Teodorovic, Selmic, Edara

7/30/2019 Teodorovic, Selmic, Edara

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Teodorović D., Šelmić M., Edara P.

BEE COLONY OPTIMIZATION APPROACH TO OPTIMIZE PLACEMENT OF TRAFFIC SENSORS

ON HIGHWAYS

1

BEE COLONY OPTIMIZATION APPROACH TO OPTIMIZE

PLACEMENT OF TRAFFIC SENSORS ON HIGHWAYS

Teodorović Dušan

Šelmić MilicaUniversity of Belgrade

Faculty of Transport and Traffic Engineering

Vojvode Stepe 305, Belgrade, Serbia

[email protected], [email protected]

Edara Praveen

University of Missouri Columbia

Department of Civil and Environmental Engineering

Columbia MO 65211, U.S.A.

[email protected]

ABSTRACT

In this paper, a model is developed to optimally place traffic detectors on freeways. The

proposed model tries to minimize the error in travel time estimation, while taking into account

the constraints of available capital and maintenance funding. A new metaheuristic, the Bee

Colony Optimization (BCO), is used to solve the formulated problem. The proposed BCO

algorithm, inspired by bees' behavior in the nature, was tested on a real-world freeway

segment in Virginia, U.S.A. The obtained results are very competitive when compared with

the results of Genetic Algorithms achieved in a previous study.

1 INTRODUCTION

Point detectors (or sensors) are deployed on roadways to collect traffic data including

volume, occupancy, and speed. The data is used by Traffic Management Centers in cities to

manage traffic and incidents and provide information to motorists about current conditions.

The spacing of detectors on freeways has a major impact on the travel time estimates derived

from the reported speeds. There exists a tradeoff between detector spacing and travel time

estimate accuracy. As detectors become more closely spaced, the data obtained from them

more closely resemble continuous data available from probes. This additional accuracy also

comes with much higher capital and ongoing costs, as all detectors require regular

maintenance to continue to report good data. Transportation agencies are therefore seeking a

method to indicate the most appropriate locations for detector deployment such that the travel

time estimate error is minimized, within the constraints of available capital and maintenance

funding.

In order to obtain accurate travel time estimates, detectors are to be located so as to

effectively sample the traffic conditions. There is limited past research on how to locate point

detectors for effective sampling. The main purpose of this paper is to propose a methodology

to discover the optimal locations of a finite set of point detectors on a freeway corridor. The

proposed model tries to minimize the error in travel time estimation, while taking into account

the constraints of available capital and maintenance funding. The performed numerical

mailto:[email protected]

mailto:[email protected]





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experiments make obvious the usefulness of the newly developed methodology. The model

formulation is solved using a new metaheuristic algorithm, the Bee Colony Optimization

(BCO). The proposed algorithm is inspired by bees' behavior in the nature. We propose an

artificial system composed of a number of precisely defined elements (or agents) and then

perform a multi-agent simulation to solve the problem. Behavioral rules are defined for artificial bees and their interactions are simulated. The artificial bees perform only activities

defined by the model and a corresponding computer program.

The paper is organized as follows. The problem of optimal placement of point detectors

is described in Section 2. The formulation of the travel time estimation problem is discussed

in Section 3. Section 4 describes the BCO technique. The implementation details of the BCO

for given problem are in Section 5, while Section 6 contains test results. The paper is

concluded in Section 7.

2 PLACEMENT OF POINT DETECTORS ON FREEWAYS

The problem of the placement of point detectors within a roadway network belongs tothe field of location theory. Many papers have been published during the past four decades

dealing with the problem of locating facilities on the network [1],[2]. “The term LocationAnalysis refers to the modeling, formulation, and solution of a class of problems that can best

be described as sitting f acilities in some given space” [2]. Location analysis tries to find

answers to the following questions: a) What should be the total number of facilities in the

network? b) Where should the facilities be located? c) What is the best allocation of clients to

the facilities?

