A network traffic control algorithm with analytically embedded traffic flow models /

Abstract

Thesis (Ph. D., Civil Engineering)--University of California, Irvine, 1997. Includes bibliographical references (leaves 149-167).

Full text

Model formulation and solution algorithm of
trac signal control in an urban network
Wann-Ming Wey *
Graduate School of Architecture and Urban Design, Chaoyang University of Technology, 168 Gifeng E. Road,
Wufeng, Taichung County, Taiwan, ROC
Abstract
The existing network trac signal optimization formulations usually do not include trac
¯ow models, except for control schemes such as SCOOT (Split, Cycle, and Oset Optimiza-
tion Technique) system that uses simulation for heuristic optimization. Other conventional
models normally use isolated intersection optimization with trac arrival prediction using
detector information, or optimization schemes based on green bandwidth approach such as
MAXBAND (Maximal Bandwidth). In this paper we present a complete formulation of the
problem that includes explicit constraints to model the movement of trac along the streets
between the intersections in a time-expanded network, as well as constraints to capture the
permitted movements from modern signal controllers. The platoon dispersion model used is
the well-known Robertson's model, which forms linear constraints. Thus it is a rare example
of a trac simulation being analytically embedded in an optimization formulation. The for-
mulation is an integer-linear program, and does not assume ®xed cycle lengths or phase
sequences. It assumes full information on external inputs, but can be incorporated in a sensor-
based environment, as well as in a feedback control framework. The formulation is an integer-
linear program that may not be eciently solved with standard simplex and branch and
bound techniques. We discuss network programming formulations to handle the linear pla-
toon dispersion equations and the integer constraints at the intersections. A special-purpose
network simplex algorithm for fast solution is also mentioned. #2000 Elsevier Science Ltd.
All rights reserved.
Keywords: Network programming formulation; Network simplex algorithm; Rolling horizon; Platoon
dispersion
Computers, Environment and Urban Systems
24 (2000) 355±377
www.elsevier.com/locate/compenvurbsys
0198-9715/00/$ - see front matter #2000 Elsevier Science Ltd. All rights reserved.
PII: S0198-9715(00)00002-8
*Tel.: +886-4-3323000 ext. 7153; fax: +886-4-3742339.
E-mail address: wmwey@mail.cyut.edu.tw (Wann-Ming Wey).

Nomenclature
Nnumber of intersections
nindex used to refer to one of the Nintersections
I
n
set of signal phases at the control intersection n;nN
O
i
set of signal phases at the start node of the from_link of a given phase i
that feeds trac for phase i
isignal phase at a given node n, i In
Li from_link of phase i
Tthe time horizon under consideration speci®ed in seconds
Knumber of discrete Ttime intervals in the optimization period
ktime interval index
Tthe sample time interval of duration (s)
Hnumber of links in the network, including entrances
Enumber of entrances and exits
qi
inkupstream in¯ow for phase iover a period [kT, (k+1)T] (veh/s)
qi
skthe ¯ow which arrives at the end of the waiting queue or at the stop-
line (veh/s)
qi
gokthe capacity ¯ow for green trac light (veh/s)
Sg
ni saturation ¯ow for green time of phase iat intersection n(veh/s)
Sy
ni saturation ¯ow for yellow time of phase iat intersection n(veh/s)
u
ni
(k) 0 if signal state is green for phase iat intersection nand time step k,
and 1 if signal state is red for phase iat intersection nand time step k
nik1 if existing signal state of phase iat intersection nis switched at end of
time step k, and 0 otherwise
Gi
min minimum green for phase i(s)
Gi
max maximum green for phase i(s)
U
ni
(k) green time used by phase iat intersection n, at the end of time step k(s)
qi
outkout¯ow of phase iat the downstream end over period [kT, (k+1)
T] (veh/s)
l
ni
(k) number of vehicles of phase iqueued up at the end of time interval kat
intersection n
d
i
(k) side entry ¯ow during phase ion the corresponding approach link (veh/s)
s
i
(k) side exit ¯ow during phase ion the corresponding approach link (veh/s)
i0exit rates within phase i
lip fraction of queued vehicles of movement ithat uses buer p
pthe queue buer number, pB
n
B
n
set of separate queue buers of node n
Q
p
set of phases that share the buer p
C
p
the storage capacity of the queue buer p
a travel time coecient between the upstream and downstream of an
intersection
356 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

1. Introduction
Increasing urban road trac congestion in major US cities signi®cantly under-
mines the mobility of urban America (Lindley, 1987). With little space for con-
struction of new roads, many eorts towards congestion relief are focused on better
utilization of existing transportation facilities through Advanced Transportation
Management Systems (ATMS), a key component of the Intelligent Transporta-
tion Systems of the future. Traditionally, the congestion problem on surface streets
was dealt with on the supply side by providing increased capacity by adding more
lanes to existing roads or adding new links to the existing transportation network.
Such a solution is no longer considered viable because of the prohibitive construc-
tion cost and the negative environmental impact. Instead, greater emphasis is placed
on trac management. The management of trac on surface street networks is
achieved primarily via signalized control of intersections. A promising enhancement
to the current control systems of pre-timed and actuated signals is the application of
ATMS technologies and techniques to isolated intersections, and to arterial and
network signal systems.
Conventional signal control strategies based on methods developed during the
1970s and 1980s are not considered uniformly eective under all possible condi-
tions (Tarno & Gartner, 1993). Many strategies fail to provide improvements
over well-timed ®rst-generation systems for congested conditions, and some of
these systems exhibit degraded performance during speci®c sets of undersaturated
conditions (Boillot et al., 1992). For example, current systems are relatively slow to
respond to sudden changes in trac ¯ow caused by incidents or large ¯uctuations
in demand. Such systems have been designed to implement small changes over time
to overcome the problem of frequent transitioning (see Michalopoulos, 1992, for
an excellent discussion of the de®ciencies in the current practice). Real-time
stochastic control based on detected trac is an option which has not been applied
in an integrated fashion at the network level due to the lack of complete analytical
network-wide optimization formulations. This paper develops such a formulation,
which also includes analytically embedded models for trac ¯ow between inter-
sections so that such a network-wide optimization can be attempted. The intent
is to use this optimization in the future as part of real-time control schemes
with possibilities for stochastic feedback update of the state variables. The focus in
the paper is on the optimization formulation, and its adaptation to a network
Fa dispersion parameter
TD total delay (veh-intervals)
Z
i
1 when phase iis oversaturated, and 0 when phase iis undersaturated
Mvery large positive value, called Big-M
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 357

