Express Service Design on a Bus Transit Network

Homero Larrain I.

Regular Service

:) :( :(

Express Service

Travel Time?

Waiting Time?

Transfers?

Operation costs?

:)

:(

:(

:)

source: brtdata.org

38 countries 158 cities 280 corridors

Express services currently in operation

Express services in the literature

Case studies: Ercolano (1984), Silverman (1998), Tétreault and El-

Geneidy (2010), El-Geneidy and Surprenant-Legault (2010), Scortia (2010).

Express service design: Jordan and Turnquist (1979), Furth (1986), Leiva et al.

(2010), Larrain et al. (2010), Sun et al. (2008), Chen et al. (2012), Chiraphadhanakul y Barnhart (2013).

Express services in the literature

Many to many

Freq. optimization

Service generation

Common lines Transfers Capacity

User Equilibrium

Jordan and Turnquist (1979)

✓ ✓

Sun et al. (2008) ✓ ✓ ✓

Leiva et al. (2010) ✓ ✓ ✓ ✓ ✓ ✓

Chen et al. (2012) ✓ ✓ ✓ ✓ ✓

Chiraphadhanakul & Barnhart (2013)

✓ ✓ ✓ ✓ ✓

Larrain et al. ✓ ✓ ✓ ✓ ✓ ✓ ✓

Capacity v/s user equilibrium

• If capacity levels are not reached, the optimization will be consistent with a user equilibrium.

• However, when capacity is taken into account, the results may not be consistent with user equilibrium.

• An iterative method was designed where the frequencies of lines were increased until they met requirements.

• The solution obtained by this method satisfies capacity constraints and user behavior.

HOW to design express services?

We are looking for a method that:

• Generates its own services.

• Reaches user equilibruim.

• Doesn’t exceed bus capacity.

• Works on PT networks.

Express service design on a network

Express service design on a corridor

Service generation

Frequency optimization

Methodology

Network

Step 1:

Frequency optimization Step 2:

Service generation Step 3:

Network problem

Modified Leiva’s model

…

…

…

…

…

…

f1

f2

f3

f4

fn

The model will give positive frequency to attractive services.

min𝑓𝑙,𝑓𝑙𝑠,𝑉𝑠𝑤 𝑐𝑙𝑓𝑙

𝑙∈ℒ

+ 𝑉𝑠𝑤 𝜃𝑤𝑡

𝜆

𝑓𝑙𝑠

𝑙∈ℒ

+ 𝜃𝑡𝑡 𝑡𝑙

𝑠𝑓𝑙𝑠

𝑙∈ℒ

𝑓𝑙𝑠

𝑙∈ℒ𝑠∈𝒮𝑤∈𝒲

+ 𝜃𝑡𝑟 𝑉𝑠𝑤

𝑠∈𝒮𝑤∈𝒲

− 𝑇𝑤𝑤∈𝒲

operating costs waiting costs in-vehicle time travel costs

transfer costs

s.t.: sign restrictions, continuity of flows, and continuity of fequencies.

Dealing with capacity (capacity heuristic)

Unconstrained solution (cap = 100 pax/hr): f1 = 10, Lcrit = 90

f2 = 8, Lcrit = 120

f3 = 5, Lcrit = 80

f4 = 0

fn = 0

Add a new restriction and solve again:

𝑓2 ≥ 8 + Δ

Iterate until reaching feasibility. Solution will satisfy user equilibrium and capacity constraints, but...

• Solution is not optimal. • It has to optimize in every iteration.

𝑓2 ≥120 ∙ 8

100

…

…

…

…

…

…

f1

f2

f3

f4

fn

Express service design on a network

Express service design on a corridor

Frequency optimization

Methodology

Network

Step 1:

Frequency optimization Step 2:

Service generation Step 3:

Network problem

Service generation

Service generation methods

Some proposed heuristics are:

• Bus stop elimination.

• Bus stop inclusion.

• Short turning service generation.

• Zonal service generation.

• Short turning service generation for congested scenarios.

• Zonal service generation for congested scenarios.

• Mixed service generation for congested scenarios.

