A FUZZY LOGIC INFERENCE APPROACH FOR THE
ESTIMATION OF THE PASSENGERS FLOW DEMAND
Aránzazu Berbey Alvarez, Rony Caballero George
Universidad Tecnológica de Panamá, Ciudad de Panama, Panama
Juan de Dios Sanz Bobi, Ramón Galán López
Universidad Politécnica de Madrid, Madrid, Spain
Keywords: Fuzzy logic inference, Passenger flow modelling, Maximum entropy theory, Matrix method of area trips O-
D.
Abstract: This paper presents a new approach that designs the flow of passengers in mass transportation systems in
presence of uncertainties. One of the techniques used for the prediction of passenger demand is the origin-
destination matrices. However, this method is limited to urban areas and rarely to explicit stations.
Otherwise, the gravity models based on friction functions can be another alternative; however, it is difficult
to fit into practical achievements. Another solution might be the application of artificial intelligence
techniques so as to include some intuitive knowledge provided by an expert to predict the flow demand of
passengers’ trips in explicit stations. This paper proposes to combine a matrix of origin-destination trips of
travel zones, with the intuitive knowledge, applying a fuzzy logic inference approach.
1 INTRODUCTION
The passenger flow modelling has been tackled by
several authors (Kiluchi et al, 1999) (Aldian et al,
2003) (Watson and Prevedouros, 2006) (Cheng et al,
2009) (Murat, 2010) (Xie et al, 2010). Some of them
have applied linear programming methods
considering small samples (Murat, 2010). It has also
been applied the maximum entropy theory (Xie et al,
2010) and results have been compared between
different methods (Watson and Prevedouros, 2006).
However, it has been shown that these methods have
limitations in certain scenarios. In this case, the use
of fuzzy logic has proven to be a promising tool
because it can integrate the railway planning
experts’ experience in multiple scenarios (Aldian et
al, 2003) (Cheng et al, 2009). This paper proposes to
integrate the effectiveness of the methods based on
origin-destination matrices, with the experience of
experts in railway planning using an artificial
intelligence support based on fuzzy logic.
2 AREA ANALYSIS
The matrix method of area trips O-D is one of the
most used methods to design the movement of
passengers. This method divides the urban
environment in areas of interests that generate and
attract trips. However, due to economic and practical
reasons, the size of the interests’ zones is usually
large in order to apply directly to urban planning
models. Therefore, it is interesting in many cases to
divide these macro areas into smaller areas, allowing
a better analysis of passenger flow.
3 SUBZONE DESIGN
Considering the attraction vectors
1
m
n
T
D
×
∈ \ and the
generation vectors
1
m
n
T
O
×
∈ \ , each element of the
attraction vectors D
TJ
and generation O
TI
can be
subdivided into n
J
subzones, becoming two new
vectors
1
s
n
D
×
∈
\ and
1
s
n
O
×
∈ \ , where
1
m
n
s
J
J
nn
=
=
∑
.
125
Berbey Alvarez A., Caballero George R., de Dios Sanz Bobi J. and Galán López R..
A FUZZY LOGIC INFERENCE APPROACH FOR THE ESTIMATION OF THE PASSENGERS FLOW DEMAND.
DOI: 10.5220/0003057701250129
In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation (ICFC-2010), pages
125-129
ISBN: 978-989-8425-32-4
Copyright
c
 2010 SCITEPRESS (Science and Technology Publications, Lda.)

Therefore, the macro areas matrix
mm
nn
T
×
∈ \ is
subdivided at the same time, obtaining sub-zone
matrix
s
s
nn
M
×
∈ \ . Consequently, it can be shown
that every element of the attraction and the
generation vectors, as well as the macro area and the
subzone matrices must meet with:
()
()
1
E
B
mJ
TJ j
jmJ
DD
=+
=
∑
(1)
()
()
1
E
B
mI
TI i
imI
OO
=+
=
∑
(2)
()
()
()
()
11
EE
BB
mI mJ
I
Jij
imIjmJ
TM
=+ =+
=
∑∑
(3)
1
1
s
n
jij s
i
A
Mj n
=
=∀=
∑
"
(4)
1
1
s
n
iij s
j
BMin
=
=∀=
∑
"
(5)
where
()
1
1
x
BK
K
mx n
−
=
=
∑
(6)
()
1
x
E
K
K
mx n
=
=
∑
(7)
0
0n =
(8)
To obtain a subzone model, a system is governed
by the principle of maximum entropy would be
applied. However, this model can be improved by
incorporating information by an expert, even if this
is inaccurate; this supposition is done with the aim to
improve the approximation, and it means to apply
the principle of maximum entropy. The information
can be included using techniques based on artificial
intelligence and more specifically based on fuzzy
logic.
