Mobile Device Programming for Finding Optimal Path
on a Fast-Bus System
Sonia Vivas and Germán Riaño
Abstract— In this project, we developed a real-time software
solution that helps users to find the optimal combination of
routes in a bus-based passenger mass transportation system,
TransMilenio (TM), in Bogotá, Colombia. The TM users face
the need to choose the best route between two points, and the
information presented, in form of maps and operation hours
for each route, usually is not enough to make that choice. Even
if a user could have all the information about the system, the
decision would not be trivial. Also, there is not a unique way to
define the optimality criteria, since besides total travel time, the
user might want to avoid changing buses which might involve
long walks in the stations, or avoid congested routes to increase
the chance of finding an available seat. This paper describes
the problem faced and the proposed solution.
I. I NTRODUCTION
Nowadays, mass transportation systems are an important
part of our lives. In any of these systems, finding the best
route between two points means a problem, because for
doing so one should not only know all the information about
the system, but be able to process it at great velocity. Usually,
this decision is defined by the expertise of the user and the
randomness of the system.
In some transportation systems, information is given to
the user, so he can make a better choice, even more, there is
software available for metro systems that allow users to find
the best route for them.
Is then clear that the problem is interesting, and it goes
further than the merely search for a shortest path, it also
involves the user necessity of ubiquity for any solution to
this problem.
With that in mind, the propose of this research was to find
a solution for the shortest path problem in the TransMilenio
(TM) system, in Bogotá, Colombia. This is a fast bus system,
which will be described later.
We developed a model of this system so it could be
used computationally. The shortest path problem was then
redefined so it could take in consideration all the extra data
available (like schedules, occupation levels, stochastic times).
An algorithm was developed for this research project, and finally a software application was developed. This application
was programmed for mobile devices, so it could be used by
any user, anywhere and at all times.
Our choice of programming for mobile devices arose from
the problem that some other solutions might have: they are
S. Vivas is with the Department of Computer Science, Universidad de
los Andes, Colombia
G. Riaño is with the Centro de Optimización y Probabilidad Aplicada
– COPA, Department of Industrial Engineering, Universidad de los Andes,
Colombia.
Fig. 1.
TransMilenio Buses(Taken form [1])
attached to a single place, even the software for computers is
not as portable as one could like, because portable computers
are not popular between most of TM users. Cellular phones,
on the contrary, are really popular, and receiving and sending
calls are the smallest tasks that users perform on their phones.
These means that any tool for these devices will be broadly
accepted. However the algorithm runs on a server, so, in
the future, other clients (like a web page) can be written to
access the information.
The results are basically two: an algorithm that finds the
shortest path, and considers many of the TM characteristics
and an application that implements the algorithm on mobile
devices.
This paper will explain the specific problem addressed and
will resume the results of the research. The remainder of the
paper is organized as follows. In Section II we describe the
characteristics of the TransMilenio system, in Section III we
review some of the shortest path problems and algorithms
that solve them, in Section IV we explain the network design
used to represent the problem and the solution algorithms
and software is described in Section V. In section VI we
present the results obtained with the program and, finally, in
Section VII we give some concluding remarks and propose
developments that can be build as extensions of this research.
II. T HE T RANS M ILENIO SYSTEM
For a better understanding of the problem, it is necessary
to explore the particular characteristics of the TransMilenio
(TM) system. TM consists on a fleet of articulated buses
that run on exclusive corridors (see Figure 1). These buses
stop at stations, which are distributed along the city. Since
the corridors have two lanes not all the buses stop in every
station along the corridor. Groups of buses cover a route path,
meaning they run on a specific corridor and stop on some
specific stations, so it is usual for two routes to take the same
corridor, but stop in a different set of stations. For sake of
clarity, think of two different routes 50 and 70, these two run
on the same route (see Figure 2), but as you can see in the
map, the set of stations in which they stop is different from
each other and the number of stops is different, affecting the
time between two stations.