programming form for ecient solution, because such speedy solutions are
essential for it to be applicable in control schemes. We also discuss the special-
purpose network simplex algorithm we developed since the network form is
non-standard.
There is a tremendous variety of problems associated with optimal intersection
control, some of which have undergone theoretical treatment (see Gazis, 1964, for a
classical optimization formulation for the oversaturated conditions). Current prac-
tices, reviewed in the next section, however, are often based on quasi-optimal tech-
niques, thanks to the ineciencies of the formulations. Application of linear
programming for intersection signal control, however, shows a potentially practical
approach (Eddelbuttel & Cremer, 1992, demonstrate an example for this). Using the
linear models available for platoon dispersion along the links, our research develops
such a linear optimization formulation. Mathematical optimization is applied to
address two interrelated problems of trac systems: trac ¯ow modeling and opti-
mal trac signal control.
The objectives of the paper are three-fold. The primary objective is to develop a
network-wide signal optimization formulation, including link-level trac movement
models as constraints. The second objective is to develop it for application in a real-
time control scheme, brie¯y discussed, but not yet implemented. The third objective
is to provide a special-purpose solution algorithm for the formulation which has
been shown to perform extremely eciently, so that practical future application
within real-time control frameworks is possible.
Another motivation in this paper is to show that the network-wide signal optimi-
zation problem when viewed on a time-expanded network form has an inherent
structure that is suitable for network programming, and thus can be solved faster.
Network programming is used extensively for inventory, cash-¯ow and other net-
works and shows orders of magnitude better solution eciency, as well as applic-
ability to orders of magnitude of larger problems, compared to standard linear
programming. Our research is perhaps the ®rst attempt to apply this to the signal
control problem. Perhaps the reason why such a technique has not been used for this
problem is that the resulting network problem is in a speci®c non-standard form
which is considered to be in a dicult class of network optimization problems. We,
however, develop some techniques to handle the non-standard nature without
aecting the solution eciency.
The paper provides a detailed review of the trac control types and existing trac
control schemes in Section 2. The new formulation for network-wide optimization is
provided in Section 3. Adaptation of the formulation to the network programming
form, the non-standard structure of the network form, as well as the special-purpose
solution algorithm are discussed in Section 4, which also provides a brief discussion
of a rolling-horizon control scheme within which the optimization can be used. The
paper concludes with a comparison of the computational performance of the
special-purpose algorithm to that of a standard linear programming algorithm for a
simulated case. The paper does not include any results based on ®eld data due to the
extensive eort involved in implementing it in the ®eld, an activity planned for
the future.
358 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

2. An overview of trac signal control
2.1. Classi®cation of trac control methods
2.1.1. Fixed-time control
Trac signals in use today typically operate based on a pre-set timing schedule.
The most common trac control system used in the USA is the Urban Trac
Control System (UTCS), developed by the Federal Highway Administration in the
1970s. UTCS generates timing schedules o-line using manual or computerized
techniques. These predetermined timing schedules are implemented by the system
according to the time of the day. The timing schedules are typically obtained by
either maximizing the bandwidth (which means the width of the through-band in
seconds indicating the period of time available for trac to ¯ow within the band) on
arterial streets or minimizing a disutility index that is generally a measure of delay
and stops. Computer programs such as MAXBAND (Little, Kelson & Gartner,
1981) and TRANSYT (i.e., Trac Network Study Tool) (Robertson, 1969) are well
established means for performing such optimization. The o-line approach used by
UTCS cannot respond adequately to unpredictable changes in trac demand.
2.1.2. Trac-responsive control without optimization
These are the adaptive control schemes where the signals are changed based on the
actuation of stop-line detectors and minimum/maximum green times. This type of
control responds to trac but attempts no optimization, network-wide or local.
2.1.3. Trac-responsive control with optimization
These techniques calculate control parameters according to prevailing trac con-
ditions. They typically respond to changing trac demand by performing incre-
mental optimization. The most notable of these are SCATS (i.e., Sydney
Coordinated Adaptive Trac System) (Lowrie, 1982, 1990; Luk, 1984; Sims, 1979)
developed in Australia, and SCOOT (Split, Cycle, and Oset Optimization Techni-
que; Hunt, Robertson, Bretherton & Royle, 1982; Robertson & Bretherton, 1991)
developed in England. SCATS is installed in several major cities in Australia, New
Zealand, and parts of Asia; recently the ®rst installation of SCATS in the USA was
completed near Detroit, MI. SCOOT is installed in even more cities around the
world, including some in the USA (e.g. cities of Oxnard and Anaheim). SCOOT uses
steady-state ¯ow patterns found by the TRANSYT models and attempts a heuristic
optimization of cycle lengths and splits.
Other notable methods under development over the past decade include UTOPIA,
PRODYN and OPAC (i.e., Optimized Policies for Adaptive Control). These are all
trac-responsive optimization schemes, with various levels of trac modeling cap-
abilities and network-wide optimization capabilities. UTOPIA system developed in
Italy was tested at Torino in 1985±86. The French system PRODYN (Henry &
Farges, 1990; Henry et al., 1983) developed in 1982, experimentally operated in
Toulouse (France) and was recently commercialized. In the meantime, the American
system OPAC (Gartner et al., 1991) had the ®rst experimental tests in 1990. OPAC
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 359

perhaps has one of the most comprehensive optimization formulations among these.
It solves the formulation using dynamic programming for an isolated intersection.
The extension for network-wide operation is under development, but is eectively
based on optimization of isolated intersections with real-time prediction of arrival
¯ows, rather than based on an integrated network-wide formulation. One of the
recent eorts at developing comprehensive trac control systems was the RHODES
(i.e., Real-Time Trac Adaptive Signal Control Logic: Prototype Control Descrip-
tion) prototype developed by Head and Mirchandani (1997), which does include a
network control formulation that is promising.
2.2. Review of intersection control algorithms
The most common approach to signalization design is to determine settings for a
®xed-cycle light that minimizes the average delay per car assuming constant arrival
rates (Miller, 1963; Webster, 1958). For pre-timed signals the most well-known
research was performed by Gazis and Potts (1963) and by Gazis (1964) for a system
of two oversaturated intersections in succession. Later researchers (Burhardt, 1971;
Gartner, 1983) based their work on Gazis' theory and further extended it for more
intersections. Dunne and Potts (1964) developed time-varying control algorithms for
an undersaturated intersection with constant arrivals which guarantee that, for any
initial state, the system eventually reaches a limit cycle for which the equilibrium
average delay per car is a minimum. In all these models, the control policy is not
responsive to the dynamics of the trac ¯ow process since there is no trac ¯ow
model or real-time trac ¯ow information involved. For real-time control, several
algorithms have been proposed (Cremer & Schoof, 1990; Gartner et al., 1992; Gor-
don, 1969; Green, 1968; Lee, Crowley & Pigantaro, 1975; Michalopoulos & Ste-
phanopolos, 1977; Miller, 1965; Papageorgiou, 1983; Ross, Sandys & Schlae¯l,
1970). For example, Miller (1965) considered an intersection with heavy trac and
assumed that at time tthe signal is green on primary approach. At this time the
controller can make a binary decision, i.e. to change the signals immediately, or
after an extension of one unit of time. However, Miller did not consider the inter-
section of adjacent intersections, and thus did not include the downstream delays in
determining an optimal extension strategy. Ross et al. (1970), basing their work on
a philosophy similar to that of Miller, developed a computer control scheme for
trac-responsive control of a critical intersection that not only minimizes the total
delay of all users of the intersection, but also minimizes the total delay accumulated
at downstream intersections. Moreover, Longley (1968) proposed a control scheme
for a two-phase congested intersection employing a `queue balancing' strategy. This
strategy seeks to hold a particular linear function of the intersection queues to a
value of zero by adjustment of the green time split.
Lee et al. (1975) also considered queues rather than delays as the objective of the
control and developed another semi-empirical strategy called ``Queue Actuated Sig-
nal Control''. This is a control policy where an approach receives green auto-
matically when the queue on that approach becomes equal to or greater than some
predetermined length, regardless of the conditions on the con¯icting approaches.
360 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

