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Transportation Research Record 1857 ■39
Paper No. 03-2627
An online routing profile updating automaton (ORPUA) approach is
introduced as a principal mechanism for operating an online hybrid
dynamic traffic assignment (DTA) system for real-time route guidance in
a traffic network. The hybrid DTA approach integrates the centralized
and the decentralized DTA frameworks by partitioning the set of guided
users into two classes according to an initial routing profile (IRP). One
class receives the centralized DTA guidance, while the other follows the
decentralized DTA routing. ORPUA takes the a priori IRP and updates
the guidance supplied to vehicles in a real-time fashion according to the
unfolding network conditions and relative performance of the two classes
of users. It does not anticipate the future network conditions; instead, it
reacts to them and optimizes the overall system performance by improv-
ing the performance of the underperforming class of vehicles. Simulation
experiments illustrate ORPUA’s potential in maintaining desirable sys-
tem performance and robustness in most of the demand-supply scenarios
considered.
One of the most challenging questions in the design and implementa-
tion of online dynamic traffic assignment (DTA) capabilities for large-
scale, real-time network operation is the system’s ability to maintain
solution quality, robustness, and responsiveness in a stochastic oper-
ating environment (1, 2). The online DTA problem of interest in this
paper entails determining routing policies to supply guidance infor-
mation to users who are about to start their trip as well as updating these
routes for users en route to ensure the performance and robustness of
the route guidance in a stochastic environment. Most proposed DTA
models for network routing applications assume deterministically
known inputs, such as origin–destination (O-D) matrices, parameters
used in traffic flow models, link cost functions, and so forth. When
unfolding realizations deviate from predicted (input) states, the per-
formance of resulting solutions may no longer be optimal. A DTA
model is considered robust if its performance degradation resulting
from varying erroneous model inputs is within an acceptable range.
The solution quality, robustness, and responsiveness are issues com-
monly concerned in many decision problems that arise in stochastic
dynamic systems. General approaches for such situations include the
following: (a) generating a priori solutions that are robust vis-à-vis the
actual realizations; (b) reoptimizing the problem periodically or as
soon as actual realizations are unveiled; and (c) hybridizing the pre-
ceding two approaches, starting from a good a priori solution, and per-
forming reoptimization for certain subproblems online.
In the first approach, a good, robust solution may not be easy to
find for the DTA problem because of the uncertainties and com-
plexity of the problem. The second approach is generally chal-
lenging from a computational standpoint for large-scale networks,
especially when real-time execution requirements are stringent.
The third type of approach could be the most suitable operational
paradigm for online DTA problems because of its potential and
flexibility to maintain the system performance at a desirable level
under various scenarios.
Only a limited number of previous studies have addressed the pre-
ceding online DTA performance issue, which remains a fertile area
for methodological development. Peeta and Mahmassani (3) intro-
duced the rolling horizon (RH) implementation of an online cen-
tralized DTA (CDTA) model with multiple user classes. The RH
approach provides a natural mechanism to confine error propagation
and enable periodic control actions and operator intervention. How-
ever, it still requires reliable short-term forecasts of O-D demands,
and the architecture demands intensive computational resources for
large networks. Chiu et al. (4) investigated the impact of demand pre-
diction errors and RH parameter settings (e.g., stage length and roll
period) on the performance of CDTA models in a RH framework.
Their experiments suggested that demand prediction errors were the
most influential factor that affects model performance and robust-
ness. Peeta and Zhou (5) also investigated the online robust CDTA
assignment where the routing policies are obtained from the a priori
solution of a stochastic programming formulation that explicitly rep-
resents the O-D demand as a stationary stochastic process. Because
of computational burden, these solutions must be obtained offline,
with a Monte Carlo approach in this case. The offline solutions are
applied online as a fixed strategy regardless of the actual unfolding
traffic states. Although this conceptually attractive approach gener-
ally outperforms solutions developed for a single average demand
scenario, it suffers limitations as a stand-alone approach for online
operation because of its computational burden and the inability to
predict and incorporate specific incident occurrences in the solution
procedure.
A commonly used approach to circumvent some of the limitations
of centralized approaches relies on decentralized control logic. In the
context of route guidance, a decentralized scheme was first proposed
by Sarachink and Ozguner (6) under simplified network flow repre-
sentations. Papageorgiou and Pavlis (7, 8) actively pursued decen-
tralized feedback control schemes for both route guidance and traffic
control. Hawas and Mahmassani (9) proposed and tested reactive
local rules with various communication protocols among controllers.