Zonal service generation

Zonal service:

Affected trips (types of users):

Social cost function:

operator costs + waiting times – travel time reduction

…

…

fa

fe

TEe

TAEe

TA

e

𝑆𝐶𝑒 = 𝑓𝑎𝑐𝑎 + 𝑓𝑒𝑐𝑒 + 𝜆𝜃𝑤𝑡𝑇𝐴𝑒

𝑓𝑎+𝑇𝐸𝑒

𝑓𝑒+𝑇𝐴𝐸𝑒

𝑓𝑎 + 𝑓𝑒− 𝜃𝑡𝑡𝑇𝐸

𝑒𝑁𝑒𝜏

Zonal service generation

Social costs approximation (no congestion):

𝑆𝐶𝑒 = 𝑓𝑎𝑐𝑎 + 𝑓𝑒𝑐𝑒 +𝜆𝜃𝑤𝑡 𝑇𝐴

𝑒 + 𝑇𝐴𝐸𝑒

𝑓𝑎+𝜆𝜃𝑤𝑡𝑇𝐸

𝑒

𝑓𝑒− 𝜃𝑡𝑡𝑇𝐸

𝑒𝑁𝑒𝜏

→ 𝑆𝐶𝑒∗ = 2 𝜆𝜃𝑤𝑡 𝑇𝐴

𝑒 + 𝑇𝐴𝐸𝑒 𝑐𝑎 + 2 𝜆𝜃𝑤𝑡𝑇𝐸

𝑒𝑐𝑒 − 𝜃𝑡𝑡𝑇𝐸𝑒𝑁𝑒𝜏

Only regular service optimal social costs:

𝑆𝐶𝑎∗ = 2 𝜆𝜃𝑤𝑡𝑐𝑎 𝑇𝑤

𝑤∈𝒲

Zonal service generation (uncongested case):

• For every possible zonal service 𝑒, compute 𝑆𝐶𝑒∗ y 𝑆𝐶𝑎

∗.

• If 𝑆𝐶𝑒∗ < 𝑆𝐶𝑎

∗, include 𝑒 in the frequecy optmization

problem initial lines set.

Zonal service generation

Caso con capacidad:

…

…

fa

fe

PM

Total frequency has to be at least enough to carry the load on the

critical arc. 𝑓𝑎 + 𝑓𝑒 = 𝑓0 =

𝑃𝑐𝑟𝑖𝑡𝑐𝑎𝑝

Defining 𝑃𝑀𝐴𝑒, 𝑃𝑀𝐸

𝑒 y 𝑃𝑀𝐴𝐸𝑒 as the portion of 𝑃𝑀 corresponding to

each type of user: 𝑓𝑎∗ =

𝑓0𝑃𝑀𝐴𝑒

𝑃𝑀𝐴𝑒 + 𝑃𝑀𝐸

𝑒 𝑓𝑒∗ =

𝑓0𝑃𝑀𝐸𝑒

𝑃𝑀𝐴𝑒 + 𝑃𝑀𝐸

𝑒

These expressions allow us to compute optimal social costs for any

zonal service.

We can find optimal solutions for congested scenarios!

Scenario N-S Direction S-N Direction Freq. (bus/hr) Max load (pax/bus)

Base No Cap. 1 oooooooooooooooooooo -------------------- 51.64 273.06

Base No Cap. 1 -------------------- oooooooooooooooooooo 51.64 58.11

Zonal No Cap. 2 oooooooooooooooooooo -------------------- 34.83 197.72

Zonal No Cap. 2 -------------------- oooooooooooooooooooo 69.82 42.98

Zonal No Cap. 2 oo------------oooooo -------------------- 12.47 210.51

Zonal No Cap. 2 oo-------------ooooo -------------------- 16.77 195.22

Zonal No Cap. 2 oo----------------oo -------------------- 1.52 128.84

Zonal No Cap. 2 oooo-----------ooooo -------------------- 4.23 264.71

Base Alg. Cap. 3 oooooooooooooooooooo -------------------- 88.13 160.00

Base Alg. Cap. 3 -------------------- oooooooooooooooooooo 88.13 34.05

Zonal Alg. Cap. 4 oooooooooooooooooooo -------------------- 33.00 157.98

Zonal Alg. Cap. 4 -------------------- oooooooooooooooooooo 89.00 33.72

Zonal Alg. Cap. 4 oo------------oooooo -------------------- 19.00 158.23

Zonal Alg. Cap. 4 oooo-----------ooooo -------------------- 37.00 158.95

Zonal Cap. 5 oooooooooooooooooooo -------------------- 41.55 160.00

Zonal Cap. 5 -------------------- oooooooooooooooooooo 88.13 34.05

Zonal Cap. 5 oo------------oooooo -------------------- 46.58 160.00

Using the heuristics

Etapa Costo social ($/hr) Reducción costo social

Base S/Cap. 8,507,920 -

Zonal S/Cap. 7,852,817 7.7%

Base Alg. Cap. 8,820,851 -

Zonal Alg. Cap. 7,977,375 9.6%

Zonal Cap. 7,915,484 10.3%

Uncongested scenarios

Congested scenarios, solved with capacity heuristic. Congested scenarios, solved with zonal generation heuristic.

Zonal capacity heuristic can beat “old” capacity heuristic.

This solution could have been reached in scenario 4 (but wasn’t).

