Traffic Detector Placement Models With Reliability Holder
J. Q. Liu, N. Zhu
College of Management and Economics, Tianjin University, China
(zh5553158@163.com)
Abstract - Traffic flow information is an important
source for a large number of transportation applications.
Therefore, the number and sites of traffic counting sensors
matters a lot. However, traffic sensors are subject to fail in
varied situations. In this paper, we extend the classic traffic
sensor location problem by considering the failure of sensors.
Models aiming to improve the reliability are proposed.
Existing sensors is also taken into account. Examples are
provided to illustrate the proposed models. Several sensor
locations patterns are found.
Keywords - Traffic detector placement; Detector failure;
Integer programming
I. INTRODUCTION
Considerable researches have been dedicated to
develop models to reconstruct and update OD matrix.
Two major methodologies are usually used for OD
matrices estimation.
(1) Data survey such as household survey, roadside
interview.
(2) By applying traffic flow counts as measurements of
link flows in a network.
The first method yields the most accurate result for
OD estimation, but it requires considerable resources and
is not affordable. Comparatively speaking, the second
alternative attracts a large number of studies in the past
decades.
For the second type of methodology, OD estimator
largely depends on traffic flow measurement. In reality,
traffic counts are considered as a convenient and cheap
way to obtain traffic information about travel pattern by
comparing with the extensive travel surveys i.e. road
interviews, household surveys. Flow information is also
used to predict OD pattern and its evolution process by
combining with current and historical information since
traffic sensors can provide recursive data over time. The
accuracy of OD estimation depends largely on the quality
and quantity of input data. Therefore the key problem
becomes how to identify the minimum number and
locations of traffic sensors.
In general, OD matrix estimation model have the
following generic form [1]:
(1)
min F1 (T , T )  F2 (v, v)
v ,T
s.t.
v  M (T )
(2)
Where T is the target OD matrix, T is the OD
matrix to be estimated, v is the vector of observed, v is
the vector of link flows to be estimated, F1 (T , T ) and
F2 (v, v) are the distance between estimated and prior OD
matrix and between estimated and observed traffic flow.
Relating to the objective function, methods can be
stratified as follows:
(a) Entropy maximization or Information Minimization
model.
Van Vuylen and Willumsen [2] proposed two models
by using information minimization and entropy
maximization principle to estimate OD matrix from traffic
counts. Maximum likelihood method and prior OD matrix
are used in these two models. Fisk [3] developed a single
mathematical model by combining maximum entropy trip
matrix estimation with a user-equilibrium model.
(b) Generalized least square estimation and statistical
approaches.
Cascetta[4] proposed a generalized least squares
estimator of OD matrix combining direct or model
estimators with traffic counts via an assignment model.
Bell [5] presented a generalized least squares(GLS)
approach to estimate OD matrix. It is proved that
inequality constraints can improve the accuracy of fitted
values, reducing their sensitivity to error in the inputs.
Spiess[6] formulated a model to estimate OD demand
which was considered as independent
Poisson
distributions with unknown means to reproduce link flow.
A maximum likelihood method was used. For other
statistical methods, see [7] for more references.
The above mentioned studies demonstrated there are
strong connections between OD estimation and link count
observation. Several scholars have been dedicated to
interpret sensor location problem as OD covering problem.
Yang [8] introduced a concept named “Maximum
Possible Relative Error”(MPRE). Accordingly Yang and
Zhou [9] proposed four basic sensor placement rule:
Rule 1. O-D covering rule: A certain portion of trips
between any O-D pair should be observed.
Rule 2. Maximal flow fraction rule: For a particular O-D
pair, link with the maximal fraction of that O-D flow
should be selected.
Rule 3. Maximal flow-intercepting rule: Under a certain
number of sensors constraint, the maximal number of O-D
pairs should be observed.
Rule 4. Link-independence rule: The resultant traffic
counts on the selected links should not be linearly
dependent.
Yang [10] further proposed a new location strategy
that offer robust traffic-counting location pattern without
the need of considering the behavioral assumption on road
users and the level of traffic congestion on the network.
Ehlert[11] extended Yang and Zhou’s[9] work, taking
existing traffic sensors and information content of prior
OD flows into account. Bianco et al. [12] formulated a
two-stage procedure. First stage is the sensor location
problem that determines the minimum number of sensors
