The document discusses express service design on bus transit networks. It proposes a methodology involving three steps: 1) optimizing frequencies ignoring capacity constraints, 2) generating express services for each route using corridor-based heuristics, and 3) solving the full network problem with capacity constraints. Testing on sample networks found the approach produced solutions with up to 10.3% lower social costs compared to baseline scenarios. The approach dynamically generates express services and optimizes frequencies to balance operator costs, user costs and capacity limits across the entire network.

1. Express Service Design on a Bus
Transit Network
Homero Larrain I.

2. Regular Service

3. :):( :(
Express Service
Travel Time?
Waiting Time?
Transfers?
Operation costs?
:)
:(
:(
:)

4. source: brtdata.org
38 countries 158 cities 280 corridors
Express services currently in operation

5. 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).

6. 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. ✓ ✓ ✓ ✓ ✓ ✓ ✓

7. 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.

8. 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.

9. 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

10. Modified Leiva’s model
…
…
…
…
…
…
f1
f2
f3
f4
fn
The model will give positive frequency to attractive services.
min
𝑓 𝑙,𝑓𝑙
𝑠
,𝑉𝑠
𝑤
𝑐𝑙 𝑓𝑙
𝑙∈ℒ
+ 𝑉𝑠
𝑤 𝜃 𝑤𝑡
𝜆
𝑓𝑙
𝑠
𝑙∈ℒ
+ 𝜃𝑡𝑡
𝑡𝑙
𝑠
𝑓𝑙
𝑠
𝑙∈ℒ
𝑓𝑙
𝑠
𝑙∈ℒ𝑠∈𝒮𝑤∈𝒲
+ 𝜃𝑡𝑟 𝑉𝑠
𝑤
𝑠∈𝒮𝑤∈𝒲
− 𝑇 𝑤
𝑤∈𝒲
operating costs waiting costs in-vehicle
time travel costs
transfer costs
s.t.: sign restrictions, continuity of flows, and continuity of fequencies.

11. 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

12. 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

13. 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.

14. Zonal service generation
Zonal service:
Affected trips (types of users):
Social cost function:
operator costs + waiting times – travel time reduction
…
…
fa
fe
TE
e
TAE
e
TA
e
𝑆𝐶𝑒 = 𝑓𝑎 𝑐 𝑎 + 𝑓𝑒 𝑐 𝑒 + 𝜆𝜃 𝑤𝑡
𝑇𝐴
𝑒
𝑓𝑎
+
𝑇𝐸
𝑒
𝑓𝑒
+
𝑇𝐴𝐸
𝑒
𝑓𝑎 + 𝑓𝑒
− 𝜃𝑡𝑡 𝑇𝐸
𝑒
𝑁𝑒 𝜏

15. Zonal service generation
Social costs approximation (no congestion):
𝑆𝐶𝑒 = 𝑓𝑎 𝑐 𝑎 + 𝑓𝑒 𝑐 𝑒 +
𝜆𝜃 𝑤𝑡 𝑇𝐴
𝑒
+ 𝑇𝐴𝐸
𝑒
𝑓𝑎
+
𝜆𝜃 𝑤𝑡 𝑇𝐸
𝑒
𝑓𝑒
− 𝜃𝑡𝑡 𝑇𝐸
𝑒
𝑁𝑒 𝜏
→ 𝑆𝐶𝑒
∗
= 2 𝜆𝜃 𝑤𝑡 𝑇𝐴
𝑒
+ 𝑇𝐴𝐸
𝑒
𝑐 𝑎 + 2 𝜆𝜃 𝑤𝑡 𝑇𝐸
𝑒
𝑐 𝑒 − 𝜃𝑡𝑡 𝑇𝐸
𝑒
𝑁𝑒 𝜏
Only regular service optimal social costs:
𝑆𝐶 𝑎
∗
= 2 𝜆𝜃 𝑤𝑡 𝑐 𝑎 𝑇 𝑤
𝑤∈𝒲
Zonal service generation (uncongested case):
• For every possible zonal service 𝑒, compute 𝑆𝐶𝑒
∗ y 𝑆𝐶 𝑎
∗.
• If 𝑆𝐶𝑒
∗ < 𝑆𝐶 𝑎
∗, include 𝑒 in the frequecy optmization
problem initial lines set.

16. 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 =
𝑃𝑐𝑟𝑖𝑡
𝑐𝑎𝑝
Defining 𝑃𝑀𝐴
𝑒
, 𝑃𝑀 𝐸
𝑒
y 𝑃𝑀𝐴𝐸
𝑒
as the portion of 𝑃𝑀 corresponding to
each type of user:
𝑓𝑎
∗
=
𝑓0 𝑃𝑀𝐴
𝑒
𝑃𝑀𝐴
𝑒
+ 𝑃𝑀 𝐸
𝑒 𝑓𝑒
∗
=
𝑓0 𝑃𝑀 𝐸
𝑒
𝑃𝑀𝐴
𝑒
+ 𝑃𝑀 𝐸
𝑒
These expressions allow us to compute optimal social costs for any
zonal service.
We can find optimal solutions for congested scenarios!

17. 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).

18. 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

19. 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.

20. 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.

21. 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

22. 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.

23. 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

24. 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

25. 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.

26. 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%

27. 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%

28. 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
Value($/h)Millions
Iteration
CT'
CTT'
CTW
CTR
COP

29. 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.)

30. 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.

31. 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.

32. 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.

33. 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.

34. Express Service Design on a Bus
Transit Network
Homero Larrain I.
Thanks!

35. Next Webinar
Traffic Safety on Bus Corridors
Presented by Nicolae Duduta, EMBARQ
Friday, September 27th at 11am EDT
Register here: http://goo.gl/XAi13S