We have to apply the following steps to set up
the subzone model:
a)
T
IJ
, O
TI
and D
TJ
have little uncertainty
within the same planning horizon.
b)
Every element of the vector O
i
and D
j
,
corresponding to a subzone
i or j, that is
part of the macro area
I or J respectively,
can be represented as a function of O
TI
or
D
TJ
vectors of the macro area in which it is
contained, and by a potential function with
exponents
C
DJ
or C
Oi
.
()
()
1
ln 2
1
ln 2
1
2
2
Dj
E
Dk
B
C
jTJ
mJ
C
J
kmJ
DD
n
=+
=
∑
(9)
()
()
1
ln 2
1
ln 2
1
2
2
Oi
E
Ok
B
C
iTI
mJ
C
I
kmJ
OO
n
=+
=
∑
(10)
c)
Every element of the trip matrix M
ij
corresponding to the subzones
i, j and that
is part of the macro area
I, J respectively,
can be represented as a function of
T
IJ
and
by a potential function with exponent
C
Mij
.
()
()
()
()
1
ln 2
1
ln 2
11
2
2
Mij
EE
Mrs
BB
C
ij IJ
mI mJ
C
IJ
rmIsmJ
M
T
nn
=+ =+
=
∑∑
(11)
d)
C
Dj
, C
Oi
and C
Mij
come from a fuzzy
inference engine based on the experience of
an expert.
It is important to note that the exponents
C
Dj
and
C
Oi
are related to the relative level of importance of
the station in a given planning horizon. If these
exponents are zero, the estimation becomes the
estimation by maximum entropy, whereas if you
have a negative or positive number it corresponds to
a station with low or high demand respectively.
On the other hand the exponent
C
Mij
establishes
the relative level of importance of the passenger
flow between two stations. Here, the estimation
using the maximum entropy also corresponds for a
value equal to zero, and a negative or positive
number corresponds to a flow of low or high
demand respectively.
4 FUZZY INFERENCE ENGINE
The exponents C
Dj
, C
Oi
and C
Mij
can be designed
using the help of fuzzy logic. In this particular case
it seems reasonable to use triangular membership
functions (Figure 1). These exponents can be
estimated not only considering the proximity of the
station by major population centres, such as:
residential areas, industrial estates, hospitals, schools
and shopping centres, but also with the presence of
transport interchanges.
ICFC 2010 - International Conference on Fuzzy Computation
126

VL Very Low
L Low
M Medium
H High
VH Very High
Figure 1: Starting fully logic functions for C
Dj
, C
Oi
y C
Mij
.
5 PROPOSED ARGORITHM
a) Estimate the exponents C
Dj
, C
Oi
and C
Mij
basing on the inaccurate available
information.
b)
Estimate the trip attraction and generation
vectors
D and O, for the stations using,
()
()
1
ln 2
1
ln 2
1
2
2
Dj
E
Dk
B
C
jTJ
mJ
C
J
kmJ
DD
n
=+
=
∑
(12)
()
()
1
ln 2
1
ln 2
1
2
2
Oi
E
Ok
B
C
iTI
mJ
C
I
kmJ
OO
n
=+
=
∑
(13)
c)
Estimate the trip matrix M
ij
, using,
()
()
()
()
1
ln 2
1
ln 2
11
2
2
Mij
EE
Mrs
BB
C
ij IJ
mI mJ
C
IJ
rmIsmJ
M
T
nn
=+ =+
=
∑∑
(14)
d)
Thus it is checked if
1
1
s
n
iij s
j
OMin
=
=∀=
∑
"
. Consequently, if
1
s
n
ij i
j
MO
ε
=
−<
∑
is met for a given value, it
is accepted. Meanwhile, if not, the values
of the matrix elements are corrected with
1
s
iij
ij
n
ij
j
OM
M
M
=
⇐
∑
(15)
e)
It is checked if
1
1
s
n
jij s
i
DMjn
=
=∀=
∑
" .
