed in the following:16

17 ( 20)18

b) Stop Cost1920

The cost associated with each stop consists of a fixed cost (scaled to an equivalent daily cost) for21installation () and a variable cost for operation and maintenance ( ), resulting in the22following equation:23

( 21)242.4 Optimization of the System25

26The preceding expressions allow for the formulation of the following multiperiod model, which27minimizes the total costs of the system as a function of the stop density in the vicinity of each28pointxand for each direction r and the headway for each period ().29


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1 {

} { } ( 22)2

Subject to:3

( 23)

4 ( 24)5 ( 25)6 ( 26)7

The objective function consists of minimizing the total daily costs, represented by the sum of the8costs of users and operators presented above. The constraints of the model are of three types: bus9

capacity, stop capacity, and equivalence of stop density in each corridor direction.1011The first set of constraints (23) requires that, for all points and time periods, the capacity12provided be sufficient to transport the demand as specified in the load profile (). In this13expression, corresponds to the capacity of one bus [users/veh]. It is important to keep in14mind that the waiting time model assumes that users can always board the first bus to arrive. In15the case of irregular headways this constraint would also require that the capacity provided be16greater to product of the load profile and the maximum headway, rather than the average17headway. So, in cases when the model returns solutions in which this constraint is binding (or18has very little slack), the assumption that passengers board the first bus to pass may be violated19and the result should be interpreted as one in which waiting time experienced by users has been20underestimated.21

22Along the same lines as the previous constraint, in the event that stops have a limited capacity of23users, ,the stop density for direction of the corridor must ensure sufficient capacity24so that users who want to board or alight at a stop can do so. This is required by the second set of25constraints (24).26

27


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An optional model constraint is to require that the density of stops be equal in both directions of1the corridor (25). Finally, there are non-negativity constraints (26).2

34

3 ANALYSIS OF THE PROPOSED MODEL56 The model proposed in the preceding section entails nonlinear optimization. Given its7

complexity, two solution phases with different goals are proposed: in the first, the optimal8headway is obtained, and in the second, the optimal stop density is obtained.9

103.1 Phase I: Optimal Headway11

12Two alternative procedures are suggested to determine the optimum headway from the proposed13model: replacing the variable delta with its first order conditions from the optimization problem,14and an iterative process with the analytical expressions for headway and stop density.15

16The first procedure consists of solving the model by replacing the stop density decision variable17with the analytic expression obtained from the first order conditions of the optimization problem,18which depends on the position in the corridor, , and the headway , that is, .19 This expression is discussed in detail below. Through this approach, the model is transformed20into a problem that has variables (according to the number of periods).21

22The second procedure consists of iterating the analytic expressions for headway and stop density23as functions of one another until reaching convergence. First, a relatively low frequency is24assigned as the initial headway. Then, the headway calculated from the preceding step is25iteratively replaced in the analytic expression for stop density by satisfying the applicable26constraints; the new value of the stop density then replaces the previous value in the equation for27the optimal headway, subject to the corresponding constraints. This is completed when the28headway reaches convergence within a specified tolerance. Once the optimal headway is found,29the stop density for each point of the corridor x is then found as part of Phase II, as explained30further below.31

32Next are presented the analytical expressions for the first order condition of the headway33variable, considering the two cases of distinct or equal stop densities in the two corridor34directions.35

363.1.1 Different Density for Each Direction37

38For this case, Equation (27) shows the optimal headway for the peak period (

), while39

Equation (28) shows this expression for the other periods.40

( 27)

41


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( 28)

1

In Equations (27) and (28), the value of if the stop time is governed by2 boardings; otherwise the stop time is proportional to boardings and alightings, and 3 . These expressions are subject to the constraint represented by Equation4(23).5

63.1.2 Equal Density for Both Directions7

8For the case of equal stop density for both corridor directions, the following constraint must be9incorporated into the model:10

( 29)

11

For this case, Equation (30) shows the optimal headway for the peak period ( ), while12 Equation (31) shows this expression for the other periods.1314

( ) ( 30)15

( )

( 31)In Equations (30) and (31), the value of will depend on whether the stop time is governed16by boardings or is proportional to boardings and alightings, subject to the constraint represented17by Equation (23).18