Point detectors are the infrastructure facilities that we are interested in this paper. With

respect to the application of Operations Research/Artificial Intelligence techniques to study

the facility location problems in transportation, the following application areas have been

studied: (1) detectors for O-D estimation [3], (2) Automatic Vehicle Identification (AVI)readers for travel time estimation [5], and (3) detectors for minimizing the travel time

variance and social costs [4].

With carefully placed detectors that are well maintained, travel time estimates can be

derived with an acceptable level of accuracy from point detection, under incident-free travel

conditions. In this paper we develop a method to indicate the most appropriate locations for

detector deployment such that the travel time estimate error is minimized, within the

budgetary constraints. The proposed approach is applicable to both regions without any (or

limited) current detector deployment and regions that currently have dense deployment. The

developed method is primarily intended for use at a planning level, to assist in determining

where to deploy detectors in an area that currently has few or no detectors, or in determining

which detectors need to be (or those that need not be) regularly maintained to obtain goodtravel time estimates in areas with dense detector deployment.

3 TRAVEL TIME ESTIMATION MODEL

Travel time data can be collected by driving probe vehicles equipped with Global

Positioning System (GPS) devices. Probe vehicles are driven on all identified freeway

sections in the study regions. A GPS device is installed in the vehicle and then a driver drives

this vehicle according to the “flow of traffic” throughout the study region. While the vehicle

is running, the GPS device automatically logs latitude/longitude points and times. In order to

obtain reliable travel times, multiple probe vehicles are usually deployed for data collection

with a headway of 5 minutes between subsequent vehicles.





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We define two notions of travel time for a freeway section: Ground Truth Travel Time

(GTTT) and Estimated Travel Time (ETT). Computation of GTTT is simple. For each probe

vehicle, the entering and exiting times for the freeway section are compared to obtain the

GTTT specific to that vehicle. The same procedure is repeated for each probe vehicle on that

section during a specified time period. ETT is calculated indirectly. Travel time for the wholefreeway section will be estimated from the travel times of constituent detector „ zones of

influence‟ travel times. The zone of influence of a detector can be defined as half the distance

upstream and downstream to the neighboring detector.

Since actual detector data are not available in this scenario, travel time for each zone of

influence is estimated from the speed data collected at the detector location using the GPS

vehicle (as a surrogate for real detector data). A key assumption in this calculation is that the

speed measured at the point detector is approximately equal to the average speed for the entire

zone of influence. Obviously, the greater the length of the zone of influence the greater the

potential for differences in speeds across the zone. The speed at a detector location is

calculated as follows: For each detector location, a number of GPS points around the detector

are defined. Speeds reported by the probe vehicle at all these points within the vicinity of thedetector location are averaged to obtain the average speed specific to that probe vehicle at that

detector location. Finally, the speeds are averaged over multiple probe vehicles as well. The

length of the zone of influence (ZOI) is divided by this average speed to obtain the travel time

value. ETT for the entire freeway section is then obtained by adding the travel time estimates

for the all constituent zones of influence.

Detector 1 Detector ( )i-1 Detector n

ZOI 1

Freeway Section

ZOI 2 ZOInZOI i-1 ZOI i ZOI i-1

Detector (i) Detector (i+1)

x i

L

Figure 1: Notations used in the Formulation

We introduce the following notation:

n - Number of detectors on the freeway section (=number of zones of influence); i -

Index for the i-th detector; xi - Position of the i-th detector, measured from the origin of

the freeway section (decision variable in the objective function); L - Length of the

freeway section; K - Total number of probe vehicle runs; ZOI i - Length of Zone of

influence of the i-th detector ( Ln

i ZOI i =∑

1=); V i - Speed reported by the i-th detector; TTi -

Travel time for ZOI i (V i

ZOI iTT i = ); ETT - Estimated travel time for the freeway section

( ∑

1==

n

iTT i ETT ); GTTT - Ground truth travel time for the freeway section; ε - Estimation

Error placement of point detectors on freeways.