The decentralized DTA (DDTA) mechanism envisions a distributed
architecture in which vehicle routing is administered by a set of
Routing Profile Updating Strategies for
Online Hybrid Dynamic Traffic
Assignment Operation
Yi-Chang Chiu and Hani S. Mahmassani
Y.-C. Chiu, Department of Civil Engineering, University of Texas at El Paso,
500 West University Avenue, El Paso, TX 79968. H. S. Mahmassani, Department
of Civil and Environmental Engineering, University of Maryland, 1173 Glenn L.
Martin Hall, College Park, MD 20742.
spatially distributed control units or agents. Each agent oversees
only a certain portion of the network and provides real-time routing
solutions that rely on limited, locally available information with vary-
ing degrees of communication among agents (9, 10). Spatial infor-
mation is available only in each agent’s control territory or in
extended areas with a certain degree of cooperation among agents.
The DDTA mechanism has been found to possess greater robustness
than a RH CDTA vis-à-vis traffic incidents and large prediction errors
inherent in the CDTA route guidance solutions (1, 10). However,
CDTA solutions typically outperform reactive local DDTA solutions
when the demand patterns are reasonably predictable.
Chiu and Mahmassani (11, 12) proposed a hybrid DTA (HDTA)
framework, which integrates the CDTA models and the DDTA capa-
bilities. The underlying concept of the HDTA framework is for the
CDTA to provide predictive route guidance, for given predicted
O-D demand, to a subset of all users in the network. Some other sub-
set of users then follows routes based on DDTA guidance generated
by online reoptimization according to unfolding traffic. The total
number of users requesting route guidance is initially partitioned
into two classes for the purpose of initial route assignment—given
to users at the time of departure—according to the initial routing
profile (IRP). The rationale is that DDTA users allow the overall
system to adapt and react to unfolding conditions, thereby compen-
sating for the negative impact of forecast errors and unpredicted
events on the CDTA solution. Note that alternative approaches to
the design of hybrid DTA systems are possible and also have been
proposed recently (13).
Determining an optimal IRP requires knowledge of the relative
quality of both CDTA and DDTA assignment solutions. The solution
quality of both CDTA and DDTA assignment solutions in fact is
determined by numerous factors that may not be known a priori before
actual online operation. For example, O-D desire, a critical factor that
affects the solution quality of the CDTA model, is difficult to predict
accurately a priori. Randomly occurring incidents that disturb net-
work supply conditions are difficult to capture when the a priori solu-
40 Paper No. 03-2627 Transportation Research Record 1857
tions are generated. As a result, a constant a priori IRP is not likely to
remain optimal and guarantee robust performance online.
A natural way to design an effective and robust HDTA framework
is to design a reasonable a priori IRP as the starting point of the assign-
ment and then dynamically adjust the routing profile vis-à-vis the
unfolding realizations. A Monte Carlo approach has been proposed to
determine a good a priori IRP (14). This paper presents an online rout-
ing profile updating automaton (ORPUA) model to adjust the online
routing profile.
As indicated in Figure 1, the a priori HDTA solves for the a priori
routing policies and the IRP. Vehicles are assigned to receive either
CDTA or DDTA guidance based on the a priori IRP. ORPUA con-
stantly monitors the performance of both classes of vehicles and
updates the online routing profile accordingly. If ORPUA determines
that the DDTA-guided vehicles are experiencing better routing per-
formance, it will switch certain CDTA-guided vehicles to receive
DDTA routing policies and vice versa. Such decisions are repeated
regularly until the end of the control horizon.
The detailed ORPUA model definitions and specifications are
discussed in the next section. The third section presents the algorith-
mic procedures of ORPUA, and the fourth section discusses the per-
formance of ORPUA through simulation experiments. Concluding
comments are provided in the last section.
ORPUA MODEL DEFINITIONS
ORPUA is defined by the quintuple <S, A, P, G, C>, where Sis
the set of possible internal states of the automaton; Ais the set of
possible actions; Pis the discrete probability distribution of the
action space and G: S→P(A)is the rule-based mapping function
that maps the internal states Sto probability distributions P; and
Cis the set of criteria to select the subset of vehicles to which the
actions are applied. The components of ORPUA are discussed in
detail hereafter.
HDTA
Newly Generated
Vehicles
A Priori Initial Routing Profile
A Priori Routing Policies
ORPUA
Performance of CDTA
and DDTA Vehicles
Traffic Dynamics
Online Routing Profile
Total Guided Vehicles
FIGURE 1 Online HDTA operations framework.