The policy assumes that no two con¯icting approaches reach the upper bound spe-
ci®ed for them simultaneously.
Another approach to critical intersection control has been suggested by Gordon
(1969). Gordon did not attempt to minimize delay at the intersection but rather to
maintain a constant ratio of the queue lengths on opposing approaches. The cycle
length is assumed constant and the splits are changed according to the demand
so that the ratio of the actual queues to the maximum link storage space on both
phases are equal.
Finally, Michalopoulos and Stephanopolos (1977) proposed an optimal control
policy for both pre-timed and real-time control. His control policy was to minimize
total system delay subject to queue length constraints.
However, it should be noted that most of the control algorithms mentioned above
suer from complex computational requirements described as above, and from the
lack of intersection-to-intersection trac ¯ow models.
3. Optimization model formulation
We describe our formulation of the network-wide signal optimization problem,
and provide the solution algorithm in Section 4. The formulation has several con-
cepts borrowed from the above algorithms, and perhaps does not consider some
constraints included in some of the above algorithms. We mention here that the
network programming solution algorithm provides a very intuitive way to incorpo-
rate the constraints of the formulation and to add or delete additional constraints
that could be considered. Brie¯y stated, the reason is that the solution uses network
paths, and even the most complicated constraints of the ¯ow and signal problem at
an intersection become intuitive and simpler to handle using the paths on the time-
expanded network. This will become clearer later, but it is useful to remember while
reading the formulation given next.
3.1. Introduction
In this research, the trac ¯ow and dynamic reduction of queues are described by
an appropriate linear model with linear capacity constraints for both road links and
¯ows in intersections. It is also assumed that origin±destination relationships are
known a-priori. As far as the network trac ¯ow is concerned, the network control
is a superposition of control at individual junctions. Because the control of neigh-
boring intersections determines the arrival process, the entire system is strongly
interactive.
The optimization model proposed here is a dynamic model with multiple-time
steps, as it is meant for real-time operations. Conventional static models like PAS-
SER II (i.e., Progression Analysis and Signal System Evaluation Routine) (Chang,
Lei & Messer, 1988) and TRANSYT-7F (Wallace et al., 1988) which use one set
of demand data during the entire control period are not appropriate for dynamic
signal control. A dynamic model should not only obtain minimum delay subject
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 361

to queue length constraints, but also should include formulas to de®ne relationships
of queue transition between time slices. In the dynamic model, the green/red lights
can change with every time slice. The signal intervals are adjusted toward minimiz-
ing delay and permitting queues to build up to a predetermined upper bound.
The optimization model uses mixed integer linear programming (MILP) to mathe-
matically model the above requirements. Some MILP formulations of signal control
already exist (Chang et al., 1988; Kim, 1990; Messer, Hogg, Chaudhary & Chang,
1986; Tsay & Lin, 1988), though not for complete networks. An MILP problem could
be solved using software packages available for mathematical programming such as
GAMS (i.e., General Algebraic Modeling Systems) (Brooke, Kendrick & Meeraus,
1988) and LINDO (i.e., Linear, Interactive aNd Discrete Optimizer) (Schrage, 1984),
which normally use the simplex algorithm and branch and bound solution techni-
ques. We, however, develop much faster algorithms as given in Section 4, for use
within a standard branch and bound solution scheme.
3.2. Road trac models and problem assumptions
A well-known platoon dispersion model (Hunt et al., 1982) is used to provide
dynamic interaction constraints among individual intersections which have their
own set of signalization constraints. This model has been empirically validated in
several urban areas around the world (since it is part of the SCOOT control
system implemented around the world). Though it is only an empirical model, it
is generally considered to represent (interrupted) trac ¯ow in signalized net-
works better than other models. The model is presented here to point to its linear
structure, which makes it particularly useful in solvable network optimal control
formulations.
The dynamic evolution of trac ¯ow on the jth road section (approach) can be
modeled by the following set of equations:
qj
outkTmin qj
gokT
qj
skTljkÿ1
1
ljk  ljkÿ1  qj
sk ÿ qj
outkT2
qj
ink  X
i
bijqi
outk 3
A discrete time approach with sample time T(the Tis selected adaptively, i.e.
the length of Tcan be set optionally depending on how adaptive the control sys-
tem needs; here the Tis chosen to be 5 s) is adopted. The following variables apply
for each road section j(j=1, ...,Js): qj
inkis the volume (veh/s) entering the jth
road section during kT4t4k1T;which depends on the turning fractions b
ij
from other links, qj
skis the volume which arrives at the end of the waiting queue or
at the stop-line, qj
outkis the leaving volume at the downstream end of the jth road
362 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