For the weekdays, there are 18 different routes, and
because some of them stop in the same station, for several
stations, there is a significant number of possible route
changes. There are multiple reasons to change from a route
to another different of the path like comfort or speed.
Beside the number of routes, the system has 94 stations.
The different routes that stop at a station, stop in different
doors, so in order to calculate the total time of a specific
path, it is necessary to include the walking time between
doors, and the waiting time in the case of a change of bus.
In order to serve the traffic of people, TM developed a
time schedule, so not all the routes are available all the time
(and nor every day).
A user can decide which routes to take using the information available: in the TM web page, where he can find the
TM map, maps in the stations and information boards, which
inform about the remaining time for a bus to arrive ( these
boards inform only about the routes that stop in the door
where they are installed); TM employees, who can inform
about the stops of a route, and give their personal opinion
about the velocity and comfort that each route could provide.
III. S HORTEST PATH P ROBLEM
The shortest path problem is interesting because it can
be used to solve a broad amount of every day problems, as
transport and telecommunication issues, turning the shortest
path problem in one of the most important problems in the
study of network flow models [2].
This problem can be defined as the need of finding an
optimal path in a specific graph. The specification of the
begin and end points, the weights of the edges and the
definition of ‘optimal’ determine the type of the problem.
The simplest definitions of the shortest path problem are the
simple shortest path and the shortest path between all pairs
of nodes.
A. Simple Shortest Path Problem
This problem can be explained considering a graph G =
(N, A), where N represent a set of nodes and A is the set
of edges, and let s ∈ N be a node chosen as origin. The
problem consist in finding all the shortest paths between s
and all the other nodes v ∈ N . Is common to consider that
the edges have an associated weight, so the shortest path will
not be the one with less number of nodes, but the one with
minimum weight, thinking in the weight of a path as the sum
of the weights of all the edges that are part of the path.
If the edge has a negative weight cycle, reachable from
the beginning node s, a shortest path could not be defined to
the nodes in the cycle, nor to any node v reachable trough
the cycle. This because in each turn of the cycle, the weights
will be lower, so if the cycle is repeated lots of times, the
weight between s and v would be inf inity.
B. Shortest Path Between All Pairs of Nodes
Using the same notation as before, we define this problem
as finding the shortest path between all pairs of nodes (i, j) ∈
N × N . [3]. This problem can be taken as n problems of
simple shortest path.
C. Problem variants
There are lots of variants to the shortest path problem [4],
between them:
• Shortest path with one begin and one end. It issues a
simple shortest path problem, with only one end.
• Shortest path with one end. It addresses the shortest
path problem between every node in the graph and a
specific end point.
• Maximum capacity path. In this problem, the weights
in the edges are considered flow capacities, so the
objective path is that in which the capacity is maximized
[5].
D. Problem Solutions
The most known algorithm to solve this kind of problem
is the Dijkstra algorithm, nevertheless there are diverse algorithms that address different specific problems, depending on
the application and the variation of the shortest path problem.
In this paper we will consider only two algorithms: Dijkstra
and Brumbaugh-Smith and Shier algorithms. The reason for
this decision is that the implemented solution only uses this
two algorithms. Other interesting algorithms for this problem
are: A*, Floyd-Warshal, Johnson’s Algorithm. [6]
E. Dijkstra Algorithm
Dijkstra algorithm is one of the simplest an most effective
shortest path algorithms. It works on non-negative weighted
graphs, solving the simple shortest path problem. The algorithm uses the greedy method, in which in order to construct
the solution for a big problem, smaller sub problems are
solved, and their size is incremented until these sub problems
reach the objective. It resembles dynamic programming, but
it does not save different partial solutions, it only saves one
(the best one).
The necessary time for the Dijkstra algorithm in a graph
with m nodes and m depends on the way that Q is implemented. If q is implemented as a linked list or an array, the time
would be O(n3 ). With a binary algorithm, the time required
will be O((m + n) log n). With a Fibonacci algorithm, the
time will be O(m + n log n). [2]
In case that the graph involves negative weights, the
Bellman Forman algorithm will provide a solution, but this
algorithm is less efficient than the Dijkstra algorithm, so it
is only used in the case of negative weights.