Consequently, if
1
s
n
ij j
i
MD
ε
=
−<
∑
is met,
it is accepted. Meanwhile, if not, the values
of the matrix elements are corrected with
1
s
j
ij
ij
n
ij
i
DM
M
M
=
⇐
∑
(16)
f)
It is checked if
()
()
()
()
11
EE
BB
mI mJ
I
Jij
imIjmJ
TM
=+ =+
=
∑∑
for
all
i, j, I, J. Consequently if,
()
()
()
()
11
EE
BB
mI mJ
ijm IJ m
imIjmJ
MT
ε
=+ =+
−<
∑∑
is met, it is
accepted. Meanwhile, if not, the values of
the matrix elements are corrected with
()
()
()
()
11
EE
BB
IJ ij
ij
mI mJ
ij
imIjmJ
TM
M
M
=+ =+
⇐
∑∑
(17)
g)
Go back to step d, if tolerances are not met.
6 RESULTS FOR SIMULATED
DATA
The following case illustrates how the proposed
algorithm works. Let’s consider the railway network
topology shown in Figure 2. In this case it is known
from previous studies that the mobility preferences
among three regions so-called A, B and C, but
without giving any specific details concerning the
connection preferences between specific stations.
The region A has the stations E1, E2 and E3; the
region B has the stations E4, E5, E6 and E7; and the
region C has only two stations E8 and E9.
O-D matrix between these regions is known and
it is described below:
A FUZZY LOGIC INFERENCE APPROACH FOR THE ESTIMATION OF THE PASSENGERS FLOW DEMAND
127

Table 1: O-D matrix between the regions.
A B C
A 9397 5282 5213 19892
B 25118 8272 5065 38455
C 22570 8732 1134 32436
57085 22286 11412 90783
Figure 2: Red topology.
We have to consider the estimated subzones
exponents’ vectors
[
]
D
C AAMAMMMMAB=
,
[
]
o
CAMMAMMAMAB=
and the exponents matrix,
D
A
M A M M M MA MB
A
BMM BM BMB
A
AMBBBMMB
M
MB MMMM B
C
M
MBM MMMMB
M
MB AM MMMB
A
AMMAA A A B
A
AM A A M A MA
B B MB MB MB MB MB M
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
=
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
After, applying the proposed algorithm, we
obtain the following matrix OD,
Table 2: Resulting O-D matrix using the proposed
algorithm.
0 2945.4 1142.1 1667.2 433.1 575.7 486.1 4203.5 85.9
2051.3 0 381.6 557.1 289.4 192.4 324.9 351.2 28.7
1467.9 1408.6 0 398.7 103.6 137.7 116.2 502.6 41.1
2036.9 1954.6 757.9 0 295.3 392.6 331.5 872.9 142.7
1978.7 1898.8 736.3 552.2 0 381.3 322 847.9 69.3
1853.7 1778.8 689.8 1034.6 268.7 0 301.7 794.4 64.9
4903.1 4705 1824.4 2736.4 710.8 944.9 0 2101.1 171.8
8445.1 8104 3142.4 3292.4 1710.5 1136.9 1920 0 783.6
1234.6 1184.7 459.4 240.6 125 166.2 140.3 350.4 0
Which is compared to the original test matrix
Table 3: Original test matrix.
0 2585 1155 920 543 460 485 3247 211
1769 0 653 510 322 306 353 538 151
1704 1531 0 519 302 267 295 901 165
2282 2194 1012 0 465 401 440 844 231
1972 1935 832 687 0 441 496 828 179
2049 2064 946 778 519 0 598 789 212
3953 4127 1752 1554 1009 884 0 1557 425
7308 7405 3766 3018 1617 1398 1556 0 805
1447 1440 1204 343 263 264 263 329 0
We can see that the results for the dominant
values of the matrix are approximated; obviously the
errors are due to the uncertain information coming
from the expert, but still much lower than using the
approximation for maximum entropy.