and location of counting points. Second stage is to update
the OD matrix. Pravinvongvuth et al. [13] proposed a
methodology for selecting the most preferred plan from
the set of Parato optimal solutions obtained from solving
the multi-objective automatic vehicle identification (AVI)
reader location problem limited by resource and OD flow
coverage. Castillo [14] dealt with problem of trip matrix
and path flow reconstruction and estimation using plate
scanning and link flow observations. Feasible subsets of
scanned links are identified. Shou-Ren Hu [15] proposed
a linear algebra approach seeking to identify the smallest
subset of links in a network which enables the accurate
estimation of traffic flows on all links of the network
under steady-state conditions. Castillo et al. [16-18]
addressed observable problem which is similar with
traffic sensor placement using algebraic techniques.
Without the failure of sensors, the ideal way is to
install sensors on every link in order to obtain reliable
estimation, but due to the limitation of budget, only partial
links could be covered by sensors. In practice, traffic
counting sensors are subject to failure. The previously
proposed models and approaches did not consider the
failure of traffic sensors. This paper is intended to choose
a desirable set of links for which should be observed by
considering the failure of traffic sensor. In particular, we
are interesting in sensor location pattern change after
considering the sensor failure.
Models proposed in this paper seek to identify
influence and pattern of traffic counting sensor placement
by taking consideration of failure of traffic counting
sensors. We proposed some models that are divided into
two categories.
One is to minimize the number of traffic counting
sensors under a specific reliability requirement. The other
is to cover as much traffic flow as possible by taking into
account of sensor failure. The rest of paper is organized as
follows: Section 2 introduces some new models via
considering failure of sensor in order to increase the
reliability or cover as much path flow as possible. Section
3 illustrates algorithm and results of resolving these
models by using a test network. Finally, some concluding
comments are provided in Section 4.
II. TRAFFIC SENSOR PLACEMENT MODEL UNDER
UNCERTAINTY
2.1 Notation
a A link in a network
w : OD pair.
la : la  1 if a sensor is installed on link a , 0 otherwise.
 aw :  aw  1 if OD pair w pass over link a ,  aw  0
otherwise.
 : the link OD pair incidence matrix.
p : Probability of sensor failure.
 : Reliability requirement.
n : Combinational result of probability of sensor failure
and reliability requirement.
cw : The number of sensor needed while there are some
sensors existed.
aw : The number of existing traffic sensors
f r : Traffic flow of route r
yr : Binary variable indicating that if route r is covered
by a traffic sensor
l * : Number of traffic sensor for placement.
 arw :  arw  1 if link a belongs to route r of OD w ,
 arw  0 otherwise.
c f : path flow vector, each element of the vector is the
path flow of a specific route.
2.2 Model formulations
The basic utility of installation of traffic counting
sensor is to estimate OD matrix. Yang et al. [8] proposed
a concept of “Maximum Possible Relative Error” (MPRE)
to measure reliability of OD estimation. In his MPRE
model, the MPRE could be infinite if any one OD pair is
not observed. It is naturally leading to the OD covering
rule (Rule 1).
The following binary integer programming model
has been proposed to minimize the use of traffic counting
sensors to cover all OD pairs (Yang 1991).
Model 1: Minimize  la
(3)
a
Subject to:
 aw  la  1 for all OD
a
pairs w.
(4)
(5)
la  {0,1}
Constraint (4) assumes the complete OD pairs
coverage. It can be shown that the resultant sensor
placement solution will cover all OD pairs. However,
traffic sensors have a probability of failure in reality, so
that more than 1 sensor is needed to assume a high
probability of obtaining traffic information.
An
extension of Model 1 by considering traffic sensor failure
is proposed as follow:
Model 2: Minimize  la .
(6)
a
Subject to:  (1   awla (1  p))  
a
for all OD pairs w .
(7)
(8)
la  {0,1}
Where p is the probability of sensor failure,  is
the reliability requirement for each OD pair. For instance,
suppose   0.05 which means enough number of
sensors is needed to assume that only 5% possibility
traffic counting information cannot be obtained.
Constraint (7) indicates failure probability of all sensors
between each OD pair is required to be less than or equal
to a reliability level represented by a scalar  . Constraint
(8) is a binary constraint indicates that each link is
prohibited for installation of more than 1 sensor. In
Model 2, we still impose the binary constraints (8). This
constraint will be relaxed in Model 4. Model 2 can be
rewritten as follows for the sake of a standard commercial
solver to solve it.
Model 3: Minimize  la
(9)
a
Subject to:

l n
aw a
for all OD
a
pair w.
la  {0,1}
(10)
(11)

 log  
Where n  
 is the minimum integer great
 log p 
log 
than or equal to
which is defined as system
log p
reliability level requirement denoting by an integer. An
intuitive exposition of constraint (10) is that several traffic
sensors have to be installed to assume the reliability to be
met. The minimum number of sensors satisfying the

 log  
reliability level requirement is 
 .
 log p 
Figure 2.1 shows the number of counting sensors are
needed when system reliability level requirement and
probability of sensor failure is given. Probability of sensor
failure is assumed to be identical and independent for all
traffic counting sensors. The horizontal axis indicates the
probability of sensor failure and vertical axis the system
reliability level requirement. From the right-upper to leftbottom part, number of sensors needed to be installed in
each OD pair is from 1 to 5, and more than 5 respectively.
The smallest part in the left-bottom is that more than or
equal to 6 sensors is needed. Value of horizontal axis
from left to right is from 0.001 to 0.5, while from bottom
to upper is same with vertical axis.
For model 3, the reliability level requirement cannot
exceed the minimum number of links between OD pairs.
Therefore, the binary constraint has to be relaxed to
integer if higher reliability level requirement is applied
which leads to Model 4.
Model 4: Minimize:  la
(12)
a
Subject to:

l n
aw a
for all OD
a
pair w.
(13)
(14)
la integer.
In practice, many cities have already some traffic
counting sensors installed. In this case, new sensors
should be located to satisfy the reliability level
requirement. Observability and reliability can be
improved for urban transportation network if some
sensors are already installed and more traffic sensors are
available.
Installed sensors might be located suboptimally; therefore the pattern of new installed sensor
could be changed by comparing with the case that no
sensors are installed previously. This leads to the
following Model 5 (Yang et al.1998).
Model 5: Minimize  la
(15)
Subject to:

a
l  cw for all OD
aw a
a
pair w.
la integer.
Where cw  n  aw ,
n
(16)
(17)
is the number of sensors
required by reliability level requirement and aw is the
number of existing traffic sensors, cw is the number of
sensors needed. cw could be less than or equal to 0 which
means the constraint associated with corresponding OD
pair is already satisfied.
III.NUMERICAL RESULTS
Fig. 2.1. Pattern of reliability level requirement and probability of sensor
failure
In order to better understand the behavior of
proposed models, Sioux-Falls network was used to test
the above 7 models. The network model has been divided
into 24 zones that represent origins and destinations.
There are 76 links and 528 OD pairs of this network. Data
about Sioux-Falls network comes from Transportation
Network Test website [19].
Computation result for Model 1 is as following:
Table 3.1
Link42, link49, link52, link62, link64, link65, link69,
link71, link73, link76 are chosen 5 times among all 6
scenarios which are marked as red rectangle in Figure 3.2
(b). All these links marked red in Figure 3.2 (b) are
considered to be relatively important. An obvious
observation is that all important links are topological
significant in the sense of link degree. In this paper, link
degree is defined as the number of links adjacent to a
specific link.
Result for Model 1
9
11
29
42
48 56
60
65
69
71
Numbers in cells of table 1 are ids of links that are
installed sensors.
3.1 Results for Model 3(binary case)
For model 3, more than 1 sensor is not allowed to be
installed on each link. We tested different reliability level
requirement n . Result show in Table 3.2.
Table 3.2
Reliability level requirement VS minimum number of sensors installed
n
# of sensors
1
10
2
20
3
28
4
38
5
48
3
1
2
1
5
2
4
8
3
4
6
7
35
11
9
13 23
10 31
9
25
33
12
36
6
58
15
5
26
11
16
21
19
24
22 47
10
29
51 49
30
16
14
42 71
72
15
74
24
22
59
66
21
62
61
20
64
(a)
3
1
56 60
63
69 65 68
75
39
18
50
19
45
70
13
18 54
46 67
23
73 76
58
57
44
55
7
17
28 43
41
38
20
52
53
37
17
8
48
32
34 40
6
12
27
14
2
1
5
2
Fig. 3.1 minimum # of sensors VS
n
No feasible solution could be found when n  7
because there are some OD pairs that only have n  6
links between them. Under the rule of at most 1 sensor
could be installed on each link, some OD pairs cannot
meet this reliability level requirement. Figure 3.1 shows
the relationship of the minimum number of needed
sensors and reliability level requirement n which satisfies
linear relationship approximately.
Some links are considered to be important in the
sense that they are always chosen to have sensors installed,
meanwhile some are considered to be unimportant
because no sensor is installed on them. For our 6
scenarios ( n equals from 1 to 6), link2, link4, link5,
link14, link17, link20, link37, link38, link39, link74 are
never chosen which are marked as red cross in Figure 3.2
(a). They are considered unimportant. Meanwhile, link9,
link11, link29, link48 are always chosen for all 6
scenarios which are marked as red circle in Figure 3.2 (b).
4
8
3
4
6
7
35
11
9
5
13 23
10 31
9
25
33
12
36
15
26
32
34 40
6
12
16
21
22 47
29
51 49
30
14
42 71
44
72
23
15
13
39
24
75
18
58
56 60
19
45
59
69 65 68
66
50
46 67
22
73 76
55
7
18 54
52
57
70
74
16
20
17
28 43
53
38
17
48
10
41
37
19
8
24
27
11
14
21
61
63
62
20
64
Fig. 3.2. “Unimportant” and “Important” links
(b)
3.2 Results for Model 4(Integer case)
Binary constraint was relaxed to be integer so that
high system reliability level requirement can be achieved.
In this section, 500 hundreds n were tested ranging from
1 to 500. Taking the sensor failure probability of 0.2 as an
example, n can be up to 29 if less than probability of
1020 failure is allowed. Thus n  500 is a very high
reliability of system requirement. Another purpose of
setting n from 1 to 500 is to find the sensor placement
pattern.
The final location patter is the same with Optimal Set.
The remaining sensors only need to be installed according
to the Optimal Set.
Case 2: Partial existing sensor location was a subset of
Optimal Set. The final location pattern is different from
the Optimal Set. Total number of sensors needed
(including existing sensors) is great than the number of
Optimal Set. Also the location pattern is different.
Case 3: None of existing sensors was in the Optimal Set.
It is necessary to compensate effect of existing sensors so
that more sensors are needed to observe all OD pairs than
case 2.
Lemma 1: The total number of sensors (including
existing sensors) is great than or at least equal to the
number of sensors needed in the no existing sensor case.
Proof: Let’s reform Model 4 in a matrix way.
Model 8: Minimize c * L
Subject to:  L  n
n i Integer
Where L is the vector, each element of L indicates
number of sensors installed on a specific link. n is
reliability level requirement vector, each element of n is
n . c is the cost vector which is assumed all one.

