37

Transportation Research Record: Journal of the Transportation Research Board, No. 2498, Transportation Research Board of the National Academies, Washington, D.C., 2015, pp. 37–45.DOI: 10.3141/2498-05

This paper presents an integrated solution method to the frequency-setting problem using optimization and assignment simulation. With a bilevel solution framework, an optimization problem is run at the upper level under the assumption that the total transit demand is fixed. The objective is to maximize wait time savings under the budget, fleet size, vehicle load, and policy headway constraints. At the lower level, an assignment and simulation algorithm is run; the algorithm models the demand response to the new frequency setting and transmits updated ridership and flow values to the upper level. The procedure is repeated until the improvement in the wait time savings converges. The platform is tested on the Chicago Transit Authority network in Illinois for the morning peak. The wait time savings in this experiment are converg-ing and are found to be comparable with the results of a stand-alone frequency-setting algorithm that finds optimal solutions. Results are comparable with the optimal results of a stand-alone platform intro-duced in another study that modeled the demand response locally by using elasticities as opposed to networkwide modeling using assign-ment. The integrated platform can be used for medium-term strategic and long-term planning decisions.

In the United States, transit ridership has witnessed an increase of more than 50% since 1971 (1), and in large cities, crowding in transit vehicles is a major issue of growing significance (2, 3). To address the demand for transit, agencies have to make decisions at the planning, strategic, and tactical levels. At the planning level, the transit network is redesigned by doing one or all of the following: adding, removing, or modifying existing routes (4–6). This redesign requires the allocation of new frequencies to the routes, which by itself is a strategic-level decision process that is performed season-ally by transit agencies without necessarily modifying the network structure (7).

This paper introduces an integrated solution method for the pure transit network frequency-setting problem. As opposed to modeling the demand response to service changes locally by using spatially

and temporally varying elasticities, this study uses transit assignment to capture the demand response in a networkwide fashion (8–10). To achieve that aim, a transit assignment-simulation platform introduced in another study (11) is integrated with the optimization framework described in Verbas and Mahmassani (8).

Early studies for the pure transit network frequency-setting prob-lem had a route-by-route approach. According to Mohring (12) and Newell (13), the optimal frequency is proportional to the square root of the waiting-cost-to-operational-cost ratio. Another set of studies has focused on the maximum load points and load profiles along a route (14, 15).

Early analytical approaches by Salzborn demonstrate the tran-sition from route-by-route solutions to networkwide solutions (16, 17), which led to one of the first network-level solution methods by Scheele (18). This study uses a bilevel approach to solve for route frequencies at the upper level and assign demand to the routes at the lower level. Similar approaches were used later by Han and Wilson (19) and Constantin and Florian (20).

Finding optimal transit frequencies along with determining the modal split was first introduced by LeBlanc (21). Another frequency-setting method with mode split and assignment was later introduced by Gallo et al. (22).

The integrated solution framework introduced in this study uses a bilevel approach. At the upper level an optimization algorithm is run to set the frequencies under constant demand, which is translated into a timetable to find least-cost hyperpaths and assign travelers to these hyperpaths at the lower level. The lower-level problem simulates the assigned travelers along with the vehicles in an iterative framework to reach user equilibrium. Ultimately, it provides the new set of boardings and flows to the upper-level problem.

The optimization algorithm to set the frequencies used at the upper level is based on the work of Furth and Wilson (7). This algorithm has been improved to capture service patterns, bus loads, and heterogene-ity of elasticities as described in detail in Verbas and Mahmassani (8), Frei and Mahmassani (9), and Verbas et al. (10). To model the demand response, a new assignment and simulation tool is used (11). For a detailed review of transit assignment, see Verbas et al. (11) and Noh (23). To assign travelers onto optimal paths, a hyperpath algorithm is developed (24). For a better understanding of hyperpaths see Verbas and Mahmassani (24), Nguyen and Pallottino (25, 26), Spiess and Florian (27), and Noh et al. (28).

Integrated Frequency Allocation and User Assignment in Multimodal Transit NetworksMethodology and Application to Large-Scale Urban Systems

I.. Ömer Verbas and Hani S. Mahmassani

Transportation Center, Northwestern University, 600 Foster Street, Evanston, IL 60208. Corresponding author: H. S. Mahmassani, masmah@northwestern.edu.