Table 4: Resulting O-D matrix using the maximum
entropy method.
0 1181.2 1228.4 526.2 353.4 438.1 351.4 2011.1 540.3
2153.9 0 1183.9 507.2 681.1 422.2 677.4 484.6 520.8
1774.2 1875.5 0 417.8 280.5 347.8 279 798.3 857.9
1915.8 2025.2 2106.1 0 593.9 736.3 590.7 522.6 1123.2
2069.3 2187.5 2274.9 477.7 0 795.3 638 564.4 606.6
2001.9 2116.2 2200.7 924.2 620.6 0 617.2 546 586.8
1970.7 2083.2 2166.4 909.8 610.9 757.4 0 537.5 577.7
3243.1 3428.3 3565.2 1129.6 1517.1 940.4 1508.8 0 885.8
3907.4 4130.5 4295.4 680.5 913.9 1133 908.9 248.2 0
Making a comparison of the percentage of the
errors of both methods shows that the method of the
proposed algorithm presents a certain amount of
errors lower than the highest entropy method.
Hereunder is the matrix that arises from the
proposed algorithm.
Table 5: Percentage errors using the proposed algorithm.
-13,94 1,12 -81,22 20,24 -25,15 -0,23 -29,46 59,29
-15,96 41,56 -9,24 10,12 37,12 7,96 34,72 80,99
13,86 7,99 23,18 65,70 48,43 60,61 44,22 75,09
10,74 10,91 25,11 36,49 2,09 24,66 -3,42 38,23
-0,34 1,87 11,50 19,62 13,54 35,08 -2,40 61,28
9,53 13,82 27,08 -32,98 48,23 49,55 -0,68 69,39
-24,03 -14,01 -4,13 -76,09 29,55 -6,89 -34,95 59,58
-15,56 -9,44 16,56 -9,09 -5,78 18,68 -23,39 2,66
14,68 17,73 61,84 29,85 52,47 37,05 46,65 -6,50
Meanwhile the matrix that arises from the
highest entropy method provides higher percentage
errors in some cells, even 100 to 200 percent.
Table 6: Percentage errors using the maximum entropy
method.
54,31 -6,35 42,80 34,92 4,76 27,55 38,06 -156,07
-21,76 -81,30 0,55 -111,52 -37,97 -91,90 9,93 -244,90
-4,12 -22,50 19,50 7,12 -30,26 5,42 11,40 -419,94
16,05 7,69 -108,11 -27,72 -83,62 -34,25 38,08 -386,23
-4,93 -13,05 -173,43 30,47 -80,34 -28,63 31,84 -238,88
2,30 -2,53 -132,63 -18,79 -19,58 -3,21 30,80 -176,79
50,15 49,52 -23,65 41,45 39,45 14,32 65,48 -35,93
55,62 53,70 5,33 62,57 6,18 32,73 3,03 -10,04
-170,03 -186,84 -256,76 -98,40 -247,49 -329,17 -245,59 24,56
ICFC 2010 - International Conference on Fuzzy Computation
128

7 CONCLUSIONS
In this paper the potential and effectiveness of this
new methodology has been proven. Clearly, the
obtained matrix has an error of 10% compared to the
original matrix. However, these results are
reasonable considering the level of uncertainty
coming from the information provided by the expert.
As a guideline for future research, it is
interesting to improve the fuzzy inference engine,
since the used in this test was quite simple. Here,
other sources of information can be incorporated
without limiting the information just to one expert.
ACKNOWLEDGEMENTS
The authors of this paper want to express their
gratitude to the National Secretary of Science and
Technology (SENACYT) of the government of the
Republic of Panama for funding this study through
the R & D project “Methodologies and Performance
Indicators of Railway Transport Systems” adjudged
by a call for the promotion of R&D activities
(MDEPRB09-001). Additionally, they want to thank
the Research Centre on Railway Technologies
(CITEF) of Universidad Politécnica de Madrid
(Spain).
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