F. Brumbaugh-Smith and Shier Algorithm [7]
This algorithm finds a solution for the bi-objective shortest
path problem. The algorithm considers a graph and two
objectives c and t. Using a labeling technique, the algorithm
steps in some nodes, from the origin node to the destination
node. In each of them, the algorithm consider the linear
Fig. 2.
Map of the TransMilenio System. Taken from [1].
maximization of the objectives, storing this answer as the
best route from the beginning node to the node that is being
reviewed, this answer is stored as the best c and t for the
node, and the id of the last node in the path that is being
stored. Not necessarily every node is explored, the nodes
reviewed are the ones that are connected in a path to the
beginning node. The algorithm returns the best route for
every reviewed node, so it is necessary to backtrack the path
from the destination node to the begin node.
G. The Extended Brumbaugh-Smith and Shier [8]
Suárez proposed an extended algorithm that allows to
incorporate dynamic behavior in the model. Suárez addressed
the TransMilenio problem, considering as the objectives the
minimization of time and of bus changes. His algorithm
requires a weighted graph with no negative cycles, the
begin and destination nodes and the time restrictions for the
edges, making the graph dynamic (the edges are or not valid
depending on the time). Besides he asks the final user for a
maximum number of bus changes that he is willing to make,
let call it M . The algorithm works as the original, but it stores
all possible solutions for each node, and before storing a
solution, reviews the time restrictions, so the solution will
only be stored if all the edges are active at the time of
the query. The algorithm returns a set of solutions for each
node, in order to find the paths it is necessary (as in the
original algorithm) to backtrack each path beginning at the
destination node. With this paths, Suárez uses the M index
to choose the ones that are valid, according to the index, and
he returns the path with less time, that is valid according to
the index, and that has the less number of bus changeovers.
IV. S HORTEST PATH P ROBLEM IN THE T RANS M ILENIO
S YSTEM
The problem we want to address is a variant of the shortest
path, that includes concepts as stochastic weights, multiple
objectives and time restrictions. We want to minimize the
time that the user expends inside the TransMilenio system,
minimize the quantity of bus changeovers, minimize the
level of occupation of the buses in the path, take in account
the dynamism of the graph and incorporate the concept of
random parameters.
This definition of the problem could mean that we need a
really complex algorithm, but the problem characteristics are
usually limited by the user will. Checking the characteristics
one by one, the limits become clear:
•
•
•
Bus changes There is a maximum quantity of bus
transfers that a user is willing to do. Checking the
map, is clear that the maximum number of necessary
bus changes between any two points is 3, nevertheless
a bigger number could be chosen in the sake of the
minimization of time, or in the minimization of the bus
occupation. In this way, the problem of minimizing the
number of bus transfers is reduced to a non-negative
restriction with a upper bound.
Occupation The occupation of a bus could be defined
as the number of persons that are in the bus divided
by the capacity of the bus. This characteristic is not
an imperative need, so the decision can be leaved to
the user to choose, transforming this restriction in just
information for the user choice.
Dynamic graph This characteristic does not depend on
T , so its mean is known to be T /2. Since the frequency of
buses change during the day, the time associated with these
edges also changes.
Since not all routes operate at all times the model has a
dynamic behavior. To model this behavior, each edge has a
state, which can be either active or inactive.
B. Model Assumptions
•
Fig. 3.
From the TM Map to the TM Graph
•
•
the user but on the time of the query, so the solution to
the problem can not have inactive edges, and that must
be secured by the solution algorithm.
Total time This characteristic is the most important one
for the users of this type of system, in fact the short
times are part of the publicity of the system, so this
will be the parameter to minimize.
V. I MPLEMENTED S OLUTION
The problem was reduced to a variation of the simple
shortest path problem, that takes in account restrictions and
dynamic edges.