38 Transportation Research Record 2498

NotatioN aNd defiNitioNs

The following notation is used in this paper:

J = set of different vehicle sizes; j = subscript for vehicle size; j ∈ J; P = set of patterns by stop order and departure time; p = subscript for pattern; p ∈ P; S = set of stops; Sp = set of stops of pattern p ∈ P; s = subscript for stop; s ∈ S; T = set of time intervals; t = subscript for time intervals; t ∈ T; te = end of time interval t ∈ T; ts p

i = scheduled departure time of pattern p’s ith trip; F = fare of one ride; H − = minimum policy pattern dispatch headway; H + = maximum policy pattern dispatch headway; dp = departure time of pattern p ∈ P; dp ∈ T; θp = subscript for vehicle capacity of pattern p ∈ P; θp ∈ J; hp = headway of pattern p ∈ P; Kp = operational cost of pattern p ∈ P; kp = unit operational cost of pattern p ∈ P per minute; τp = travel time of pattern p ∈ P; τ′p = surrogate travel time of pattern p ∈ P; τ′p = min{τp, 30}; Bp = vehicle capacity (maximum allowed number of passengers

in a vehicle) of a pattern p ∈ P; Mt,j = fleet size of vehicles with size j ∈ J at time t ∈ T; ep,s = number of boardings at stop s ∈ Sp on pattern p ∈ P; ap,s = number of alightings at stop s ∈ Sp on pattern p ∈ P; fp,s = passenger flow right after stop s ∈ Sp on pattern p ∈ P; gp,s = vehicle load right after stop s ∈ Sp on pattern p ∈ P; Dp,s = “stay constant,” fraction of flow fp,s−1 from previous stop

s − 1 that stays on board at stop s ∈ Sp on pattern p ∈ P; wp,s = wait time savings at stop s ∈ S on pattern p ∈ P; βp,s = elasticity of boardings ep,s with respect to pattern dispatch

headway hp; and Γ = available subsidy.

The interval length in this study is 30 min. A bus route consists of one or more service patterns p. A service pattern is the subset of stops served from the entire stop range of a route (8). A common example for a pattern is the “short-turn” pattern that serves the higher-demand segment of a route and is overlapped by the “full-length” pattern (29). In this paper, every pattern p has a dispatch time interval dp and a dispatch headway hp. For a general case, assume that the ridership

ep,s for a service pattern p at a stop s is elastic to the dispatch headway hp by the relation shown in Equation 1:

e h eh

hp s p p s

p

p

p s

( ) =

β

(1), ,0

0

,

where e0p,s is the observed ridership at the base scenario for the

given base-scenario headways of h0p. Assuming that the average wait-

ing time per passenger is half of the headway, the wait time savings for a single passenger is (H+ − hp)/2 because as long as there is service for a pattern, the policy headway H+ is guaranteed. How-ever, changing the headway does not just improve the waiting time; it also changes the ridership under the assumption that ridership is elastic with respect to headway (7, 8). This assumption yields Equation 2:

w h e u due

hH hp s p p s

p s

p s p

p

h

H

p s

p s p s

p

∫( ) ( )( ) ( )( )( )= =β +

− β+ β β

+

1

2 2 1(2), ,

,0

,0 ,

, ,

Figure 1 shows the change in ridership and wait time savings under the elastic and inelastic cases. The flow fp,s for pattern p at a stop s is a function of the boardings and alightings from the first stop of the pattern up to the current stop. In a recursive formulation, the flow fp,s was derived in Verbas and Mahmassani by using the constant Dp,s, which gives the fraction of flow fp,s−1 from the previous stop s − 1 that stays on board at stop s as shown in Equation 3 (8):

Da

fp s

p s

p s

= −−

1 (3),,

, 1

The flow fp,s is defined as shown in Equation 4:

f h eh

he

h

hDp s p p s

p

pp m

p

pp s n

n

s m

m

sp s p m

∏∑( ) =

+

β β

− +=

−

=

−

(4), ,0

0 ,0

0 , 111

1, ,

Equation 4 leads to Equation 5 for the vehicle load gp,s:

g h

eh

he

h

hD

h

p s p

p sp

pp m

p

pp s n

n

s m

m

s

p

p s p m

∏∑( ) =

+

β β

− +=

−

=

−

30 (5),

,0

0 ,0

0 , 111

1, ,

FIGURE 1 Ridership and wait time savings under elastic and inelastic cases.