The estimation error equals:





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GTTT n

iTT iGTTT ETT ε -∑

1==-= (1)

GTTT n

i V i ZOI iε -∑

1== (2)

We can express ZOI i in terms of the decision variable xi as follows:

ni for

x x

L

ni for x x

i for x x

ZOI

nn

iii

2

1,...,3,22

12

1

11

21

(3)

The relative error r equals:

GTTT r

(4)

Cumulative relative error CRE equals:

∑1=

=

K

k GTTT k

ε

k εCRE (5)

where:

k - index for the speed profile (i.e., GPS travel time run)

k - travel time estimation error for the k -th profile

GTTTk - ground truth travel time for the k -th profile.

Upon substituting relations (1), (2), (3), and (4) into relation (5), the Cumulative relative

error CRE equals:

∑

1=

∑

1-

2=-

)2

1-+nx

(-L

+2

1--1++

2 1

2+1

= K

k GTTT k

n

t GTTT k

nk V

n x

V tk

xt xt

V k

x x

εCRE

(6)

where, V ik is the speed reported by the i-th detector for the k -th profile.

The mathematical formulation of the problem of optimal placement of point detectors

on freeways reads:

Minimize





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∑

1=

∑

1-

2=-

)2

1-+nx(-L

+2

1--1++

2 1

2+1 K

k GTTT k

n

t GTTT k

nk V

n x

V tk

xt xt

V k

x x

(7)

subject to:

ni L xi ,...,2,1=≤ (8)

ni L xi ,...,2,1=≤ (9)

In Eq. (7), the numerators of the first three terms are linear combination of two decision

variables. The denominator has the speed term V ik , which is a function of the distance xi. This

means that the objective function is non-linear. The problem (7)-(9) is combinatorial by its

nature.

We divide freeway section into discrete segments. This means that the detectors can be

deployed only at the mid points of these discrete segments. Taking into account thecombinatorial nature of the considered problem, we decided to solve the problem using a

metaheuristic.

4 BEE COLONY OPTIMIZATION

The Bee Colony Optimization (BCO) is a meta-heuristic for solving combinatorial

optimization problems. The BCO algorithm belongs to the class of stochastic swarm

optimization methods. The proposed algorithm is inspired by the foraging habits of bees in the

nature. The communication systems between individual insects contribute to the configuration

of the ‘‘collective intelligence” of the social insect colonies. Swarm intelligence [6] is the part

of Artificial intelligence based on studying actions of individuals in various decentralizedsystems.

The BCO is inspired by bees' behavior in the nature. The basic idea behind the BCO is

to create the multi agent system (colony of artificial bees) capable to successfully solve

difficult combinatorial optimization problems. The artificial bee colony behaves partially

alike, and partially differently from bee colonies in nature.

a. Bees in the Nature In spite of the existence of a large number of different social insect species, and

variation in their behavioral patterns, it is possible to describe individual insects‟ as capable of performing a variety of complex tasks. The best example is the collection and processing of

nectar, the practice of which is highly organized. Each bee decides to reach the nectar source by following a nestmate who has already discovered a patch of flowers. Each hive has a so-called dance floor area on which the bees that have discovered nectar sources dance, in that

way trying to convince their nestmates to follow them. If a bee decides to leave the hive to get

nectar, it follows one of the bee dancers to one of the nectar areas. Upon arrival, the foraging

bee takes a load of nectar and returns to the hive relinquishing the nectar to a food store. After

it relinquishes the food, the bee can (a) abandon the food source and become again

uncommitted follower, (b) continue to forage at the food source without recruiting the

nestmates, or (c) dance and thus recruit the nestmates before the return to the food source. The

bee opts for one of the above alternatives with a certain probability. Within the dance area, the

bee dancers “advertise” different food sources.





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10. Output the best solution found.

5 THE BCO APPROACH

In this paper, we propose the BCO heuristic algorithm tailored for problem of trafficsensors placement. As of the authors‟ knowledge this is the first implementation of Swarm

intelligence to a given problem.

At the beginning of a process, we assume that all bees are in the hive. The hive is an

artificial location, it is not connected to potential sensors location. We allow every artificial

bee to fly out from the hive and to generate NC constructive moves. After that, every bee

returns to the hive. Bees exchange information about the quality of the partial solutions

generated.

a. Constructive moves in forward pass

Authors define the probability ( pi) of choosing a certain alternative based on its utility

(U i) to a user. The probability that specific bee chooses node to be sensor location is:

∑

1=

= K

k k U

iU i p , i=1,2,…,n (10)

where U i is utility of the i-th node and K is the number of “free” nodes (not previously

chosen).