section, qj
gokis the capacity ¯ow during green trac light and ljkÿ1is the
number of vehicles waiting in the queue. qj
sand qj
in are connected through the pla-
toon dispersion equation (Eq. (4)) stated as follows:
qj
sk  Fq j
ink  1ÿFqj
skÿ1 4
where F1=1;it is a smoothing factor determined by a speci®c platoon
dispersion factor and the travel time coecient between the upstream and down-
stream of an intersection (F=1 means no dispersion). And is a constant param-
eter (normally around 0.5) which may have a value ranging from 0.2 to 0.5,
depending upon whether there is very little or extensive platoon dispersion along the
road link. The other parameter which is equal to 0.8(average travel time in T
units). However, one thing that needs to be noted is that many studies have focused
on the analysis and the calibration of the For factor. Most of these investigations
have suggested that platoon dispersion should not be generalized with the standard
default parameter settings that are suggested for this model, but that instead these
parameters should each time be customized to match the unique road condition on
each link.
In order to control trac during a limited period of interest eciently, detailed
information about the demand structure and the factors determining intersection
capacities, which often are the bottlenecks causing the congestion, can be useful. The
trac control model assumes that the trac assignment is known a-priori, which
means that the demand can be speci®ed for individual routes. Rather than using the
traditional concept of turning fractions at intersections, we use actual volumes for
each movement at the intersection. Note that it is easy to ®nd these by multiplying
approach arrivals by turning fractions as well. An additional reason is our path-
based approach. Recently proposed path-based assignment algorithms (Jayakrish-
nan, Tsai, Prashker & Rajadhyaksha, 1994; Sun, Jayakrishnan & Tsai, 1996) have
been found to perform much faster than the link-¯ow-based assignment algorithms,
which points to the attractiveness of our approach if path-based static or dynamic
assignment is used in real time for prediction of path ¯ows in an ATMS.
Roads entering the network have to be included in the model, because their queue
lengths are determined by the signals at the intersections. On the other hand, it is
assumed that vehicles can leave the networks as soon as they pass the last intersec-
tion on their routes. The entering links (or external approaches) are numbered 1,...,
Ewhile interior links (or internal movement) receive indices E+1, ..., H. Each exit
gets the same index as the associated entrance.
We also assume that:
1. the lost time (yellow and all red) of intersection Nis given;
2. the saturation ¯ows S
i
, I I
n
, are known; and
3. the turning movement fractions are known and may be time variant.
With regard to the turning movement fractions (last assumption above), it should
be noted that they may be estimated in real time by known algorithms (e.g. Cremer,
1991).
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 363

3.3. Model formulation of network trac signal control
Notations used in the model formulation are given in the nomenclature.
Fig. 1 shows the main variables on a from_link of phase iconnecting two inter-
sections m, n such that iO
m
and iI
n
. We de®ne qi
inkand qi
outkto be the in¯ow
and out¯ow, respectively, of phase iover a period [kT, (k+1)T] where Tis
the sample time interval and k=1, 2,... is a discrete time index. Similarly we de®ne
d
i
(k) and s
i
(k) to be the demand ¯ow and the exit ¯ow, respectively, occurring within
phase i.
The control objective of the dynamic model is to minimize total delay in the net-
work. Total delay is the sum of delays on all phases. The full formulation of the
mathematical program for minimization of delay is below, and the description of
the constraint set follows:
Minimize
Delay TD TX
N
n1X
i2InX
K
k1
lnik 5
subject to
lnik50;8iIn;nN;kK6
lnik5lni kÿ1  qi
sk ÿ qi
outkT;8iIn;nN;kK;Li4E7
lnik4MZni k;8iIn;nN;kK;where M is some large number 8
lnik ÿ flni kÿ1  qi
sk ÿ qi
outkTg4M1ÿZni k;
8iIn;nN;kK;Li4E9
Fig. 1. An urban road link.
364 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

lnik5lni kÿ1  TfF 1ÿi0  X
j2Oi
qj
outkÿ  1ÿF  qi
s
kÿ1  dik ÿ qi
outkg ;
8iIn;nN;kK;E14Li4H
10
lnik ÿ flni kÿ1  TfF 1ÿi0  X
j2Oi
qj
outkÿ  1ÿF  qi
s
kÿ1  dik ÿ qi
outkgg4M1ÿZni k ;
8iIn;nN;kK;E14Li4H
11
qi
outk  1ÿuni kSg
ni1ÿni k  Sy
ni nik
Sg
ni nik  uni k;8iIn;nN;kK12
unik  ni kÿ1  uni kÿ1 ÿ 2nikÿ1uni kÿ1;8iIn;nN;kK13
Unik  Uni kÿ1  T1ÿni kÿ1 ;8iIn;nN;kK14
Gi
min4Uni k4Gi
max ;8iIn;nN;kK15
X
i2QP
liPlni k4CP;8iIn;nN;kK;PBn 16
un1k  un2k51;un1k  un3k51;...un7k  un8k51;
8nN;kK17
The number of the vehicles discharged during the green time depends on whether
the corresponding phase is oversaturated or not. If the phase is oversaturated, then
it is equal to the capacity ¯ow during green. If the phase is undersaturated, then it
depends on the sum of the vehicle arrivals at the end of the waiting queue and the
existing queue lengths at the end of time interval kÿ1 (l
i
(kÿ1)). It should be noted
that whether a phase is oversaturated or undersaturated cannot be predetermined.
The model automatically determines the state of saturation during the optimization
procedure. The above equation also implies that the queue lengths occurring at
the end of each time interval must be non-negative. These considerations result in
Eqs. (6)±(11). Note that integer variables, Z
i
are introduced for modeling the non-
negativity of the queue lengths. Unfortunately, the model now becomes a complex
MILP problem because of the integer variables Z
i
, and u
i
described next.
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 365

For an internal link, the queue length l
ni
(k) is obtained by adding the balance
between the new arrivals and the departures to the queue length l
ni
(kÿ1) at the end
of the previous interval. This gives us Eqs. (10) and (11), which includes the platoon
dispersion variables described next. For the external approaches in the network (i.e.
link number L
i
4E), the queue lengths at the end of time interval k,l
ni
(k), can be
represented as the sum of any queues transferred from the previous time interval and
the dierence between input and output at the current time interval, which results in
Eqs. (7) and (9).
The most important factor to be considered in a network of signals is the
movement of vehicles from upstream intersections to downstream intersections,
which requires the use of a reliable trac model that can accurately re¯ect the
movement of vehicles in the network. Research has already been conducted on
the applicability of platoon dispersion as a reliable trac movement model in
urban street networks. Most of the research has shown that Robertson's model of
platoon dispersion is reliable, accurate, and robust (Axhausen & Korling, 1987;
Castle & Bonniville, 1985). While the arrival patterns for the external approaches
of the intersections are derived exogenously, the arrival patterns on the internal
movements are obtained from the departures of the upstream intersections using
Robertson's platoon dispersion equations. The Fvariable in Eqs. (10) and (11)
incorporates the platoon dispersion model into the constraints. In the presence of
long queues, the assumption of a constant travel time may lead to some inaccuracy
with regard to the queue evolution in each time step, but the delay calculation
should be suciently accurate, as the time spent by the vehicles will still be cap-
tured over a number of time steps. Note that for consideration of further control
measures such as route guidance and variable message signs control, the turning
movements are externally speci®ed, through variables appearing in Eqs. (10)
and (11).
The signal state of any phase i,u
ni
(k), at time step k, is given by Eq. (13). The ®rst
term represents the control decision at the end of time step kÿ1. The second term
signi®es the signal state of phase iat time step kÿ1.u
ni
(k) is a binary variable. If the
signal state was red for time step kÿ1, i.e. u
ni
(kÿ1)=1, and the control decision at
the end of the time step was to switchover, i.e. 
ni
(kÿ1)=1, then the signal state for
time step kmust be 0, which corresponds to a green state. The green time already
used up by phase iat intersection n, at the end of time step k, is computed by Eq.
(14). The ®rst term denotes the green time by the end of step kÿ1. Based on the
control decision, 
ni
(kÿ1), green time is either increased by a duration of Tsec-
onds or is increased by none. The standard minimum and maximum green time
requirements are re¯ected in Eq. (15). The maximum queue length (capacity) con-
straint is ®xed in Eq. (16).
Constraints are also needed to ensure that con¯icting phases are not given
the green indication during any split. Since us are constrained to be binary vari-
ables in the formulation, it is easy to ensure that no more than one of any combi-
nation of two non-permissible phases has its corresponding uset to 0 (i.e. green
time). In this formulation, the standard NEMA (National Electrical Manu-
factures Association) numbering convention for the dierent movements are used
366 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