hp

Headway (min)

Elastic

Inelastic

Nu

mb

er o

f R

ider

s

H+

Verbas and Mahmassani 39

The operational cost per pattern is the product of the unit cost, pattern travel time, and frequency as shown in Equation 6:

K kh

p p pp

= τ30

(6)

ModeLiNg deMANd ReSpoNSe

This paper introduces an integrated method for solving the transit frequency-setting problem and uses a formulation similar to that presented in Verbas and Mahmassani (8). One of the objectives of the stand-alone frequency-setting method presented in that study was to maximize ridership and waiting time savings subject to the constraints of budget, policy headways, fleet size, and vehicle loads. The ridership response to the changes in headways was captured locally by elasticities, and the entire problem was solved optimally with a mathematical program.

The integrated framework introduced in this paper captures the ridership response to service change in a networkwide fashion as opposed to locally. This objective is achieved by integrating a simu-lation and assignment platform with the frequency-setting platform (11). The transit simulation and assignment platform itself uses an iterative framework. For a given network, traveler demand, and transit schedule, (a) it calculates the time-dependent multimodal least-cost hyperpaths (24); (b) then using a gap-based formulation and demand data (origin, destination, and departure time), it assigns travelers on these least-cost hyperpaths (11, 30); and (c) for a given assignment scenario and vehicle schedule, it simulates the travelers and vehi-cles in the network instance-by-instance (11). These three steps are repeated iteratively until the time-dependent user equilibrium is reached (11).

The assignment-simulation platform described above is multimodal including the transit modes (bus and rail) and the walking mode. Hence, walking is not just an access or egress mode in this model. It can compete with the transit modes for short distances granted its generalized cost is less than any other option. This framework is integrated with the transit frequency-setting optimization framework by using a bilevel approach in which the frequencies are set at the upper level with a mathematical program using a formulation simi-lar to the stand-alone frequency-setting problem. The objective is to maximize waiting time savings subject to the constraints of budget, policy headways, fleet size, and vehicle loads.

The demand response to the new headways is captured at the lower level with the transit simulation and assignment platform described above. [See Verbas et al. for further details (10).] The lower-level problem provides the number of boardings and the flows at the pat-

tern stop level to the upper-level problem. In this integrated approach, the total demand is assumed to be fixed although the total number of boardings is subject to change as the number of transfers or walking trips changes as a response to the new service. In comparison with the closed-form analytical method of the stand-alone method, in which the solution is a local optimum, the integrated approach is a heuristic method, in which success can be measured by the improvement of the objective function without guaranteeing optimality. Table 1 provides a summary of the comparison between the two methods.

iNtegRAted fReqUeNcy-SettiNg ALgoRithM

Figure 2 presents the integrated framework for setting frequencies with optimization and modeling the demand response via assignment and simulation. The algorithm starts with simulating the base sce-nario at the lower level, which is assigning the current demand to an existing service and simulating the vehicles and the assigned travel-ers to obtain the necessary input for the optimization problem at the upper level. As mentioned in the previous section, the assignment-simulation platform uses an iterative process itself (11). For the rest of this paper, the iterations in that platform will be designated as “inner iterations,” whereas the iterations of this integrated frequency-setting algorithm will be called “outer iterations” or simply “iterations.”

The decision variable at the upper level is pattern headways hp for every pattern p ∈ P. The lower-level problem transmits the num-ber of boardings ep,s for every stop s ∈ Sp of every pattern p ∈ P. The calculation is done by summing vehicle boardings for every vehicle that belongs to pattern p ∈ P at each stop of the vehicle trip. The pattern stop pair flows fp,s are calculated in the same fashion. As the demand response is modeled at the lower level, the number of boardings ep,s = e0

p,s is constant for the upper-level problem of frequency setting; hence the elasticity of ridership with respect to headway is set to zero: βp,s = 0. The calculation of wait time savings introduced in Equation 2 then becomes

w h e u due

H hp s p p sp s

h

H

p

p

∫ [ ]( ) ( )= = −+

+

1

2 2(7), ,

,

Equation 7 gives half of the area of the rectangle shown in Figure 1. Similarly, the calculation of vehicle loads is also straightforward as the pattern stop pair flows fp,s are fixed for the upper-level problem:

g hf

h

p s pp s

p

( ) = 30 (8),,

TABLE 1 Comparison Between Stand-Alone and Integrated Frequency-Setting Approaches

