Homero Larraín - The limited stop bus service design problem with stochastic passenger assignment

The limited-stop bus service
design problem with stochastic
passenger assignment
Homero Larrain
Pontificia Universidad Católica de Chile

The results presented in this webinar are taken from
“Un algoritmo bi-nivel de diseño de servicios limited-
stop con asignación determinística y estocástica”, a
thesis by Guillermo Soto, developed in co-guidance
with Juan Carlos Muñoz.
Acknowledgement

BRT and limited-stop services
The limited-stop service design problem
New ideas for the LSDP
Testing the new ideas
Conclusions

Bus Rapid Transit (BRT) is a high-quality bus-based
transit system that delivers fast, comfortable, and
cost-effective services at metro-level capacities. It
does this through the provision of dedicated lanes,
with busways and iconic stations typically aligned to
the center of the road, off-board fare collection, and
fast and frequent operations.
Institute for Transport and Development Policy
What is BRT?

What is BRT?
Limited-stop services are a key element in a well designed BRT system.

:):( :(
Travel times
Waiting times
Transfers
Operator costs
:)
:(
:(
:)
Limited-stop services

source: brtdata.org, 2007
42 countries 165 cities 358 corridors
Limited-stop services around the world

Express services in the literature
Case-study oriented works:
Ercolano (1984), Silverman (1998), Tétreault and El-
Geneidy (2010), El-Geneidy and Surprenant-Legault
(2010), Scortia (2010).
Design models:
Jordan and Turnquist (1979), Furth (1986), Leiva et al.
(2010), Larrain et al. (2010, 2015), Sun et al. (2008), Chen
et al. (2012), Chiraphadhanakul y Barnhart (2013).

Given a public transport corridor and a trip demand matrix
for a period of time, the objective of the problem is to define
a set of services (characterized by the stops they serve and
omit) and their operational frequencies, so as to minimize
the costs of the operator and the users.
It is assumed that frequencies of the system are high. Thus,
there is no scheduling involved, just frequency setting.
The limited-stop service design problem

Designing limited-stop services is a challenging problem:
1. The problem is non-linear: operator costs are directly
proportional to service frequencies, waiting costs are
inversely proportional to them.
2. Binary decisions are involved: to serve or skip a stop in a
particular service.
3. Passenger assignment is not trivial: it depends on the
level of service of the set of available options for a trip,
i.e., adding stops to a service might make it unattractive.
Main challenges

Our previous research
Leiva et al. (2010) introduced an algorithm that solves the
express service design problem over a corridor:
• There is an a priori set of candidate services.
• Passengers are allowed to transfer.
• Passenger assignment is deterministic.
• Bus capacity is taken into account via a greedy heuristic.
Larrain et al. (2013) extended this work:
• Services are generated heuristically.
• An algorithm for networks is proposed.
• Many other improvements were implemented.

LSDP
CFOAP LSGP
FOAP
LSDP: Limited-Stop service
design problem.
CFOAP: Capacitated frequency
optimization and assignment
problem.
LSGP: Limited-Stop service
generation problem.
FOAP: Frequency optimization
and assignment problem.
Problem framework
The problem at the core of the LSDP is to optimize the
frequencies of a set of services, while predicting passenger
behavior (i.e., which services they will use).

…
…
…
…
…
…
f1
f2
f3
f4
fn
min
𝑓 𝑙,𝑓𝑙
𝑠
,𝑉𝑠
𝑤
𝑙∈ℒ
𝑐𝑙 𝑓𝑙 + 𝜃 𝑊𝑇
𝑤∈𝒲 𝑠∈𝑆
𝑉𝑠
𝑤
𝜆
𝑙∈𝒮 𝑓𝑙
𝑠 + 𝜃 𝑇𝑇
𝑤∈𝒲 𝑠∈𝒮
𝑉𝑠
𝑤 𝑙∈ℒ 𝑡𝑙
𝑠
𝑓𝑙
𝑠
𝑙∈ℒ 𝑓𝑙
𝑠 + 𝜃 𝑇𝑟
𝑤∈𝒲
𝑇 𝑤
′
− 𝑇 𝑤
[operator costs + waiting time + travel time + transfers]
s.t.: non-negativity, flow conservation, bus conservation, frequency bounds.
Objective: Minimizing social costs
Frequency optimization and assignment
problem (FOAP)

Limitations of the current approach
The algorithms by Leiva and Larrain solve the FOAP using a
commercial solver. Their current approach comes with the
following limitations:
1. The associated MINLP problem can only be solved
reasonably for instances of limited size.
2. Deterministic behavior leads to an all-or-nothing
assignment, which can be unrealistic, and might make
the problem even harder to solve.
3. Passenger assignment comes from minimizing social cost.
This is valid only in absence of capacity constraints, so a
greedy heuristic is implemented for the CFOAP.