Internal States S
Sis defined by the double <M, H>. Mis defined as the set of perfor-
mance measures. Mis defined on, but not limited to, the set [mt, md],
where mt is the travel time measure and md is the travel distance mea-
sure. The set H={Ht
mt, Ht
md} contains the sets of distribution estima-
tors for the travel time and travel distance at time t. Ht
mt is defined as
the pair of travel time distribution estimators for CDTA-guided and
DDTA-guided vehicles leaving from all origins to all destinations at
departure times τbefore t. That is,
where the distributions are characterized by respective mean esti-
mators µ
ˆmt,C, µ
ˆmt,L; standard deviation estimators, s
ˆmt,C, s
ˆmt,L; and sam-
ple sizes nmt,C, nmt,L. Ht
md is defined as the set of the travel distance
distribution estimators for CDTA-guided and DDTA-guided vehi-
cles leaving from all origins to all destinations at all departure times
τbefore t. That is,
where the distributions are characterized by respective mean esti-
mators µ
ˆmd,C, µ
ˆmd,L; standard deviation estimators s
ˆmd,C, s
ˆmd,L; and
sample sizes nmd,C, nmd,L.
As mentioned, the performance measure set Mcould be defined
over a large measure space, depending on the control objectives of
the updating automaton. In this problem, the experienced travel time
mt is the most important measure of effectiveness (MoE), and the
experienced travel distance md provides complementary information
to mt under certain conditions. Therefore, mt is defined as the primary
MoE and md is defined as a secondary MoE. The characteristics of
these two MoEs are discussed next.
mt is defined as the primary performance measure in S. At any
time t, for a vehicle departing at time τ, the travel time measure mt
is defined as follows:
For a vehicle that departs at time τ, the travel distance measure
is taken as md =∑di, where diis the length of link iin the path the
vehicle has already traversed by time t.
The distribution estimators of the travel time measure mt exhibit
certain properties, as follows. First, because Ht
mt is the travel time
distribution for all vehicles that depart between the same O-D pair
at the same time τ, if t−τis less than the minimum required travel
time between the O-D pair, there will not be any vehicle departing
at time τthat would have reached the destination at time t. There-
fore, the travel time distribution estimators for CDTA and DDTA
class vehicles taken at time twould have the same mean t−τand no
dispersion—that is, the trip time distributions are hmt,C
i,j,τ,t(t−τ, 0, nmt,C)
for CDTA class vehicles and hmt,L
i,j,τ,t(t−τ, 0, nmt,L) for DDTA class
vehicles. In other words, at 0 ≤t−τ≤φ
i,j, where φi,j is the minimum
travel time (travel at free-flow speed) between origin iand destina-
tion j, the travel time distribution estimators for CDTA and DDTA
vehicles are identical. This property suggests that the mt distribution
estimator will not be able to identify performance discrepancy be-
mt t t
mt t
=−
=− ≤
τ
ατ α α
if the vehicle is still in the network at time
if where is the time at which the
vehicle reaches the destination
,
Hh snh sn
iIjJ t
md
tmd C
ij t md C md C md C md L
ij t md L md L md L
=
(
)
(
)
[]
{
∈∈ ≤≤
}
,
,,, ,, , ,
,,, ,, ,
ˆ,ˆ,, ˆ,ˆ,
,,
ττ
µµ
τ0
Hh snh sn
iIjJ t
mt
tmt C
ij t mt C mt C mt C mt L
ij t mt L mt L mt L
=
(
)
(
)
[]
{
∈∈ ≤≤
}
,
,,, ,, , ,
,,, ,, ,
ˆ,ˆ,, ˆ,ˆ,
,,
ττ
µµ
τ0
Chiu and Mahmassani Paper No. 03-2627 41
tween the two classes until some vehicles reach the destinations and
thereby introduce some deviation into the distribution. Under these
circumstances, it would be too late for ORPUA to act. As a result, a
secondary performance measure is introduced for the situation that
0 ≤t−τ≤φ
i,j.
The travel distance measure md is used as a secondary performance
measure when 0 ≤t−τ≤φ
i,j. Because md ∝f(mt, v), md is a function
of mt and of the average speed on the traversed portion of the path.
Because mt is identical for CDTA and DDTA class vehicles when
0 ≤t−τ≤φ
i,j, speed becomes a pertinent measure. The higher the speed
vehicles experience, the longer the distance traversed within the same
mt time period; therefore, md is used as a suitable performance mea-
sure when 0 ≤t−τ≤φ
i,j. His dynamically updated on a regular basis
for online operation; it keeps only those distribution estimators that
involve vehicles still present in the network at time t.
Action Set A
The action set Acontains three sets of switching actions [at
CL, at
LC, at
NS],
where at
CL ={aCL
i, j,τ,t, ∀i∈I, j ∈J, 0 ≤τ≤t} contains a set of binary
indicators aCL
i,j,τ,t. When aCL
i,j,τ,t=1, some of the CDTA-guided vehicles
leaving from origin ito destination jat time τwill be switched to
receive DDTA guidance from time tonward. Another action, at
LC =
{aLC
i,j,τ,t, ∀i∈I, j ∈J, 0 ≤τ≤t}, contains a set of binary indicators aLC
i,j,τ,t.