Platform Stand-Alone Integrated

Objective Max. ridership and waiting time savings Max. waiting time savings

Decision variables Pattern headways Pattern headways

Constraints Budget, fleet, policy, and load Budget, fleet, policy, and load

Scope of decision making Networkwide Networkwide

Demand response Locally by elasticities Networkwide via assignment and simulation

Solution method Analytical via optimization Heuristic via bilevel programming

Note: Max. = maximum.

40 Transportation Research Record 2498

The mathematical program at the upper level is presented next. The objective is to maximize wait time savings subject to budget, fleet, vehicle load, and policy headway constraints.

w hh

p s p

s Sp Ppp

∑∑ ( )∈∈

max (9),

subject to

K Fep p s

s Sp Pp P p

∑∑∑ − ≤ Γ∈∈∈

(10),

hM t T j Jp

pt j

p P d t jp p

∑ ′τ≤ ∀ ∈ ∈

{ }∈ = θ =

; (11),

,

g B p P s Sp s p p≤ ∀ ∈ ∈; (12),

H h H p Pp≤ ≤ ∀ ∈− + (13)

The objective function presented in Equation 9 is the sum of wait time savings over all patterns and stops. Equation 10 is the net oper-ational cost constraint. The first term is the total operational cost of all patterns, and the second term is the product of total ridership and the average fare F, that is, the total revenue. Their difference is bounded by the available subsidy Γ. Equation 11 is the fleet size constraint. The surrogate travel time is τ′p = min{τp, 30}. If τp is less than or equal to 30 min, then τp /hp is the number of required transit vehicles for pattern p. However, if τp is greater than 30 min, keep-ing in mind that pattern p is active only for one half-hour interval dp, then only 30/hp vehicles are needed. Equation 12 is the vehicle load constraint. The number of passengers in a transit vehicle can-

not exceed the number Bp. Finally, Equation 13 presents the policy headway constraint.

Once the upper-level problem is solved, new optimal headways are used to create a timetable for the lower-level problem. For a given pattern departure time interval pair, the first trip is placed to

tst hp p=

+2

(14)10

where t0 is the starting time of the departure time interval and tsp1

is the departure time for a trip belonging to pattern p ∈ P. For the following trips i > 1, the following rule is used:

ts

ts h ts h t

tsh h

ts h tip

ip

p ip

p e

ip p p

ip

p e

=+ + ≤

+ + + >

− −

−′

−2 2

(15)1 1

1 1

In case ts pi−1 + hp > te, where te is the end of the time interval,

the headway is averaged with the next time interval’s headway hp′, where p′ ∈ P |dp′ = dp + 1.

experimeNts

The following integrated frequency-setting experiment is run on the Chicago Transit Authority network in Illinois presented in Figure 3. Ten outer iterations are run for the 5 to 9 a.m. (morning peak) period. The network has the following properties:

• 1,072 zones,• 13,754 nodes,• 63,602 links,

FIGURE 2 Integrated transit frequency-setting algorithm (N 5 no; Y 5 yes).

Verbas and Mahmassani 41

The following weights and parameters are used at the lower level:

• Relative weight of waiting XW = 2,• Relative weight of walking WW = 1,• Walking speed is set to 4.6 ft/s,• Relative weight of standing WS = 2,• Biking is disabled,• Base transfer penalty PX = 5 min,• Simulation interval = 1 s,• Assignment interval = 30 min, and• Least-cost path calculation interval = 30 min.

The upper-level problem is solved by using the AMPL/KNITRO platform (33). The procedures for the upper- and lower-level prob-lems are run 10 times each (outer iterations), whereas the assignment and simulation platform is run with five iterations (inner iterations) at the lower level. Figure 4 presents the change in average waiting time over the five inner iterations for each of the 10 outer iterations and the base scenario. As seen in the figure, after one outer iteration, the progress of the waiting time in the following outer iterations is comparable.