to identify the dierent phases. Thus, the inadmissible combinations of
NEMA phases are phases numbered 1 and 2 (EW left and opposing through), 1
and 3 (EW left and SN left), 1 and 4 (EW left and NS through), and so forth,
through 7 and 8 (NS left and opposing through). These constraints are shown in
Eq. (17).
4. Development of solution process
In this section we present an ecient solution approach, based on a modi®cation
of the network simplex algorithm, to solve the model formulation proposed in the
previous section. Large-scale MILP problems such as above are normally solved
with branch and bound techniques with repeated simplex solutions of the linear
programming problem within the MILP. We next explain the ecient network
simplex algorithm that can replace the standard simplex for linear programming
problems of network form.
4.1. Network simplex algorithm
In the network simplex algorithm, we need not explicitly maintain the matrix
representation (known as the simplex tableau) of the linear program and can
perform all the computations directly on the network. If the dynamic optimal trac
signal model is solved by a simplex tableau, there are some disadvantages. In the
simplex tableau approach, we need to check each possible pivot to ®nd out
the minimum objective value. As a result, the enormously large number of pivots can
make the computation prohibitively expensive when the number of time periods and
the number of intersections on the network increase signi®cantly.
A specialized network simplex algorithm is used to eciently operate at any given
time interval. It is shown that the algorithm is more ecient in those problems for
which its structure is a large-scale network form (for more detailed description
about the eciency of this method and its illustrated examples, see Ahuja, Magnanti
& Orlin, 1993, and Kennington & Helgason, 1980).
The network simplex algorithm maintains a feasible spanning tree structure and
moves from one spanning tree structure to another until it ®nds an optimal struc-
ture. At each iteration, the algorithm adds one arc to the spanning tree in place of
one of its current arcs. The entering arc is a nontree arc violating its optimality
condition. The algorithm: (1) adds this arc to the spanning tree, creating a negative
cycle loop; (2) sends the maximum possible ¯ow in this cycle until the ¯ow on at
least one arc in the cycle reaches its lower or upper bound; and (3) drops an arc
whose ¯ow has reached its lower or upper bound, giving us a new spanning tree
structure. Because of its relationship to the primal simplex algorithm for the linear
programming problem, this operation of moving from one spanning tree structure
to another is known as a pivot operation.
The network simplex algorithm maintains a feasible basis structure at each itera-
tion and successively modi®es the basis structure via pivots until it becomes an
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 367

optimum basis structure. The special structure of the basis enables the simplex
computations to be performed eciently.
4.2. Modi®ed network simplex algorithm and the system control logic
4.2.1. Graphical presentation of network programming formulation
The model presented in Section 3 provides the coupling constraints in the network
optimization formulation. Each intersection has its own constraints in terms of the
phase sequences and con¯icting movements. The arrival ¯ows in each time period at
each intersection are dependent on the departure ¯ows from other intersections
through these platoon dispersion coupling constraints. The research has identi®ed a
key aspect of the formulation, namely that the linear optimization sub-problem
involves extremely sparse matrices, thanks to the network structure. One key reason
for this is the nature of the platoon dispersion model which is similar to a multi-
period inventory ¯ow model.
Note that the Robertson's platoon dispersion equation means that the trac ¯ow
(qj
sk), which arrives during a given time step at the downstream end of a link, is a
weighted combination of the arrival pattern at the downstream end of the link dur-
ing the previous time step (qj
skÿ1) and the departure pattern from the upstream
trac signal seconds ago (qj
inkÿ). However, the recurrence platoon dispersion
Eq. (4) can be transformed as the following form:
qj
sk  X
1
i
F1ÿFiÿqj
inkÿi;18
where Fis a dispersion parameter and is a travel time coecient between the
upstream and downstream intersection. The summation can be truncated after a
reasonable number of terms (say 10±15).
The transformed platoon dispersion equations directly translate to links on a
time-expanded network, as in Fig. 2. A complete graphical representation of
the network structure including the constrained turning movements at the inter-
section for multiple time steps is too complicated to show here; however, the net-
work inventory-¯ow nature of the linear platoon dispersion equations should be
clear from Fig. 2. The network size of the whole trac signal control problem can
be reduced after each iteration of computation to a simpler and smaller network.
This reduction process simpli®es the diculty of incorporating directly net-
work simplex algorithm to platoon dispersion-based network trac signal control
problem.
The optimization algorithms employ the two standard principles of implicit enu-
meration: network simplex algorithm and branch and bound. Branch and bound is
mainly used to obtain integer solutions when combined with some other relaxation
method. Alternatively, the network simplex algorithm is mainly applied to solve
minimum cost ¯ow problems (i.e. delay minimization); so it appears as subroutines
in branch and bound methods. The elements of a network simplex algorithm are
368 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

demands, nodes, arcs, costs, and capacities which determine the value of the objec-
tive function at each iteration.
4.2.2. Non-standard network ¯ow problem and special-case simplex
The network problem presented above poses one diculty, however, and this is
the ®xed split required from the upstream node to the platoon dispersion links. The
standard network simplex algorithm does not involve such `®xed nodal splits frac-
tions' [F,F(1ÿF), F(1ÿF)
2
, etc., as in Fig. 2, are these fractions]. This poses a di-
culty during the basis update step of network simplex. The reason is that when we
attempt to change the ¯ow on any network loop (cycle) involving any of the platoon
dispersion arcs, the ratio of ¯ows among the platoon dispersion arcs change. This
has to be avoided to retain the split ratios, which means the simplex basis update
schemes no longer apply. Thus, the class of network problems with such ®xed nodal
splits are considered to be a class of problems with much higher diculty which do
not have easy solutions.
We develop a technique to handle this problem and the basic idea is straight for-
ward. We can see that the arcs with costs in the network are only the inventory
(queuing) arcs shown vertically in Fig. 2. This means that a restricted version of
basis update for such a graph that is not based on normal augmenting cycles (loops)
can be developed. This essentially involves the selection of a few inventory arcs (i.e.
upstream intersections' arcs) and updating the current solution of split fraction arcs
in the whole network. Once the inventory arc's ¯ow (note that this is not actual
trac ¯ow, but rather queue length) is updated, the exit ¯ows and the platoon dis-
persion links can be updated. The algorithm operates in a decomposed fashion,
Fig. 2. Graphical representation of network programming formulation.
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 369