The utility of having a detector at any particular location depends on several factors that

affect travel time estimates. Factors such as the presence of a natural bottleneck at thatlocation (e.g., a lane reduction) that leads to recurring congestion during the peak traffic

periods, level of traffic volumes, etc, can be used to determine the utilities. In this paper, all

potential detector locations are assumed to have equal utilities. However, the proposed model

is still applicable when the utilities of locations differ from each other.

Using Eq. (10) and a random number generator, we assign detector locations to bees.

b. Bee’s partial solutions comparison mechanism

All bees return to the hive after generating the partial solutions. All these solutions are

then evaluated by all bees. Every generated partial solution is characterized by the travel time

estimation error. In this paper, we choose as criteria for comparison, the maximum travel time

error over all travel time runs. We denote by E b the maximum travel time error over all traveltime runs in the case of the partial solution created by the b-th bee. Further, it is normalized as

follows:

[ ] BbbO E E

b E E bO ,1=1,0∈,

min- max

-max= (11)

where:

b E - travel time error value created by the i-th bee

max E , min E - maximum/minimum travel time error value over all partial solutions

generated so far





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The probability that the b-th bee (at the beginning of the new forward pass) is loyal to

its previously discovered partial solution is expressed as follows:

Bbu

bOO

eub p ,...,2,1=,

-max

=1+ (12)

where u is the ordinary number of the forward pass.

In relation (12), the better the generated partial solution (higher Ob value), the higher the

probability that the bee will be loyal to this discovered partial solution. Also, the greater the

ordinary number (u) of the forward pass, the higher the influence of the already discovered

partial solution. In other words, at the beginning of the search process bees are “more brave”to search the solution space. The more forward passes they make, the bees have less courage

to explore the solution space. The more we are approaching the end of the search process, the

more focused the bees are on the already discovered solutions.

Using relation (12) and a random number generator, every artificial bee decides to become an uncommitted follower, or to continue flight along the already known path.

c. Recruiting Process In the case when at the beginning of a new stage bee does not want to expand

previously generated partial solution, the bee will go to the dancing area and will follow

another bee. Within the dance area the bee-dancers (recruiters) “advertise” different par tial

solutions. We have assumed in this paper that the probability the recruiter b‟s partial solutionwill be chosen by any uncommitted bee equals:

∑

1=

= R

k Ok

Obb p , b=1,2 ,…, R (13)

where Ok is objective function value of the k-th advertised solution and R - the number of

recruiters.

Using relation (13) and a random number generator, every uncommitted follower join

one bee dancer (recruiter). Recruiters fly together with a recruted nestmates in the next

forward pass along the path discovered by the recruiter. At the end of this path all bees are

free to independently search the solution space and generate the next iteration constructivemoves.

6 APPLICATION OF BCO TO A CASE STUDY

The problem of optimal placement of point detectors was studied in the case of

Northern Virginia region in Virginia. The data related to probe vehicle runs are taken from the

previous work of Edara et al. [13]. In Northern Virginia, an approximately 11-mile section of

I-66 from exit 43 to exit 57 in both directions was studied. The I-66 corridor is one of the

busiest freeways in Virginia with over 90,000 vehicles per day traveling to and from

Washington DC. Three GPS equipped probe vehicles were driven on three week days during

both the morning and evening peak periods in both the east and west directions. Vehiclesdeparted at 5 minute headways. This resulted in a total of 37 travel time runs during the





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morning peak and 40 runs during the evening peak. Currently, detectors are deployed at

approximately one-half mile spacing resulting in a total of about 20 detectors over the study

segment.