Figures 5 through 8 present the progression of the average gap, average cost, average waiting time, and number of boardings. The average gap presented in Figure 5 goes from 13.5 to 3.8 min. Gap is a measure of user equilibrium, which is defined as the deviation between the traveler’s experienced cost and the least possible cost for the trav-eler’s origin–destination pair at the departure time interval (30). In the case of user equilibrium, this gap becomes zero, as “no single traveler is able to improve his or her experience by switching the path.”

The average cost presented in Figure 6 starts at 76 min and con-verges to 59 min. The average cost is the average generalized cost a traveler experiences between his or her origin and destination zone (24):

XW WW

WS

PX

= × + ×

+

+ ×

+ ×

cost Total_Waiting_Time Total_Walking_Time

Total_Seated_in_Vehicle_Time

Total_Standing_in_Vehicle_Time

Number_of_Transfers (16)

The average waiting time converges to 5.1 min; it was 9.4 min at the base scenario. The number of boardings is increased by 9% from 556,112 boardings to 605,641. The overall transit demand has not changed, and the increase in boardings is the result of shifts from the walking mode and the increased number of transfers.

ComparisoN experimeNt

To be able to compare the performance and the output of this inte-grated frequency-setting platform with stand-alone transit frequency setting, the following is performed:

Step 1. With the changes in headways and boardings between the eighth and ninth iteration, arc elasticities βp,s are estimated:

e e

e e

h h

h hp s

p s p s

p s p s

p s p s

p s p s

β =−+

×+−

(17),,

9,

8

,9

,8

,9

,8

,9

,8

Step 2. Headways h9p, boarding e9

p,s, alighting a9p,s and flow values

f 9p,s at the pattern stop pairs ( p, s) of the ninth transit assignment and

simulation run are used as the base scenario values.

FIGURE 3 Chicago Transit Authority network.

• 134 routes:Eight rail routes and126 bus routes,

• 498 patterns (5 to 9 a.m.):24 rail patterns and474 bus patterns,

• 2,420 pattern departure time pairs (5 to 9 a.m.) (decision variables):131 rail pattern departure time pairs and2,289 bus pattern departure time pairs,

• 5,316 vehicle trips (5 to 9 a.m.):376 rail trips and4,940 bus trips, and

• Transit traveler demand (5 to 9 a.m.):418,820 travelers and556,112 boardings.

The network and the vehicle schedule are prepared with general transit feed specification data (31); the time-dependent passenger demand data are obtained from a forecasting project (32). A trav-eler’s exact departure time is set by randomly picking a second from a uniform distribution spanning a 30-min interval, which is the reso-lution of the available data. This procedure is performed once, and the same departure times are used for all experiments.

Gap

–TR

(min

)

Outer Iterations

FIGURE 5 Change in average gap (5 to 9 a.m.) over outer iterations (TR 5 traveler).

Gap

–TR

(min

)

Outer Iterations

FIGURE 6 Change in average cost (5 to 9 a.m.) over outer iterations.

FIGURE 4 Change in average waiting time (5 to 9 a.m.) over inner iterations.

Wai

ting

Tim

e–Bo

ardi

ng (m

in)

Inner Iterations

Verbas and Mahmassani 43

Step 3. A wait time savings maximization is performed by using the stand-alone formulation with elasticities:

e

hH h

h

p s

p s p

p

s Sp Ppp s

p s p s

p

∑∑ ( )( ) ( )( )β +

− β+ β β

∈∈

max2 1

(18),9

,9 ,

, ,

subject to

kh

Feh

hp p

pp s

p

ps Sp Pp P

p s

p

∑∑∑ τ −

≤ Γβ

∈∈∈

30(19),

90

,

hM t T j Jp

pt j

p P d t jp p

∑ ′τ≤ ∀ ∈ ∈

{ }∈ = θ =

; (20),

,

eh

he

h

hD

h

B p P s S

p sp

pp m

p

pp s n

n

s m

m

s

p

p p

p s p m

∏∑

+

≤ ∀ ∈ ∈

β β

− +=

−

=

−

30 ;

(21)

,9

0 ,9

0 , 111

1, ,

H h H p Pp≤ ≤ ∀ ∈− + (22)

The objective function presented in Equation 18 is the sum of wait time savings over all patterns and stops under elastic conditions. Equa-tion 19 is the net operational cost constraint, in which the ridership is elastic with respect to headways. The first term is the total operational cost of all patterns; the second term is the product of total ridership and the average fare F, that is, the total revenue. Their difference is bounded by the available subsidy Γ. Equation 20 is the fleet size con-straint. Equation 21 is the vehicle load constraint, in which the flow term in the numerator is changing as a result of the change in boardings and alightings from the first stop up to the current stop s. The number of passengers in a transit vehicle cannot exceed the number Bp. Finally, Equation 22 presents the policy headway constraint.