The solution considers a graph that models the TransMilenio system, and uses the algorithms of Dijkstra and of
Brumbaugh-Smith and Shier. The model and the algorithm
will be described in this section.
•
•
•
•
•
The time needed to go from one station to another is a
random variable, and it is represented by the mean and
the standard deviation.
The time needed to change from one bus to another is
a random variable, and it considers the time needed to
walk between doors and the time needed to wait the
new bus.
The hour in which a route is scheduled to begin is taken
as the moment in which the first bus of the route start
running from the two portals of the route.
The system is modeled as it was at December 31 of
2005.
We only consider the express and common routes from
Monday to Friday, de super express and the weekend
routes are not taken in consideration.
The time and occupation levels are estimates, and
are based on official TransMilenio information and in
intuition on busy traffic hours.
The stochastic nature of the edges is vaguely exploited, although the model uses media and deviation, the
stochastic variables could be deeply exploded to obtain
even more realistic solution.
A. TransMilenio as a Graph
In order to model the TransMilenio system, we consider
a set of nodes and edges for the graph, and a set of routes
for the algorithm.
• Nodes: There are two types of nodes: The first represent
pair (station, route), allowing to model bus changes, and
the second station nodes, identified by the station name.
• Edges: We defined three types of edges: entrance,
movement and bus transfer. The first ones link a station
node and a regular node, reflecting the time of waiting
for the first bus to arrive; the second ones consider
tuples with same route and different station, reflecting
the movement time; the third ones consider tuples with
the same station and different routes, reflecting bus
changeovers. In the Figure reffig:tmmtg a example of
the model is shown.
Graph Characteristics: Since we want to involve the
characteristics of TransMilenio, we need to define some
details for the graph: The movement edges had an associated
weight, specified by a media and a deviation. This weight
represents the time needed to go from one station to another
in a specific route. The transfer edges consider the time
needed to walk between two doors (the doors where the
buses stop), and the waiting time for the new bus. This time
is random. If the buses come deterministically, one every T
minutes this is a uniform random variable between 0 and
C. Proposed Algorithm
Taking into account the Suárez [8] algorithm, the introduction of the new model characteristics should be easy.
Nevertheless the model defined is really big, and could have
lots of cycles. Besides the idea of implementing the solution
on mobile technology force us to use an efficient algorithm,
and because of the size of the model, the Suárez algorithm
is not as efficient as we may want.
The proposed algorithm consider as beginning node a
station node, because the user knows in which station he
is, but he does not necessarily know which should be the
first route to use.
Under the need of efficiency, we use an important assumption: we consider a perimeter when searching for the
shortest path between two stations a and b. The perimeter
contains 3 different sets: the begin and end stations (a and
b), the stations that are physically between a and b, and the
stations that are less than 3 stations away (physically) from
the stations in the last two sets. This allows the algorithm to
ignore some stations (and nodes) making the search quicker.
The algorithm also checks that a path does not return
to a station, in other words, unless we are dealing with a
bus changeover, the path can not have two or more nodes
with the same station. The presence of infinite cycles is also
checked, in the every bus change there is a possibility for the
labeling algorithm to collapse, because the nodes involved in
the change could be checked repeatedly forever.
The maximum number of bus transfers is chosen by the
user between 1 and 5.
The proposed algorithm requires a graph, the begin and
end station, the maximum number of bus transfers, the time
schedule of the routes and the physical station distribution.
The algorithm starts defining the perimeter and then start
a cycle in which for each labeled node s, s is unlabeled,
the route changes are reviewed (to avoid infinite cycles) and
then, according to the perimeter, a node n is chosen from
the set of next nodes from s is checked, to validate the route
schedule, and if it is valid, the number of bus changeovers
is checked so it will be inside the limits; if it is, the solution
is added to a list of solutions for the node that is being
reviewed. The solution is formed by the time, the deviation,
the previous node of the path and the occupation level. The
node n is labeled.