Proposed solutions for the limitations
1. The associated MINLP problem can only be solved
reasonably for limited sized instances.
Implement a bi-level solution approach for the FOAP.
2. Deterministic behavior leads to an all-or-nothing
assignment, which can be unrealistic, and might make
the problem even harder to solve.
Model passenger assignment as a stochastic process.
3. Passenger assignment comes from minimizing social cost.
This is valid only in absence of capacity constraints, so a
greedy heuristic is implemented for the CFOAP.
Solve the capacity problem using a GRASP algorithm.

A bi-level solution approach for the FOAP.
1.

FOAP
Frequency optimization and passenger assignment problem
Frequency optimization Passenger assignment
A bi-level approach for the FOAP
Given a passenger
assignment (i.e., the demand
for each service), what are
the optimal frequencies?
For a set of given
frequencies, how will
passengers assign to
available services?

LSDP
CFOAP LSGP
FOAP
FOP PAP
LSDP: Limited-Stop service design
problem.
CFOAP: Capacitated frequency
optimization and assignment
problem.
LSGP: Limited-Stop service
generation problem.
FOAP: Frequency optimization and
assignment problem.
FOP: Frequency optimization
problem.
PAP: Assignment problem.
A bi-level approach for the FOAP
By separating the problem into frequency optimization and
passenger assignment, we can solve bigger instances, and
also work with different behavioral models.

Stochastic passenger assignment.
2.

Deterministic assignment: an overview
You are at stop A waiting for a bus to go to stop B. The
regular service shows up first. ¿Should you take it?
A B
Regular service: 𝑡 𝑟 = 60𝑚𝑖𝑛, 𝑓𝑟 = 10𝑣𝑒ℎ/ℎ
Express service: 𝑡 𝑒 = 50𝑚𝑖𝑛, 𝑓𝑒 = 6𝑣𝑒ℎ/ℎ
Option 1: Take it.
Waiting time: 0
Travel time: 60𝑚𝑖𝑛
Option 2: Leave it.
Waiting time: 5𝑚𝑖𝑛
Travel time: 50𝑚𝑖𝑛
For every O/D pair in our corridor we can solve a similar
problem, where we find the attractive set of services to
perform that trip.

The assignment process consists of three steps:
1. Finding the attractive services and the expected travel
time for every O/D pair.
2. On the resulting network, assign trips to their shortest
paths.
3. Compute the ridership for each service.

𝑙1: 𝑡𝑡1, 𝑓1
𝑙2: 𝑡𝑡2, 𝑓2
𝑙 𝑛: 𝑡𝑡 𝑛, 𝑓𝑛
𝑡𝑡1 ≤ 𝑡𝑡2 ≤ ⋯ ≤ 𝑡𝑡 𝑛
…
The attractive set of lines for A-E, 𝑆𝐴𝐸, is the set that minimizes
the expected travel time:
𝐸𝑇𝑇𝑆 𝑤
𝑤
= 𝜙 ∙
𝑘
𝑙∈𝑆 𝑤
𝑓𝑙
+
𝑙∈𝑆 𝑤
𝑡𝑡𝑙
𝑤
∙ 𝑓𝑙
𝑙∈𝑆 𝑤
𝑓𝑙
[expected waiting + travel times]

𝑙1: 𝑡𝑡1, 𝑓1
𝑙2: 𝑡𝑡2, 𝑓2
𝑡𝑡1 ≤ 𝑡𝑡2 ≤ ⋯ ≤ 𝑡𝑡 𝑛
…
𝐸𝑇𝑇𝐴𝐸, 𝑆𝐴𝐸

2. On the resulting network, assign trips to their shortest
paths.
𝑙1: 𝑡𝑡1, 𝑓1
𝑙2: 𝑡𝑡2, 𝑓2
𝑡𝑡1 ≤ 𝑡𝑡2 ≤ ⋯ ≤ 𝑡𝑡 𝑛
…
𝐸𝑇𝑇𝐴𝐸, 𝑆𝐴𝐸

3. Compute the ridership for each service.
100
+50
+50
+50+50
80
+30+30
+20
etc…

There are many ways to model stochasticity in passenger
assignment on this problem. In our model we introduce it in
two ways:
• Stochastic choice for the set of attractive lines (step 1).
• Stochastic route choice (step 2).
Stochastic assignment