Step 4. Call this alternative outer iteration “10b” as opposed to the iteration 10 of the integrated platform. After the new headways hp

10b are obtained, Equations 14 and 15 are used to obtain new timetables for the assignment-simulation platform.

Step 5. Run the assignment-simulation platform with the time-tables obtained in Step 4, and compare these Iteration 10b results with the Iteration 10 results of the integrated platform.

Wai

ting

Tim

e–Bo

ardi

ng (m

in)

Outer Iterations

FIGURE 7 Change in average waiting time (5 to 9 a.m.) over outer iterations.

Tim

e–Bo

ardi

ng (m

in)

Outer Iterations

FIGURE 8 Change in boardings (5 to 9 a.m.) over outer iterations.

44 Transportation Research Record 2498

Figure 9 presents the headways optimized by using the integrated and stand-alone algorithms. To measure their difference from each other, the concordance correlation coefficient is used. This coef-ficient is a measure that evaluates how much two value sets deviate from the 45-degree line, which is found to be 0.9955 (34). This result suggests that the sets of headways by the two algorithms are comparable with each other to a high degree.

As presented in Table 2, the performance measures of the stand-alone and integrated platforms are also comparable. The average gap differs by 13 s, while the average cost and average waiting time differ by 3 and 7 s, respectively. There are two numbers of boardings for the stand-alone results. The first is the result of the assignment-simulation platform, which is comparable with the results of the integrated framework. The ridership value in parentheses is the esti-mated value as a result of the stand-alone optimization algorithm with the estimated elasticities βp,s.

As mentioned, the integrated platform uses a fixed transit demand. The change in ridership is the result of a modal shift between the walking mode and the transit modes, as well as the change in the number of transfers, which is modeled by using the assignment-simulation platform. However, the ridership shown in parentheses is estimated by using elasticities at the pattern stop pairs. In this case, the demand is modeled locally as a response to the change in headways. However, this method is not capable of capturing the reason for the change: Did the ridership increase as a result of a shift from another pattern (assignment)? Did a traveler decide to

wait for a second transit vehicle instead of walking (modal shift and transfers)? Did the improvement in service attract travelers from the passenger car mode (modal shift)? These reasons are unknown in the stand-alone formulation. Therefore, the integrated model has a wider range of use, which is discussed below. The similarity of the two presented models in relation to the decision variables and performance measures does not invalidate the importance of the network interactions. Agencies are interested not only in the overall performance measures but also in the local effects of their deci-sions, which can be modeled more accurately with the integrated framework.

As a result, the stand-alone formulation presented in detail in another study can be used in the following ways (8):

• As a tactical tool to make short-term frequency changes to cer-tain sections of the network while keeping the rest of the frequen-cies fixed and

• As a strategic tool to make medium-term frequency changes in the overall network in an optimal manner compromising the accu-racy of the demand response.

The integrated platform can be used in the following ways:

• As a strategic tool to make medium-term frequency changes in the overall network with a very detailed demand-response captur-ing traveler experience including transfers, which are very impor-tant for transit agencies. The compromise is that the optimality of the solution cannot be proved. Nevertheless, improvement in the objectives and a comparison of results with those of the stand-alone platform can be used as a benchmark.

• As an integral part of a planning tool for long-term decision making:

– Further integration with a highway simulation-assignment platform and an activity-based model would help to capture the entire modal share of the population in a city. Hence, the change in total transit demand as a response to the change in transit ser-vice, which was omitted in this study, would be captured. The quality of service and the user experience in the highway modes would be modeled.

– Further integration with a transit network design tool would help in assigning frequencies and modeling the demand response in case an agency decides to make network design changes, such as introducing an express bus service, adding a new rail line, intro-ducing more short-turn patterns, extending the length of some of the patterns, and changing the route of certain lines.