The result of the cycle is a set of paths for each reviewed
node, and the path from de begin node to the end node is
backtracked as in the Brumbaugh-Smith and Shier original
algorithm.
Since this algorithm takes a little much time for a mobile
application, we use first Dijkstra algorithm to provide a faster
answer, and then we allow the user to run this algorithm.
D. Implementation
Fig. 4.
•
•
•
The solution was implemented using JAVA as programming language, creating a solution that is able to run in most
of the operatives systems. We used the J2ME (Java Micro
Edition) [9] as the environment for mobile applications.
The Mobile Information Device Profile (MIDP), is the Java
runtime environment for the most popular compact mobile
information devices, such as cell phones and mainstream
PDAs.
The application was implemented in a server - client
architecture, principally because of the size restrictions of
the MIDP package. The communication between the server
and the client is based on sockets and threads.
The client was implemented as a light MIDP application.
It has classes that structure the user interface and the path
management. The client is concerned with the translation of
the server data into clear information to the client.
In the server is where the algorithm was implemented. The
server reads some xml archives, and with that information it
loads the TransMilenio graph.
Both the server and the client are implemented in a way
that allows to easily extend the software and to use the
classes that are already implemented as base for new classes
that implement the methods to serve any particular problem.
The application is based on the work of K. Knizhnik, who
developed a mobile application for finding the shortest path
in the Kiev metro [10].
VI. R ESULTS
To describe the application we will consider its use cases.
The initial screen can be seen in Figure 4.
•
•
•
•
Initial Screen and Choose Station Screen
To choose the begin and destination stations The application allow the user to display all the stations either
by runway, by route or in alphabetic order. Once both
stations are chosen, the application displays the shortest
path using the Dijkstra algorithm (without concerning
about restrictions). See Figure 4.
To see the description of the shortest path Once the
shortest path is calculated, the user can review this
solution. The application shows the time, the deviation
of the time, the number of bus changeovers, the stations
in which the routes stop, the description of the bus
changeovers, the routes that must be taken in this path
and the level of occupation of each of these routes.
To choose a maximum number of bus changeovers The
system display a “It does not matter” sign and the
numbers from 1 to 5, so the user can choose. If the
user choose that it does not matter, the preset value is
3 bus changeovers.
To see more solutions Once the begin and end stations
are selected, and the maximum number of changeovers
is set, the user can check the solutions obtained by the
implemented algorithm. When the user select to see
more solutions, the application displays a set of the 10
better possible solutions, described in terms of time,
time deviation, number of bus changeovers and media
of the occupation levels.
To select a solution When the user is seeing the possible
solutions, he can select a solution, returning to the main
menu. The time of the selected solution is displayed in
the screen.
To see the description of a solution The information
displayed is the same as in the case of the seeing of the
description of the shortest path. Figure 5
To see the map of a solution This choice displays
the TransMilenio map in which the chosen solution is
marked. Figure 5
The algorithm has a poor efficiency in matter of time, nevertheless the effectiveness is complete, the paths are found
correctly and the model restrictions are taken in account. In
order to determine the algorithm efficiency, we conducted
10 tests between every pair of stations at a high traffic time.
The tests were conducted in a AMD Athlon XP 1800, with
480 MB in ram. The shortest time for the algorithm to
find an answer was observed in pairs of physically close
•
•
•
•
•
Fig. 5.
Path Description and Path Map Screens
Improve the algorithm efficiency, by means of data
structures.
Include the routes that have not been considered.
Run a market research in order to find out the user
preferences when choosing a path.
Consider the possibility of link the application with a
satellite in order to know where are the buses, and, with
this, obtain real times.
Consider dynamic times, times that change in time. For
example, in a busy traffic hour, the waiting for a bus
time will be longer than in a low traffic hour. Same as
in the case of rain, or in the case of a traffic accident.