starting from peripheral nodes and moving inwards, updating platoon arcs and
in¯ows to other nodes. Note that the platoon dispersion arc ¯ows (which in turn
depend on the signal settings) are adjusted after every iteration, and the network
simplex operates essentially independently for each intersection. After about four or
®ve iterations, all the nodes are normally reached in reasonably sized networks, and
the signal settings converge in a few more additional iterations (enough to adjust
¯ows along loops in the network).
This approach of the solution process is implemented in iterations as described in
the previous section by using network simplex algorithm as the subroutine of branch
and bound method. In every iteration, ®rst of all, the branch and bound is per-
formed to prevent con¯icting movements. As Eq. (18) shows, we can easily ®nd the
¯ow rates of the initial platoon (i.e. qj
ink) during time step kby ignoring the split
phenomenon on the road link. This is followed by splitting these platoon ¯ow rates
as the initial arriving ¯ows of downstream intersection. The algorithm continues by
®nding the second iteration's ¯ow rates as the initial platoon ¯ow rates of next iter-
ation's under the implementation of branch and bound, and so on.
A platoon dispersion-based problem with ¯ow split constraints is constructed with
the introduction of an arrival ¯ow prediction/estimation concept. Based on the
branch and bound-and-network-simplex algorithm, the solution of the ®rst iteration
(usually the minimum cost ¯ow computation of upstream intersections in a network)
facilitates an initial solution of the second iteration. The solution procedure for each
of the iterations can be stated as the following system control strategy.
The following steps describe the basic control strategy governing the proposed
models for network trac signal control with platoon dispersion constraints. Given
the aforementioned system and all functions of its key elements, the operational
procedures are summarized below.
.Step 0. All the external intersections included in the set Nare initialized with
external entry ¯ows and initial branch and bound signal settings.
.Step 1. For intersection n(in current set N) for each phase icarry out the
branch and bound steps. If all intersections are completed, go to step 7.
.Step 2. Check the minimum and maximum green constraints for all time steps
k:
Condition 1: If green time is less than the minimum green time (i.e.
Uik4Gi
min), then extend the green. For instance, if 
i
(k)=0, then U
i
(k) is
updated.
Condition 2: If U
i
(k), the green time already used up by phase iat time step
k, has reached the maximum green time Gi
max (i.e. Uik5Gi
min), then the
green is terminated immediately, i.e., 
i
(k) is changed to 1. If the integer
variables for all time step kremain the same as in the previous iteration,
update nto the next intersection and go to step 1. Else go to step 3.
.Step 3. Examine and prevent the con¯icting movements which happen at
intersections using the branch and bound method by setting the constraints
uik  ujk51;where i,jI
n
and u
i
and u
j
are binary integers which are either
0 or 1.
370 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

.Step 4. Using the network simplex-based algorithm to get the minimum cost
¯ow solution (i.e. delayed vehicles, TD) of the network for the green extension
option evaluated by the branch and bound. (Here network simplex is operated
at the given intersection and could even conceptually be replaced with vehicle
clearance rules if they can guarantee optimality.)
.Step 5. If the performance index is the bene®t of giving a green which is com-
pared with that of terminating it by computing the tradeos incurred in vehicle
delays and it is negative, then the optimal decision is not favorable to the
intersection and a switchover decision is possible. Otherwise, the current green
can be extended for another Tseconds, i.e. Uik1  Uik  T:
.Step 6. Go to step 2 to ensure of switchover being within the min±max
constraints.
.Step 7. Update platoon dispersion arc ¯ows and the corresponding in¯ows to
existing intersections as well as new intersections joining the set N. If the pla-
toon dispersion arc ¯ows are same as in the previous outer iteration (over the
last set of intersections N), i.e. no signal settings changed, then STOP; else, go
to step 1.
In the proposed models, the control decision is made every 5 s (i.e. duration of a
time step) depending on the comparison of bene®ts between extending green and
terminating it. The control strategy makes use of real-time trac state conditions,
instead of pre-stipulated strategies.
4.2.3. Real-time aspects Ð implementation
Although this paper assumes full information on external inputs, the for-
mulation can be applied in a rolling horizon control scheme in a sensor-based
environment. The ¯ow detectors can be located at the upstream junction output
and at about 120 feet from the stop line an online updating scheme for the
turning fractions as well as the platoon dispersion factors is possible, in addition
to updating the side entry/exit ¯ows and external entry ¯ows. One thing needs to
be noticed here is that vehicles are presumed to travel undelayed to the stop line
before joining a vertical queue (if present). As a consequence of this assumption,
detectors upstream of the intersection give advance information on vehicle arri-
vals at the stopline, the degree of forewarning depending on the undelayed tra-
vel time from the detector to the stop line. In addition, the more we go away
from the present time, the more the demand forecasts and the model calcula-
tions become inaccurate. In practice, the extend of forewarning is limited and is
unlikely to exceed 15 seconds. Any such implementation could present unfore-
seen challenges and sensitivity analyses are necessary to address questions
regarding the choice of parameter values (e.g. roll period, stage length) and their
in¯uence on the eectiveness of the solution procedure. Also of signi®cance is
the deployment of detectors and issues such as whether the upstream output
detectors are needed at all. These issues and issues pertaining to the computa-
tional eciency of the solution procedure are the focus of current research
eorts on this topic.
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 371

5. Comparison of solution eciency via numerical tests
The preliminary studies on the signal timings from this optimization have shown
that the results are sound. Since other such network-wide optimization formulations
are currently unavailable for implementation, no comparisons with other formula-
tions are possible. We also point out that since our algorithm is not heuristic, the
solutions will be optimal within the assumptions of the model. Thus, such compar-
isons will only be contrasting the assumptions of the models, not how well the
assumptions re¯ect reality. In this section, however, we present results which are
most important in determining the practicability of the algorithm, namely, its ability
to be computationally solved in a fast manner.
5.1. Conventional solution approach via commercial programs
Some restrictive versions of the network signal control mixed integer model with a
limited number of intersections and time steps were solved using LINDO program.
The optimal solution of simple cases were obtained by the LINDO. Even in the
simple case of only three time intervals, the number of constraints exceed 10,000 and
the number of variables exceed 2000. This huge dimensionality renders the solution
process dicult, especially in the execution of simplex tableau process and the
branch and bound techniques. The computation time needed is as many as 30 min
on a personal computer (120 Mhz Pentium with 16MB RAM), and can even extend
beyond a day for an increase in any such related components as number of inter-
sections, time intervals, or number of phases. While providing an initial check on the
correctness of the formulation, the computation results for the simple cases clearly
demonstrate that standard treatment by mathematical programming are not practi-
cally feasible with such a complex formulation.
5.2. Comparison of network simplex programming algorithm with LINDO program
As indicated in the previous section, the LINDO program was used as a tem-
porary tool to ®nd the solution to a series of test cases to ensure validity of the
model formulation. The characteristics and performance of the ecient network
simplex programming algorithm is introduced and applied to solve such problems
below.
For illustration, we provide an example of an urban network. A simple network
composed of one controlled four-approach junction surrounded by controlled junc-
tions is selected as an application example. The network characteristics are shown in
Fig. 3. The link lengths were all 1000 feet each and the average vehicle speed was 30
mph, respectively. This structure allows a ¯uctuated demand on the controlled links
according to the varied green and red durations of the neighbor junctions. The
entering network demand is assumed to be known along the time horizon. This is, of
course, not a realistic con®guration from a trac point of view but this allows us to
study the principal algorithm's behavior. Trac ¯ow generation is according to a
Poisson distribution. Also, the percentage of turning movements and exit rates
372 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