Selection of the set 𝑆 𝑤 is no longer deterministic. We model
this process using a Logit model. The cost of the O/D pair is
now represented by its expected maximum utility.
𝑙1: 𝑡𝑡1, 𝑓1
𝑙2: 𝑡𝑡2, 𝑓2
𝐸𝑀𝑈 = −
1
𝜃
ln
𝑙=1
𝑛
exp −𝜃 ∙ 𝐸𝑇𝑇 1,…,𝑖
…
𝐸𝑀𝑈 𝑤
Pr 1 Pr 1,2 Pr 1,2, … , 𝑛
…
𝐸𝑇𝑇{1}
𝐸𝑇𝑇{1,2}
𝐸𝑇𝑇{1,2,…,𝑛}
Pr 1, … , 𝑖 =
exp −𝜃 ∙ 𝐸𝑇𝑇 1,…,𝑖
𝑗=1
𝑛
exp −𝜃 ∙ 𝐸𝑇𝑇 1,…,𝑗
Stochastic assignment: attractive services

To model route choice, now we use Dial’s algorithm (1971).
𝐸𝑀𝑈 1,3 𝐸𝑀𝑈 2,4
𝐸𝑀𝑈 1,2 𝐸𝑀𝑈 2,3 𝐸𝑀𝑈 3,4
𝐸𝑀𝑈 1,4
𝑇 1,4 𝑇 1,4
Stochastic assignment: route choice
𝑇 1,4 ∙ Pr 𝑝1
𝑇 1,4 ∙ Pr 𝑝2
𝑇 1,4 ∙ Pr 𝑝3
Total demand for each service can be obtained like before, but
taking into account the probability of choosing each possible
subset 𝑆 𝑤.

A GRASP capacity algorithm.
3.

Dealing with bus capacity
CFOAP greedy heuristic
1 solve an instance of the FOAP, obtaining 𝑓 𝑛
.
2 while there is a line 𝑙 that shows a capacity deficit:
3 Select 𝑙′
as the line with the greater capacity deficit.
4 Increase the lower bound for 𝑓𝑙′.
5 Re-solve the FOAP, and update 𝑓 𝑛+1
.
This is a greedy heuristic that can be easily extended to a
GRASP algorithm (but we need a fast solution for the FOAP).
Leiva / Larrain use the following heuristic for solving the
capacitated design problem:

Dealing with bus capacity
CFOAP greedy heuristic
.
3 Select 𝑙′
as the line with the greater capacity deficit.
.
CFOAP GRASP heuristic
.
3 Select 𝑙′
randomly from the lines in deficit.
.

Case study
We optimized the design for three corridors: Pajaritos and
Grecia in Santiago, and Caracas in Bogotá. For each case we
generated a 20, 40 and 80 stop version of the O-D matrices,
thus defining 9 scenarios.
0
5000
10000
15000
20000
T 3 6 9 12 15 18 21 24 27 30 33 36 39 41 44 47 50 53 56 59 62 65 68 71 74 77 80
Pajaritos Grecia Caracas
1 40 41 80
Demand(pax/h)
Stops
20.000
15.000
10.000
5.000
0
Corridor loads

The experiment
Algorithm
Optimization
strategy
Passenger
assignment
Capacity
heuristic
Transfers
allowed
A0 Simultaneous Deterministic Greedy No
D / g / N Bi-level Deterministic Greedy No
D / G / N Bi-level Deterministic GRASP No
D / g / T Bi-level Deterministic Greedy Yes
D / G / T Bi-level Deterministic GRASP Yes
S / g / T Bi-level Stochastic Greedy Yes
S / G / T Bi-level Stochastic GRASP Yes
These algorithms were applied on each of the nine scenarios
described before.

• We confirm that by separating the problem we don’t lose the
quality of the solution.
• Transfers seem to have a bigger impact as demand grows.
• Greedy algorithms sometimes converge to suboptimal solutions,
but in general they perform reasonably well.
• In the stochastic case there are some unexpected trends, likely
due to suboptimality of the solutions. However, the algorithm still
manages to beat the all-stop solution.
General results: savings
CTC ($/h) Corrected percentage savings (CPS)
Scenario All-stop A0 D / g / N D / G / N D / g / T D / G / T S / g / T S / G / T
P20 16,920 48.5% 48.5% 48.5% 48.5% 48.5% 65.6% 65.6%
P40 33,847 56.2% 56.5% 56.5% 56.5% 56.5% 48.8% 48.8%
P80 68,353 63.7% 63.8% 63.8% 63.8% 63.8% 48.2% **
G20 18,070 22.2% 22.2% 22.2% 22.2% 22.2% 48.7% 50.0%
G40 36,589 31.3% 31.2% 31.2% 31.3% 31.3% 27.6% 27.6%
G80 73,873 35.7% 35.6% 35.6% 38.0% 38.0% 22.8% **
C20 22,142 13.2% 17.8% 17.8% 14.7% 18.3% 0.0%* 0.0%*
C40 36,540 30.2% 31.3% 31.6% 34.4% 34.5% 1.6% 2.6%
C80 172,995 79.4% 79.3% 79.6% 81.4% 81.4% 71.0% **