R EFERENCES
stations. Although is difficult to resume in one average all
the possibilities, we can assure that in close stations, the time
of search is between 0.5 and 5 seconds. The problem comes
between physically far stations (from one portal to another),
the average time is between 1 and 3 minutes. The average
time for a consult between two stations is 0.7 minutes. This
tests were done with a limit of bus changeovers set to 2.
If this number is augmented, the time grows exponentially,
meaning that with 5 bus changeovers, the average time will
be 3 minutes.
VII. C ONCLUSIONS AND F UTURE W ORK
The principal objective of the realized study was to give
to the TransMilenio user a tool for facilitate the decision of
which path to take between two stations of the system. The
solution was an application, implemented for mobile devices,
that find a set of good solutions and leave the final decision
to the user.
The algorithm takes in account the time schedules of the
routes, the stochastic nature of the displacement times, and
a limit in the number of bus changeovers. The algorithm
considers also a perimeter, which allows to leave apart some
stations in the quest for the best path.
The final solution set has 10 solutions that minimized time,
minimized number of changeovers and considered the level
of occupation.
The algorithm efficiency is diminished by the graph complexity. Of course there is a bias introduced by the machine
in which the test were realized. The run times could be
improved in a better machine.
The application was developed thinking in expansion
possibilities, therefore any limitation that could be inserted
in this work can be easily overcome.
This research van be continued in a number of ways:
• Exploit the stochastic nature of the graph, to obtain
more realistic times.
[1] Transmilenio S.A., “http://www.transmilenio.gov.co,” 2005.
[2] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows. Upper
Saddle River, NJ, USA: PrenticeHall, Inc, 1993.
[3] T. T. Shane Saunders, “Efficient algorithms for solving shortest
pathson nearly acyclic directed graphs,” Research and Practice in
Information Technology, vol. 41, 2005.
[4] D. P. Bertsekas and J. N. Tsitsiklis, “An analysis of stochastic shortest
path problems,” Mathematics of Operations Research, vol. 16, pp.
580–595, 1991.
[5] P. Charnsethiku and K. Virojsailee, “The linear constrained maximun
capacity path problem,” Industrial Engineering Departament, Kasetsart
University, Bagkok, Thailand, Tech. Rep., 2000.
[6] S.-J. Lee, “Routing and multicasting strategies in wireless mobile ad
hoc networks,” Ph.D. dissertation, University of California, 2000.
[7] J. Brumbaugh-Smith and D. Shier, “An empirical investigation of some
bicriterion shortest path algorithms,” European Journal of Operational
Research, vol. 43, pp. 216–224, 1989.
[8] J. P. Suárez, “Aproximacion al problema de ruta más corta con
transbordos,” Master’s thesis, Universidad de los Andes, 2005.
[9] S. Microsystems, “Java technology,” Jan. 2006. [Online]. Available:
http://www.java.sun.com/
[10] K. Knizhnik, “J2ME applications for mobile phones,” Moscow
State University, http://www.garret.ru/ knizhnik/mobile.html, Tech.
Rep., Sept. 2005. [Online]. Available: http://www.garret.ru/∼knizhnik/
mobile.html
AUTHORS BIOGRAPHIES
Sonia Vivas received a B.Sc. (2005) in Systems Engineering
from Universidad de los Andes. She is a B.Sc. student in Industrial Engineering and and this is her graduation project. She
is also a M.Sc. student in Computer Science at Universidad de
los Andes. She works with the Formal Methods Group of the
Computer science Department. She can be reached by e-mail at
[email protected].
Germán Riaño received a M.Sc. and a Ph.D. from Georgia
Institute of Technology, U.S.A. in 1996 and 2002, respectively.
He also holds B.Sc. degrees in Industrial Engineering and Physics
form Universidad de los Andes in Colombia, where he is currently
working as an Assistant Professor. He is a founding member of
the Centro de Optimización y Probabilidad Aplicada (COPA). His
research interests include the development of tools for stochastic
modeling. He can be reached by e-mail at [email protected]
and his web address is http://wwwprof.uniandes.edu.
co/˜griano/.