occurring in internal links are assumed to be zeros here in order to simplify this test
example and ensure that a LINDO solution for comparison was possible.
Two-phase operations are used in the evaluation and comparison. The saturation
¯ow rate of each of the phases is time-invariant and set to be 1800 vph. Five demand
intervals of 5 s each and zero initial queues are used in the comparisons. A minimum
green time of duration 10 s (including yellow and all-red time) and a maximum green
time of 60 s are used in the control simulation.
With the optimization control for this test network, the trac arrival ¯ows of the
external approaches on each of the phases are using three cases of average arrival
¯ows (800 vphpl, 1000 vphpl and 1200 vphpl) to test our control model. The delay
(veh-s) and computation times (s) are provided in Table 1. The percent change
column gives the amount of delay or computation time reduction achieved by the
network simplex programming algorithm as a percentage of the delay and compu-
tation time due to LINDO program. As shown by Table 1, for the ®ve-node net-
work, 1200 vphpl case, the algorithm reaches the objective value of 937.84 veh-s
utilizing 17 s on a personal computer (120 Mhz Pentium with 16MB RAM) by using
network simplex programming algorithm. Compared with the same delay value and
computation time 38 min (2280 s) obtained by LINDO program, the computation
time reduction is signi®cantly achieved to 99.25% in this case. Note that the network
simplex-based algorithm is applied here to a problem without rolling horizons. In a
rolling horizon implementation, better initial solutions will cause each problem to be
solved much faster than the times implied by the 17 s here.
The signi®cant result here is how the solution technique is orders of magnitude
faster than normal linear programming solutions. These results show that the
Fig. 3. Link-node diagram of urban network with ®ve intersections.
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 373

Table 1
Comparisons of solution eciency for three simulated arrival ¯ows
Trac volume
(vphpl)
LINDO program Network simplex
programming algorithm
Percent change
Delay (veh/s) Computation time (s) Delay (veh/s) Computation time (s) Delay (veh/s) Computation time (s)
800 310.24 2250 310.24 15 0 99.33
1000 742.56 2280 742.56 15 0 99.34
1200 937.84 2280 937.84 17 0 99.25
374 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

formulation, though it is large, can be solved in a manner fast enough for real-time
operation. In fact, we could say that the solution technique is very ecient as we
solved a problem that has never been solved before, and showed that our funda-
mental assumption that the hypothesized network structure can be exploited for a
model that runs hundreds of time faster than a non-network model. The solution
times are expected to improve signi®cantly after we remove certain computational
overhead as well. We expect real-time solution speeds for networks of up to a few
tens of nodes.
6. Conclusions and future research
This research presents a systematic approach to network trac signal control
problem. The approach involves formulation and solution of a linear optimiza-
tion problem for multiple intersections in a network, connected by explicit con-
straints capturing trac movement on the links connecting the intersections.
Our model formulation is distinct from other models in that the trac arrival
platoon is explicitly incorporated into the network signal control model. The pla-
toon dispersion constraints directly translate to links of a time-expanded network,
and thus the problem has the form of a linear multi-commodity network ¯ow prob-
lem. Even though, the ®xed nodal splits for the platoon constraints render it a non-
standard problem, we develop ecient special-purpose version of network simplex
to solve the problem, in conjunction with a branch and bound scheme for the integer
signal constraints. The results show extremely fast solutions from this algorithm.
Several operational aspects of the algorithm and its potential application remain
to be studied. One option of particular interest is a hierarchical operation where the
algorithm provides strategic control for an extended trac network with an updat-
ing period of several seconds and complemented by an inferior, short-term reacting,
possibly decentralized direct-control layer, and a superior adaptation layer that
provides updated demand forecastings, estimation, and detection information
(Papageorgiou, 1984; Stephanedes & Chang, 1993). Another possibility (and per-
haps necessity) is to use the algorithm for smaller subnetworks based on real-time
computational concerns, and to coordinate multiple subnetworks using a super-
visory program.
Real-world numerical results and evaluation is to follow in the future. The current
and next steps related to this research include the following issues: (1) integration of
the available demand-forecasting algorithms; (2) use of observed ¯ows and queues
with stochastic feedback and ®ltering schemes for detected data; and (3) application
of the overall methodology to real trac signal networks.
References
Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network ¯ows theory, algorithms, and applications.
New Jersey: Prentice Hall.
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 375