CoNClusioN

This paper has introduced a bilevel solution method to the frequency-setting problem integrated with assignment simulation. At the upper level, an optimization problem with the objective of maximizing wait-ing time savings under the budget, fleet size, vehicle load, and policy headway constraints was run. The ridership and flows at every pattern stop pair were assumed to be fixed; these results were obtained from the lower-level problem. The lower-level problem received the new pattern headways and converted them into a timetable. An assignment and simulation algorithm that modeled the demand response to the new frequency setting was run. The procedure was repeated until the improvement in waiting times became insignificant.

Headways—Integrated (min)

Head

way

s—St

and-

Alon

e (m

in)

FIGURE 9 Comparison of headways of integrated and stand-alone methods.

TABLE 2 Comparison Between Stand-Alone and Integrated Frequency-Setting Results

Platform Stand-Alone Integrated

Average gap (s) 240 227

Average cost (s) 3,561 3,558

Average waiting time (s) 304 311

Number of boardings 607,757 (614,506) 605,641

Verbas and Mahmassani 45

This integrated platform was tested on the Chicago Transit Author-ity bus and rail network for the morning peak. The waiting time sav-ings, as well as the average gap of the assignment problem at the lower level, were converged to improved values. Results were found that are comparable with the results of a stand-alone platform intro-duced in another study, which modeled the demand response locally by using elasticities as opposed to networkwide modeling using assignment (8). It is concluded that the stand-alone platform is useful for short-term tactical and medium-term strategic decisions, whereas the integrated platform can be used for medium-term strategic and long-term planning decisions.

RefeReNceS

1. Public Transportation Fact Book. American Public Transportation Association, Washington, D.C., 2013.

2. San Francisco Transportation Plan 2014 Final Report. San Francisco County Transportation Authority, Calif., 2013.

3. Transit Overload: A Station-by-Station Examination of Subway Ridership Growth over the Past Decade. In New York by the Numbers—Economic Snapshots of the Five Boroughs. Center for an Urban Future, New York, 2009.

4. Ceder, A., and N. H. M. Wilson. Bus Network Design. Transportation Research Part B, Vol. 20, No. 4, 1986, pp. 331–344.

5. Baaj, M. H., and H. S. Mahmassani. A Hybrid Route Generation Heu-ristic Algorithm for the Design of Transit Networks. Transportation Research Part C, Vol. 3C, No. 1, 1995, pp. 31–50.

6. Shih, M.-C., H. S. Mahmassani, and M. H. Baaj. Planning and Design Model for Transit Route Networks with Coordinated Operations. In Transportation Research Record 1623, TRB, National Research Council, Washington, D.C., 1998, pp. 16–23.

7. Furth, P. G., and N. H. M. Wilson. Setting Frequencies on Bus Routes: Theory and Practice. In Transportation Research Record 818, TRB, National Research Council, Washington, D.C., 1981, pp. 1–7.

8. Verbas, I.. Ö., and H. S. Mahmassani. Optimal Allocation of Service Fre-

quencies over Transit Network Routes and Time Periods: Formulation, Solution, and Implementation Using Bus Route Patterns. In Transporta-tion Research Record: Journal of the Transportation Research Board, No. 2334, Transportation Research Board of the National Academies, Washington, D.C., 2013, pp. 50–59.

9. Frei, C., and H. S. Mahmassani. Riding More Frequently: Estimating Dis-aggregate Ridership Elasticity for a Large Urban Bus Transit Network. In Transportation Research Record: Journal of the Transportation Research Board, No. 2350, Transportation Research Board of the National Acad-emies, Washington, D.C., 2013, pp. 65–71.

10. Verbas, I.. Ö., C. Frei, H. S. Mahmassani, and R. C. Chan. Stretching

Resources: Sensitivity of Optimal Bus Frequency Allocation to Stop-Level Demand Elasticities. Public Transport, Vol. 7, No. 1, 2015, pp. 1–20.

11. Verbas, I.. Ö., H. S. Mahmassani, and M. F. Hyland. Dynamic Assignment-

Simulation Methodology for Multi-Modal Urban Transit Networks. Presented at 94th Annual Meeting of the Transportation Research Board, Washington, D.C., 2015.

12. Mohring, H. Optimization and Scale Economies in Urban Bus Transpor-tation. American Economic Review, Vol. 62, No. 4, 1972, pp. 591–604.

13. Newell, G. F. Some Issues Relating to the Optimal Design of Bus Routes. Transportation Science, Vol. 13, No. 1, 1979, pp. 20–35.