One of the main purposes of developing proposed methodology was to generate tradeoff

plots between the travel time error and the number of detectors which would give the optimal placement of detectors for different levels of available funding. Tradeoff plots were generated

by varying the actual number of detectors (q) from 2 to 19 in increments of 1. The BCO

algorithm parameters were set to the following values: the total number of bees B engaged in

the search process was equal to 10; the number of moves, NC, during one forward pass was

equal to 1; the number of iterations I within one run was equal to 100. All the tests were

performed on a AMD Sempron (tm) Processor with 1.60 GHz and 512 MB of RAM under

Windows OS. Results of the BCO runs are shown in Figure 2. For a given number of

detectors, the obtained optimal placement would result in a travel time estimation error for

each travel time run. The maximum of these errors versus the detector deployment is plotted

in Figure 2. Edara et al. [13] used Genetic Algorithms (GA) to discover the optimal detector

deployment. The maximum error versus the detector deployment obtained by the Geneticalgorithms (GA) is also plotted in Figure 2.

Figure 2: Maximum Travel Time Estimation Error Plot (BCO vs GA)

As it can be seen from the plot, the maximum error value is high when only a few

detectors are deployed; however, as the deployment increases the error value decreases. After

reaching a certain level of deployment any further increase in the number of detectors may not

decrease the error.

Table 1: Optimal detectors locations Number

of Detectors

Deployed

Maximum TT Estimation Error (Minutes)

RecommendedPlacement of Detectors

2 6.7827 13 19

3 5.7032 3 13 19

4 4.4583 8 12 18 20

5 3.9362 8 11 17 18 20

6 2.7712 5 8 10 12 18 20

7 2.9008 3 5 10 11 12 18 20

8 2.5474 2 3 9 11 15 17 19 20

9 2.1731 1 3 7 8 10 14 17 19 20

10 1.9631 2 3 5 9 12 14 17 18 19 20

11 1.9433 1 3 4 7 10 13 15 16 17 19 20

12 1.6342 1 4 6 7 10 11 12 14 17 18 19 20

13 1.4200 1 4 6 7 9 10 11 12 14 17 18 19 20

14 1.2423 1 2 4 7 8 9 10 12 13 14 17 18 19 20

0

1.7

3.3

5

6.7

8.3

10

2 4 6 8 10 12 14 16 18 20

GA Solutions

BCO Solutions

Number of Detectors

M a x i m

u m T

T E s t i m a t i o n E r r o r ( m i n u t e s )





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10

15 1.2584 1 2 3 4 5 7 10 12 14 15 16 17 18 19 20

16 1.2959 1 2 3 4 5 6 7 10 12 14 15 16 17 18 19 20

17 1.3334 1 2 3 4 6 7 9 10 12 13 14 15 16 17 18 19 2018 1.3621 1 2 3 4 5 6 7 9 10 12 13 14 15 16 17 18 19 20

19 1.7793 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20

From Figure 2 it can be inferred that deploying 14 detectors would result in the least

maximum TT error (~ 1.24 minutes) and an acceptable error distribution. We can further

conclude that the 20 detectors currently deployed are more than is needed to provide

reasonably accurate travel time estimates. We show in Table 1 the optimal detector locations

obtained by the BCO algorithm for different numbers of detectors along with the

corresponding maximum travel time errors.

7 CONSLUSIONS

The model developed could be used for planning purposes. In a very short time it is

possible to generate a large number of different solutions for detectors placement, and observe performances (numbers of detectors employed, and travel time estimation error) achieved for

each one. In this way, state and local departments of transportation can decide where to add

new detectors. The proposed model also suggests which of the existing detectors should go on

with getting maintenance, given agencies‟ financial resource constraints.In this paper, the Bee Colony Optimization was used to solve the problem of optimally

locating traffic detectors on freeways. The BCO belongs to the class of constructive methods.

The proposed BCO algorithm was tested on a real-world case study in Virginia with travel

time data collected using highly accurate GPS-equipped probe vehicles. The obtained results

were compared with the results of a Genetic Algorithm methodology achieved in a previous

study. The comparison indicated that the BCO results were better than or equal to the GA

results for majority of sensor deployments. The promising results obtained in this work encourage further investigation of the Bee Colony Optimization to similar combinatorial

optimization problems.

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