Axhausen, K. W., & Korling, H. G. (1987). Some measurements of Robertson's platoon dispersion factor
(Transportation Research Record 1112, 71-77). Washington, DC: Transportation Research Board.
Boillot, F., Blosseville, J. M., Lesort, J. B., Motyka, V., Papageorgiou, M., & Sellam, S. (1992). Optimal
signal control of urban trac networks. In International conference on road trac monitoring and con-
trol (6th), Institution of Electrical Engineers, London, UK.
Brooke, A., Kendrick, D., & Meeraus, A. (1988). GAMS: a user's guide. New York: World Bank.
Burhardt, K. K. (1971). Urban trac system optimization. PhD thesis, University of Minnesota, Minnea-
polis, MN.
Castle, D. E., & Bonniville, J. W. (1985). Platoon dispersion over long road links (Transportation Research
Record 1021, 36-44). Washington, DC: Transportation Research Board.
Chang, E. C. P., Lei, J. C., & Messer, C. J. (1988). Arterial signal timing optimization using PASSER II-
87 Ð microcomputer user's guide (Report TTI-2-18-86-467-1). College Station, TX: Texas Transporta-
tion Institute, Texas A&M University System.
Cremer, M. (1991). Origin±destination matrix: dynamic estimation. In M. Papageorgiou, Concise ency-
clopedia of trac and transportation systems (pp. 310±315). Oxford: Pergamon Press.
Cremer, M., & Schoof, S. (1990). On control strategies for urban trac corridors. Control, computers,
communications in transportation (pp. 213±219). Oxford, New York: Pergamon Press.
Dunne, M. C., & Potts, R. B. (1964). Algorithm for trac control. Oper. Res., 12, 870±871.
Eddelbuttel, J., & Cremer, M. (1992). A new algorithm for optimal signal control in congested networks.
Journal of Advanced Transportation, 28(3), 257±297.
Gartner, N. H. (1983). OPAC: a demand-responsive strategy for trac signal control. Transportation
Research Record, 906, 75±81.
Gartner, N. H., Tarno, P. J., & Andrews, C. M. (1991). Evaluation of optimized policies for adaptive
control strategy (Transportation Research Record 1324, 105±114). Washington, DC: Transportation
Research Board.
Gazis, D. C. (1964). Optimum control of a system of oversaturated intersections. Oper. Res., 12, 815±831.
Gazis, D. C., & Potts, R. B. (1963). The oversaturated intersection. Yorktown Heights, N.Y.: International
Business Machines Corporation, Thomas J. Watson Research Center.
Gordon, R. L. (1969). A technique for control of trac at critical intersections. Transpn. Sci., 4, 279±287.
Green, D. H. (1968). Control of oversaturated intersections. Ops. Res. Quart, 18, 161±173.
Head, L. J., & Mirchandani, P. (1997). RHODES-integrated trac management system (Report No. AZ-
SP-9701). AZ: Arizona Department of Transportation.
Henry, J. J., & Farges, J. L. (1990). PRODYN Control, Computers, Communications in transportation
(pp. 253±255). Oxford; New York: Pergamon Press, 1990.
Henry, J. J., Forges, J. L., & Tual, J. (1983). The PRODYN real time trac algorithm. Fourth IFAC-
IFIP-IFORS conference on control in transportation systems (pp. 305±310), Germany, Baden-Baden 307.
Hunt, P. B., Robertson, D. I., Bretherton, R. D., & Royle, M. C. (1982). The SCOOT on-line trac signal
optimisation technique. Trac Engineering and Control, 23, 190±192.
Jayakrishnan, R., Tsai, W. K., Prashker, J. N., & Rajadhyaksha, S. (1994). A faster path-based algorithm
for trac assignment. Transportation Research Record, 1443, 75±83.
Kennington, J. L., & Helgason, R. V. (1980). Algorithms for network programming. New York: John
Wiley.
Kim, Y. (1990). Development of optimization models for signalized intersections during oversaturated con-
ditions. PhD thesis, Texas A&M University, College Station, TX.
Lee, B., Crowley, K., & Pigantaro, L. (1975). Better use of signals under oversaturated ¯ows. Transpor-
tation Research Board, SR153, 60±72.
Lindley, J. A. (1987). Urban freeway congestion: quanti®cation of the problem and eectiveness of
potential solutions. ITE Journal, 57(1), 27±32.
Little, J., Kelson, M., & Gartner, N. (1981). MAXBAND: a program for setting signals on arteries and
triangular networks (Transportation Research Record 795). National Research Council, Washington,
DC, pp. 40±46.
Longley, D. (1968). A control strategy for a congested computer controlled trac network. Transporta-
tion Research, 2, 391±408.
376 Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377

Lowrie, P. R. (1982). SCATS principles, methodology, algorithm. IEE conference on road trac signalling
(IEE Publication 207, pp. 67±70). London: IEE.
Lowrie, P. R. (1990). SCATS Ð a trac responsive method of controlling urban trac (sales information
brochure). Sydney, Australia: Roads and Trac Authority NSW, Trac Control Section.
Luk, J. (1984). Two trac-responsive area trac control methods: SCATS and SCOOT. Trac Engi-
neering and Control, 25(1), 17±18.
Messer, C. J., Hogg, G. L., Chaudhary, N. A., & Chang, E. C. P. (1986). Optimization of left turn phase
sequence in signalized networks using MAXBAND 86±vol. 2. User's manual (Report FHWA/RD-84/,
FHWA). Washington, DC: US Department of Transportation.
Michalopoulos, P. G. (1992). Intersection control through video image processing: executive summary.
Prepared for the Minnesota Department of Transportation, Department of Civil and Mineral Engi-
neering, University of Minnesota, St. Paul, MN.
Michalopoulos, P. G., & Stephanopolos, G. (1977). Oversaturated signal systems with queue length con-
straints-I. Transportation Research, 11, 413±421.
Miller, A. J. (1963). Settings for ®xed-cycle trac signals. Oper. Res. Q., 14, 373±386.
Miller, A. J. (1965). A computer control system for trac networks. In J. Almond, Proceedings of the
second international symposium on the theory of road trac ¯ow (pp. 200±220). Paris: Organisation for
Economics Cooperation and Development.
Papageorgiou, M. (1983). Applications of automatic control concepts to trac ¯ow modeling and control.
Berlin; New York: Springer-Verlag.
Papageorgiou, M. (1984). Multilayer control system design applied to freeway trac. IEEE Transactions
on Automatic Control, 29, 482±490.
Robertson, D. I. (1969). TRANSYT: a trac network study tool (TRRL Report LR 253). London:
TRRL.
Robertson, D. I., & Bretherton, R. D. (1991). Optimizing networks of trac signals in real-time: the
SCOOT method. IEEE Trans. on Vehicular Technology, 40(1), 11±15.
Ross, D., Sandys, R., & Schlae¯l, T. (1970). A computer control scheme for critical intersection control in
an urban network. Menlo Park, CA: Stanford Research Institute.
Schrage, L. E. (1984). LINDO: user's manual, Palo Alto, CA: Scienti®c Press.
Sims, A. (1979). The Sydney coordinated adaptive trac system. Proceedings of the ASCE engineering
foundation conference on research priorities in computer control of urban trac systems, 12±27.
Stephanedes, Y. J., & Chang, K.-K. (1993). Optimal control of freeway corridors. ASCE Journal of
Transportation Engineering, 119, 504±514.
Sun, C., Jayakrishnan, R., & Tsai, W. K. (1996). Computational study of a path-based algorithm and
its variants for static trac assignment. Paper presented at Transportation Research Board annual
meeting. Washington, DC: TRB, National Research Council.
Tarno, P. J., & Gartner, N. (1993). Real-time, trac adaptive signal control. ATMS Conference (pp. 157±
167), St. Petersburg, FL.
Tsay, H., & Lin, L. (1988). A new algorithm for solving the maximum progression bandwidth (Paper pre-
sented at Transportation Research Board). Washington, DC: TRB, National Research Council.
Wallace, C. E. et al. (1988). TRANSYT-7F user's manual. University of Florida, Gainesville, FL: Trans-
portation Research Center.
Webster, F. V. (1958). Trac signal settings (Road Research Technical Paper 39). London: Her Majesty's
Stationery Oce.
Wann-Ming Wey / Comput., Environ. and Urban Systems 24 (2000) 355±377 377