14. Ceder, A. Bus Frequency Determination Using Passenger Count Data. Transportation Research Part A, Vol. 18A, No. 5-6, 1984, pp. 439–453.

15. Ceder, A. Public Transport Timetabling and Vehicle Scheduling. Pergamon Imprint/Elsevier Science, Ltd., New York, 2003.

16. Salzborn, F. J. M. Optimum Bus Scheduling. Transportation Science, Vol. 6, No. 2, 1972, pp. 137–148.

17. Salzborn, F. J. M. Scheduling Bus Systems with Interchanges. Transpor-tation Science, Vol. 14, No. 3, 1980, pp. 211–231.

18. Scheele, S. A Supply Model for Public Transit Services. Transportation Research Part B, Vol. 14B, No. 1-2, 1980, pp. 133–146.

19. Han, F. H., and N. H. M. Wilson. The Allocation of Buses in Heavily Utilized Networks with Overlapping Routes. Transportation Research Part B, Vol. 16B, No. 3, 1982, pp. 221–232.

20. Constantin, I., and M. Florian. Optimizing Frequencies in a Transit Network: A Nonlinear Bi-Level Programming Approach. International Transactions in Operational Research, Vol. 2, No. 2, 1995, pp. 149–164.

21. LeBlanc, L. J. Transit System Network Design. Transportation Research Part B, Vol. 22B, No. 5, 1988, pp. 383–390.

22. Gallo, M., B. Montella, and L. D’Acierno. The Transit Network Design Problem with Elastic Demand and Internalisation of External Costs: An Application to Rail Frequency Optimisation. Transportation Research Part C, Vol. 19C, No. 6, 2011, pp. 1276–1305.

23. Noh, H. Capacitated Schedule-Based Transit Assignment Using a Capacity Penalty Cost. PhD dissertation. University of Arizona, 2013.

24. Verbas, I.. Ö., and H. S. Mahmassani. Finding Least Cost Hyperpaths in

Multi-Modal Transit Networks: Methodology, Algorithm, and Large-Scale Application. Presented at 94th Annual Meeting of the Transporta-tion Research Board, Washington, D.C., 2015.

25. Nguyen, S., and S. Pallottino. Equilibrium Traffic Assignment for Large Scale Transit Networks. European Journal of Operational Research, Vol. 37, No. 2, 1988, pp. 176–186.

26. Nguyen, S., and S. Pallottino. Hyperpaths and Shortest Hyperpaths. Combinatorial Optimization, Lecture Notes in Mathematics, Vol. 1403, 1989, pp. 258–271.

27. Spiess, H., and M. Florian. Optimal Strategies: A New Assignment Model for Transit Networks. Transportation Research Part B, Vol. 23B, No. 2, 1989, pp. 83–102.

28. Noh, H., M. Hickman, and A. Khani. Hyperpaths in Network Based on Transit Schedules. In Transportation Research Record: Journal of the Transportation Research Board, No. 2284, Transportation Research Board of the National Academies, Washington, D.C., 2012, pp. 29–39.

29. Furth, P. G., and F. B. Day. Transit Routing and Scheduling Strategies for Heavy Demand Corridors. In Transportation Research Record 1011, TRB, National Research Council, Washington, D.C., 1985, pp. 23–26.

30. Lu, C. C., H. S. Mahmassani, and X. Zhou. Equivalent Gap Function-Based Reformulation and Solution Algorithm for the Dynamic User Equilibrium Problem. Transportation Research Part B, Vol. 43B, No. 3, 2009, pp. 345–364.

31. Google Developers. General Transit Feed Specification (GTFS). https://developers.google.com/transit/gtfs/reference. Accessed March 1, 2014.

32. Mahmassani, H. S., L. Jiang, A. Frei, T. Hou, A. Zockaie, and M. Saberi. Modeling and Forecasting of Toll Revenues. Final report. FHWA, Wash-ington, D.C., 2013.

33. Nocedal, J., and S. J. Wright. Numerical Optimization. Springer, 2006.34. Lin, L. I. K. A Concordance Correlation Coefficient to Evaluate

Reproducibility. Biometrics, 1989, pp. 255–268.

The Standing Committee on Transportation Network Modeling peer-reviewed this paper.