Diseño de servicios expresos en una red

Express service design on a corridor

Frequency optimization

Methodology

Network

Step 1:

Frequency optimization Step 2:

Service generation Step 3:

Network problem

Service generation

Express service design for a network

What’s the difference of working over a network?

Frequency optimization:

• Problem of scale.

Service generation:

• Our methods can only work for a corridor.

We can apply service generation over routes, and frequency optimization over the network.

Algorithm overview

1. Choose a set of initial attractive routes for the network. Express

services will be designed over these routes.

2. Optimize frequencies (ignoring congestion) for the initial solution

where every route is served by a regular service.

3. While certain convergence criteria is not met:

For every route:

a. Isolate the demand for the services contained on the route,

and generate services for the resulting corridor.

b. Optimize the frequencies for the current set of services,

ignoring capacity.

4. Apply the capacity algorithm.

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

Route selection

We could use the existing routes, or use a route design model.

1 2 3 4 5

6 7 8 9 10

1 2 3 4 5

11 12 13 14 15

5

6 7 8 9 10

11 12 13 14 15

Route 1

Route 2

Route 3

Performance indicators

How to measure the benefits of different scenarios?

Indicator Meaning

𝑶𝑪 Operator costs.

𝑻𝑻𝑪 Total in-vehicle travel time costs.

𝑾𝑻𝑪 Total waiting time costs.

𝑻𝑹𝑪 Total transfer costs.

𝑼𝑪 User costs: 𝑼𝑪 = 𝑻𝑻𝑪 +𝑾𝑻𝑪 + 𝑻𝑹𝑪.

𝑺𝑪 Social costs: 𝑺𝑪 = 𝑶𝑪 + 𝑼𝑪.

𝑭𝑻𝑻𝑪 Fixed in-vehicle travel time costs.

𝑺𝑪′ Corrected social costs: 𝑺𝑪′ = 𝑺𝑪 − 𝑭𝑻𝑻𝑪.

Initial solution

We start feeding the model with express services for every route, and

optimizing it with Leiva’s adapted model.

𝒍 Stops Freq.

(bus/hr) Max load (pax/bus)

1 1 2 3 4 5 6 7 8 9 10 46,8 154,3

2 10 9 8 7 6 5 4 3 2 1 51,4 38,7

3 1 2 3 4 5 11 12 13 14 15 49,8 180,8

4 15 14 13 12 11 5 4 3 2 1 45,2 20,3

5 10 9 8 7 6 5 11 12 13 14 15 37,7 97,2

6 15 14 13 12 11 5 6 7 8 9 10 42,3 84,9

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

Indicator Value ($/hr)

𝑶𝑪 1.100.337

𝑻𝑻𝑪 13.018.830

𝑾𝑻𝑪 1.100.487

𝑻𝑹𝑪 0

𝑼𝑪 14.119.317

𝑺𝑪 15.219.655

𝑭𝑻𝑻𝑪 8.191.080

𝑺𝑪′ 7.028.575

Service generation

Isolating route 1:

• Route demand is conformed by trips (or trip stages)

asigned to services that are completely contained by the

route.

• On nodes where services from other routs begin or end we

have to force frequency continuity by adding exogenous

frequencies.

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

Node 1 exogenous freq. = 1 to 15 regular service freq. – 15 to 1 regular service freq.

𝐹1 = 𝑓3 − 𝑓4 = 49,8 − 45,2 = 4,57𝑏𝑢𝑠/ℎ𝑟

Service generation

1 2 3 4 5

6 7 8 9 10

𝒍 Stops Freq.

(bus/hr) Max load (pax/bus)

1 1 2 3 4 5 6 7 8 9 10 28,3 98,1

2 10 9 8 7 6 5 4 3 2 1 39,9 16,3

3 1 2 3 4 5 11 12 13 14 15 49,8 179,7

4 15 14 13 12 11 5 4 3 2 1 45,2 20,7

5 10 9 8 7 6 5 11 12 13 14 15 37,7 97,2

6 15 14 13 12 11 5 6 7 8 9 10 42,3 84,1

7 10 9 8 5 1 45,7 35,0

8 1 10 29,7 90,8

9 1 2 4 5 6 8 9 10 23,0 76,0

Indicator Value ($/hr) 𝑶𝑪 1.324.254

𝑻𝑻𝑪 12.104.130

𝑾𝑻𝑪 1.194.934

𝑻𝑹𝑪 0

𝑼𝑪 13.299.064

𝑺𝑪 14.623.318

𝑭𝑻𝑻𝑪 8.191.080

𝑺𝑪′ 6.432.238

It. Savings 8,5%

Ac. Savings 8,5%

Not a user equilibrium!

New services.

Frequency optimization

With current services, the whole network frequencies are optimized,

and user equilibrium is reached once again.

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

𝒍 Stops Freq.