Fig. 3.3. Pattern of sensor placement of integer constraint
Figure 3.3 is a scatter plot of how many times each
links was chosen among 500 experiments where n is
ranging from 1 to 500. All links can easily be divided into
two categories. Part of them has a high probability to be
chosen which are considered to be important meanwhile
some of them are rarely to be chosen. The times each
links was chosen can be used as a rank of links.
Observation 1: Important links are inside the network.
Most links on the border of the network are considered to
be unimportant.
Observation 2: For a directed graph, two links
connecting two same nodes are both considered important
or unimportant.
Observation 3: The sensor location pattern is highly
dependent on OD pair pattern.
3.3 Results for Model 5(Existing sensors case)
Three types of existing traffic counting sensor
location pattern were tested in this paper. Optimal traffic
counting sensor location pattern without existing sensor is
denoted as Optimal Set. The Optimal Set is defined as not
only the location of observed links, but also the number of
sensors installed on each link.
Case 1: Existing sensors were a subset of Optimal Set. In
this case, the optimal location pattern does not change.
Let’s denote existing sensor vector as L which is
known exogenously in following Model 9. The Model 5
can be reformed as:

Model 9: c *( L  L)

Subject to: ( L  L)  n
n i Integer
Let’s use L*8 and L*9 denote the optimal solution of

Model 8 and Model 9. Since L*9  L is also a feasible

solution of Model 8, it is obvious that c * L*8  c *( L*9  L)
by definition of Model 8. The lower bound of total
number of traffic counting sensors is optimum of Model 8.
Lemma 2: There is an upper bound of number of sensor
needed no matter the number and location pattern of
existing sensors.
Proof: The objective of Lemma 2 is to
prove c * L*8  c * L*9 .
Let’s rewrite Model 9 as following:
Model 9’: c * L

Subject to:  L  n   L
n i Integer
Since L*8 is the optimal solution of Model 8, then

L*8  n  n   L , L*8 is also the feasible solution of
Model 9’. According to the definition of optimal value of
Model 9’, we obtain c * L*8  c * L*9 which means c * L*8 is
the upper bound of number of new sensors needed.
IV.CONCLUSION
In this paper, Extensions to existing classic traffic
counting sensor location problem are proposed. One
intuitive inspiration is from the failure of traffic counting
sensors. All the models proposed aim to improve the
performance of sensors location problems for the purpose
of OD estimation.
Firstly, more sensors are allowed in be placed on the
same link to increase the reliability of observation.
Results show that links with higher topological degree are
more important and can be used as weights to measure
links. Secondly, locations of existing sensors are
considered. Upper bound number of new traffic counting
sensors needed for the purpose of fully OD coverage is
provided.
Future directions include considering the budget
constraints and propose new algorithm for large size
network.
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