193.2 Phase II: Optimal Stop density20

21Once the optimal headway is determined, it is possible to establish the optimal stop density,22which, in this phase, only depends on the position along the corridor. Each case is considered23below.24

253.2.1 Different Density for Each Direction26

27The specified total cost function is minimized by setting the derivative with respect to equal28to zero and solving for the optimal stop density.29

30It is interesting to note that modeling the system by using a continuous approximation allows for31a solution for each point x independently of the other points of the corridor. That is, for the32


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expression to be minimized, each integrand must be minimized. So it is sufficient to minimize1

the expressionfor the total cost function .23

Therefore, Equation (32) describes the form of obtaining the optimal stop density through the4first derivative of the total cost with respect to the variable

and setting the resulting5

expression to zero.67 ( 32)8

The density of stops for the multiperiod case is presented in Equation (33), which is subject to the9constraint of Equation (24).10

11

(

)

( 33)

123.2.2 Equal Density for Both Directions13

14Equation (34) is the expression for the case in which the stop density is equal in both directions15subject to the constraint of Equation (24).16

17 ( ) ( ) ( ) ( ) ( ) ( ) ( 34)18

The proposed model was applied to the Avenida Grecia corridor, located in the east sector of19 Greater Santiago, Chile.2021

4 APPLICATION TO CASE STUDY2223

4.1 Description of Case Study2425

The corridor selected for the case study, Avenida Grecia, is located in the eastern sector of26Greater Santiago, crossing the municipalities of uoa and Pealoln, with a length of 10.427kilometers. There are currently 22 stops in the westbound direction of the corridor, and 21 in the28eastbound direction (Figure 2).29

30


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1(a)

(b)

Figure 2. Spatial distribution of stops along Avenida Grecia for the: (a) westbound2direction and (b) eastbound direction.3

456

One set of parameters used in the model consists of the continuous functions of passenger7boardings and alightings. These were obtained through the database of the DICTUC Study8(2010), with counts of boardings and alightings made in December, 2009 and March, 2010.9These profiles are included in the Appendix.10

11For stop access and egress walking speed, 3.6 [Km/Hr], or 1 [m/s], a standard value within12transportation engineering studies, was used.13

14

For bus acceleration and deceleration, the values suggested by the American Association of State15 Highway and Transportation Officials, 1994 in Saka (2001) were used. AASHTO suggests an16acceleration of 0.5 [m/s2] and a deceleration of 2 [m/s2].17

18The time lost when the bus is at a stop can be decomposed into a fixed stop time and boarding19and alighting times, the latter of which are proportional to the number of passengers who20complete each action (see Fernndez et al. (2009; 2010)). In Fernndez et al. (2010), onboard21fare payment for trunk routes is considered, with resulting average values of 7.06 [seconds/stop]22


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for fixed stop time, 1.55 [seconds/user] for boarding and 0.99 [seconds/user] for alighting. In the1case of fare prepayment, the resulting average values are 5.77 [seconds/stop] for fixed stop time,21.74 [seconds/user] for boarding and 1.26 [seconds/user] for alighting.3

4For the value-of-time parameters, the values determined in Raveau (2009), which were calculated5

for Santiago, were used. These values are 4.09 [US$/user-Hr] for walking time, 2.73 [US$/user-6 Hr] for waiting time, and 1.64 [US$/user-Hr].78

Without more detailed information available, it was assumed that the cost of bus acceleration and9deceleration () is equal to the cost incurred when traveling a unit distance at cruising speed10(). It was also assumed that the idling cost for a vehicle ( ) is zero, as in Fernndez et al.11(2005).12

13For stop construction costs, information presented in the SECTRA Study (2002) was used. This14study states that construction costs are US$52,200 for a Salamanca-style stop and the cost of a15high-capacity stop is US$228,400.16

17 Lastly, a minimal value is used for stop maintenance because stops are not managed by a18company, except for those that have fare prepayment. Specifically, maintenance cost for each19stop is assumed to be one person who earns minimum wage and cleans for one hour each day.20

214.2 Modeling Results22

23Table 1 shows the both the currently observed headways and the optimal headways suggested by24the model, using 3 periods (morning peak, evening peak, and off-peak). As seen in the table,25differences between the observed and suggested values range from 1.1 minutes in the morning26peak (which corresponds to 65%) up to 5.8 minutes for the evening peak (which corresponds to27232%).28