(bus/hr) Max load (pax/bus)

1 1 2 3 4 5 6 7 8 9 10 16,2 122,4

2 10 9 8 7 6 5 4 3 2 1 14,2 15,8

3 1 2 3 4 5 11 12 13 14 15 50,0 184,7

4 15 14 13 12 11 5 4 3 2 1 44,7 14,5

5 10 9 8 7 6 5 11 12 13 14 15 38,8 94,8

6 15 14 13 12 11 5 6 7 8 9 10 44,1 84,4

7 10 9 8 5 1 62,9 28,0

8 1 10 29,7 90,8

9 1 2 4 5 6 8 9 10 26,0 86,8

Indicator Value ($/hr)

𝑶𝑪 1.259.106

𝑻𝑻𝑪 12.003.896

𝑾𝑻𝑪 1.259.148

𝑻𝑹𝑪 67.050

𝑼𝑪 13.330.094

𝑺𝑪 14.589.200

𝑭𝑻𝑻𝑪 8.191.080

𝑺𝑪′ 6.398.120

It. Savings 0,5%

Ac. Savings 9,0%

Some iterations later

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

𝒍 Stops Freq.

(bus/hr) Max load (pax/bus)

1 1 2 3 4 5 6 7 8 9 10 24,4 54,6

2 10 9 8 7 6 5 4 3 2 1 11,1 19,2

3 1 2 3 4 5 11 12 13 14 15 0,0 0,0

4 15 14 13 12 11 5 4 3 2 1 41,9 19,5

5 10 9 8 7 6 5 11 12 13 14 15 0,0 0,0

6 15 14 13 12 11 5 6 7 8 9 10 0,0 0,0

7 10 9 8 5 1 44,0 35,8

8 1 10 29,9 90,3

9 1 4 5 6 8 9 10 43,4 113,1

10 10 9 5 1 20,5 21,2

11 15 14 5 2 1 51,0 22,4

12 1 2 3 4 5 13 14 15 25,2 68,6

13 1 2 14 15 18,0 99,3

14 1 2 3 14 15 27,4 139,4

15 15 12 5 10 45,9 75,5

16 10 14 15 28,2 63,1

17 10 7 6 5 12 14 15 18,5 51,8

18 10 9 7 6 5 11 12 13 14 15 21,4 49,5

Indicator Value ($/hr)

𝑶𝑪 1.648.084

𝑻𝑻𝑪 9.588.485

𝑾𝑻𝑪 1.648.064

𝑻𝑹𝑪 209.325

𝑼𝑪 11.445.875

𝑺𝑪 13.093.959

𝑭𝑻𝑻𝑪 8.191.080

𝑺𝑪′ 4.902.879

Ac. Savings 30,2%

Some iterations later

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

0

1

2

3

4

5

6

7

8

0 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b

Val

ue

($

/h)

Mill

ion

s

Iteration

CT'

CTT'

CTW

CTR

COP

Final solution L5(67,3)

L1(23,0)

L4(15,1)

L2(49,1) L3(29,1)

L10(34,7)

L12(44,7) L11(28,9)

L6(23,7)

L9(55,2)

L7(56,0) L8(43,4) Line(Freq.)

Conclusions

We have presented a model that’s able to find a solution to the

express service design problem which:

• Generates it’s own services.

• Is consistent with user equilibrium.

• Does not exceed capacity.

• Works on networks.

Conclusions

On the frequency optimization problem:

Frequency continuity restrictions, transfer node limiting and

improvements to the capacity algorithm has made the problem easier

and faster to solve. Still,

• The bi level approach used in the network algorithm can be used

to separate passenger assignment from frequency optimization.

• Other forms of capacity constraint, such as the maximum capacity

of bus stops, can be implemented in the model.

• Some other user behavior assumptions can be tested, such as

optimal strategies.

Conclusions

On the service generation problem:

We found different ways to generate services to feed the frequency

optimization problem, yielding savings around 10%. Furthermore, we

found some cases where the problem can be solved to optimality in

presence of congestion, which opens new possibilities. However,

• Numerical solutions could improve the generation formulas, by

avoiding approximations.

• Other service configurations could be studied, besides zonal and

short turning services.

• The results for the generation heuristics could be tested against

the services that an expert would design.

Conclusions

On the network express service design problem:

Our model is able to generate and optimize the services over a corridor,

which can carry benefits as high as a 30% cost reduction. Still,

• The route selection problem at the begining can be improved and

automated.

• The model has not still been tested on large networks. However,

the bottelneck of the algorithm occurs in the frequency optimization

problem, which can take instances of larger size than the ones we’ve

tried.

• A computational tool is on our plans.

Express Service Design on a Bus Transit Network

Homero Larrain I.

Thanks!

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