The effect of express services is much greater when transfer
costs are reduced!
The impact of transfer costs
Corrected percentage savings (CPS)
Scenario
(Transfer)
D / G / N
No transfers
D / G / T*
Transfers
D / G / B*
Bidirectional
D / G / FT*
Free Transf.
D / G / FB*
Free Bid. Tr.
P20 48,50% 48.5% (0) 48.5% (0) 48.7% (430) 48.7% (430)
P40 56,50% 56.5% (12) 56.5% (18) 57.3% (1169) 57.5% (1,882)
P80 63,80% 63.8% (511) 64.2% (2,192) 65.5% (10655) 66.9% (17,459)
G20 22,20% 22.2% (0) 22.2% (0) 23.6% (2100) 23.6% (2,100)
G40 31,20% 31.3% (187) 31.3% (412) 35.8% (7445) 35.8% (8,221)
G80 35,60% 38.0% (3041) 39.6% (7,074) 45.1% (16331) 47.6% (29,391)
C20 17,80% 18.3% (2085) 18.3% (2,085) 22.9% (11718) 22.9% (11,718)
C40 31,60% 34.5% (6786) 34.5% (6,786) 39.5% (19731) 39.5% (19,274)
C80 79,60% 81.4% (15950) 81.5% (17,383) 83.0% (31278) 83% (32,208)
* Number of transfers is given in parenthesis.

Solution times
The bi-level approach finds good solutions in just a fraction of
the time!
Scenario A0 D / g / N D / G / N* D / g / T D / G / T* S / g / T S / G / T*
P20 1 3 59 (10) 4 88 (10) 9 189 (10)
P40 32 11 232 (10) 11 378 (10) 161 973 (5)
P80 984 48 1,318 (10) 8 197 (10) 1,912 **
G20 3 28 222 (10) 9 267 (10) 127 1005 (10)
G40 31 4 86 (10) 3 79 (10) 40 349 (5)
G80 169 28 411 (10) 9 235 (10) 9,289 **
C20 17 10 128 (10) 3 135 (10) 10 135 (10)
C40 219 59 2,127 (10) 22 191 (10) 298 1573 (5)
C80 1,382 149 4,427 (10) 110 1364 (10) 17,241 **
* Number of iterations is given in parenthesis.

The impact of stochastic assignment
0
20
40
60
80
100
120
140
T 3 6 9 12 15 18 21 24 27 30 33 36 39 41 44 47 50 53 56 59 62 65 68 71 74 77 80
0
20
40
60
80
100
120
140
T 3 6 9 12 15 18 21 24 27 30 33 36 39 41 44 47 50 53 56 59 62 65 68 71 74 77 80
Bus capacity
All Stop
All stop
Deterministic assignment Stochastic assignment
Load(pax/bus)
1 40 41 80 1 40 41 80
Bus load per service
Stochastic assignment gives more robust solutions, spreading
demand among services instead of saturating a few of them.

What if we got assignment wrong?...
What if we design for deterministic passenger
assignment, but passengers really behave
stochastically?

0
120
240
360
480
600
T 3 6 9 12 15 18 21 24 27 30 33 36 39 41 44 47 50 53 56 59 62 65 68 71 74 77 80
Deterministic design facing stochastic behavior
All Stop
All stop
L17
L36
L23
1 40 41 80
Bus capacity
Load(pax/bus)
What if we got assignment wrong?...
Deterministic assignment underestimates regular-like service
demands!

• We formally introduce the Limited-stop Service Design
Problem and propose a solution framework for it.
• We greatly improve solution times for the deterministic
version of the Capacitated Frequency Optimization and
Assignment Problem, which solves the LSDP for a given set
of services.
• Stochastic assignment leads to more realistic and robust
solutions, but makes the problem harder to solve. The bi-
level approach finds solutions for this variant of the
problem, but there is room for improvement.
Conclusions

• Allowing for transfers during service design can lead to
better solutions. We also can conclude that reducing the
nuisance associated with transfers can improve the
performance of limited-stop services.
• Future research should tackle the issue of capacity at a bus
stop level, which is an active constraint in existing systems,
but adds a new layer of complexity to the problem.
• Another future avenue for research consists in solving the
design problem at a BRT network level, which would allow
to better consider during the design phase the complex
travel patterns of the system.
Conclusions

Homero Larraín - The limited stop bus service design problem with stochastic passenger assignment

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