29Table 1. Observed and Optimal Headways30

31Period Observed Headways

[min]

Optimal Headways

[min]

Morning Peak 1.7 2.8Evening Peak 1.8 3.1Off-Peak 2.5 8.3

32Figure 3 shows the result of the discretization of the stop density function using 3 periods for the33

case of the corridor in the westbound (Figure 3.a) and eastbound (Figure 3.b) directions. The34 circles on the stop density curve represent the stops resulting from the discretization and the35vertical lines delineate the coverage area of each stop, formed so that the areas they define with36the curve and the x-axis will all have a unit area and contain a stop. On the horizontal axis of the37figure are plotted circles that represent the location of the corridors currently existing stops. As38a reference, the figure includes the position of relevant corridor landmarks such as the39Municipality of Pealoln (P), intersection with Avenida Tobalaba (Tb), intersection with40


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Avenida Amrico Vespucio (AV), Pedaggico (Pd), Estadio Nacional (EN), and the intersection1with Avenida Vicua Mackenna (VM).2

3

(a)

(b)

Figure 3. Discretization of the stop density for the multiperiod case (3 periods) for the4corridor in: (a) the westbound direction and (b) the eastbound direction.5

6Figure 4 shows the result of the discretization of the stop density function using 3 periods, for the7case in which the stop density is equal in both directions. While the number of stops obtained is8the same as in the previous case, the demand function differs from the two profiles shown above,9so the proposed stop location is different from the previous cases and the current configuration.10

11For the corridor in the westbound direction (Figure 3.a), the optimal number of stops is 23.7,12

taking the capacity constraints of buses and stops into account; the average distance between13stops is decreased by 7.2% compared to the current configuration. In the eastbound direction14(Figure 3.b), the optimal number of stops is 23.8 and the average distance between stops is15decreased 11.6% from the current configuration.16

17

(x)

[stops/km]

(

x)

[stops/km]

Multiperiod Discretization, Westbound Direction

Multiperiod Discretization, Eastbound Direction


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Figure 4. Discretization of stop density for the multiperiod case (3 periods) for the corridor1in both directions.2

34

5 CONCLUSIONS56

In this project, a continuous and deterministic model was developed to identify the optimal7location of bus stops in a corridor by using a multiperiod structure of trips. The model was8applied to the case of the Avenida Grecia corridor. Additionally, the model developed solves the9stop location and frequency determination problems simultaneously, considering the cycle time10as a function of the passengers who board or alight the buses at stops in each period.11

12The proposed set of stop locations results in a reduction of total costs of about 20%. The13proposed headways are longer than the current ones (over 60%), especially in the off-peak14period. The discretization process determined the location of stops in the corridor so that they15would have areas under the curve that are all equal to one. This showed interesting results, in16that the distance between stops decreased between 7.2% and 11.6% compared to the current stop17configuration. The model proposed accordingly allows for the determination of stop densities18that, after being discretized, conform to a set of stops with a better spatial distribution than the19current stops.20

21Even though bus stops cannot be located anywhere along the corridor, and therefore the specific22locations obtained by this approach may not be feasible, the model provides a very valuable23output by identifying the number of bus stops to be installed and their approximate locations. It is24important to recognize that the mathematical form of the objective function is quite flat around its25optimal solution, and therefore small perturbations in the proposed solution have very little26

impact in the total cost.2728

In the case of Avenida Grecia, the model is clearly robust in its results, since both the stop29density and the headways are not very sensitive to variations in parameters. The maximum30variation that is observed is the 5% that results from a 10% change in the value of walking access31time.32

33

(x)

[sto

ps/km]

Multiperiod Discretization, Both Directions


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Even though the application presented here considers an avenue that already has a public transit1corridor, the model is designed as a decision-making tool for stop locations in corridors that have2not yet been built. The model could later be extended to consider the impact on stop density of a3limited stop express service operating in the corridor.4

5

ACKNOWLEDGEMENTS67We thank the financial support of the ACT-32 Project Real-Time Intelligent Control for8Integrated Transit Systems and FONDECYT project 1110720. This research was also supported9by the Across Latitudes and Cultures- Bus Rapid Transit Centre of Excellence funded by the10Volvo Research and Educational Foundations (VREF).11

1213

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