When aLC
i,j,τ,t=1, some of the DDTA-guided vehicles leaving from ori-
gin ito destination jat time τwill be switched to receive CDTA guid-
ance from time tonward. The action set at
NS ={aNS
i,j,τ,t, ∀i∈I, j ∈J, 0 ≤
τ≤t} contains a set of binary indicators aNS
i,j,τ,t. When aNS
i, j,τ,t=1, no
switching actions will be activated for either CDTA or DDTA
vehicles that leave from origin ito destination jat time τ.
Probability Distribution Function Set P(A)
Pis the probability function set defined on action set A. One has P=
{pi,j,τ,t∀i∈I, j ∈J, 0 ≤τ≤t}, where pi,j,τ,t={p(aCL
i, j,τ,t), p(aLC
i,j,τ,t),
p(aNS
i, j,τ,t)} is the probability mass function for the three actions defined
in A. The mass conservation for the three actions is maintained—
namely, p(aCL
i, j,τ,t) +p(aLC
i,j,τ,t) +p(aNS
i, j,τ,t) =1, ∀i∈I, j ∈J, 0 ≤τ≤t.
Mapping Function G
The mapping function set G: S→P(A)are the rule-based updating
functions that map the internal states Sto the probability of the
switch actions. This function set introduces a mechanism for guid-
ance switching decisions based on the actual performance of both
centralized and locally controlled vehicles at each time instant t. G
is defined as follows.
For an origin i, destination j, departure time τ, and current time t,
assume the required travel time between origin iand destination j
for departure time τis t
~i,j,τ. t
~i,j,τis unknown a priori but could be
approximated by taking the average travel time for those who depart
from ito jduring [τ−∆t, τ] and have exited the network by time t.
The decision processes in Gare described as follows: p(aCL
i,j,τ,t) =1,
if one of the following conditions is satisfied; otherwise p(aCL
i,j,τ,t) =0.
Condition 1: is statistically tested by using the
estimators and
µµ
µµ
mt L mt C
mt L mt L mt L mt C mt C mt C
sn sn
,,
,, , ,, ,
ˆ,ˆ,ˆ,ˆ,()
<
(
)
(
)
1
where 0 ≤ξ≤1 is a parameter that represents the threshold as the
percentage of t
~i,j,τ, for checking the secondary MoE (distance).
The preceding rule states that if the average travel time for DDTA
class vehicles traveling from origin ito destination jat time τis sta-
tistically tested to be less than that of CDTA class vehicles that travel
between the same O-D pair and depart at the same time, then the
updating function considers it has sufficient information to assert that
the quality of DDTA routing policies is superior to that of the CDTA.
As a result, the probability of activating action aCL
i,j,τ,tis assigned to be
1 (which means some CDTA vehicles will be supplied with DDTA
guidance). If the difference in the average travel time measure mt
of the two classes of vehicles is not statistically significant, then
the updating function does not have sufficient information and
needs to seek additional verification to make correct decisions.
Additional verification is done by examining the alternative per-
formance measure md, the travel distance estimate, as explained
earlier in this section.
Under the condition that p(aCL
i,j,τ,t) =1, an activation function
π[p(aCL
i,j,τ,t) =1] is defined as follows to indicate which of the pre-
ceding conditions triggered p(aCL
i,j,τ,t) =1. That is,
p(aLC
i,j,τ,t) =1, if one of the following conditions is satisfied, otherwise
p(aLC
i,j,τ,t) =0.
Condition 1: is statistically tested by using the
estimators and
µµ
µµ
mt C mt L
mt L mt L mt L mt C mt C mt C
sn sn
,,
,, , ,, ,
ˆ,ˆ,ˆ,ˆ,()
<
(
)
(
)
3
πτ
pa
CL
ij t,,,
(
)
=
[]
=
10
1
if condition (1) is met
if condition (2) is met
Condition 2: is statistically tested by using the
estimators and and
or and is statistically
tested by using the estimators
and
,,
,,,
µµ
µµ
µµ ξµµ
µ
µ
τ
mt L mt C
mt L mt L mt L mt C mt C mt C
mt L mt C i j md L md C
md L md L md L
sn sn
t
sn
=
(
)
(
)
(
)
<>
(
)
ˆ,ˆ,ˆ,ˆ,;
ˆˆ ˜;
ˆ,ˆ,
ˆ
,, , ,, ,
,,,
,, ,
mdmd C md C md C
sn
,, ,
,ˆ,()
(
)
2
42 Paper No. 03-2627 Transportation Research Record 1857
Similarly, under the condition that p(aLC
i,j,τ,t) =1, an activation func-
tion π[p(aLC
i,j,τ,t) =1] is defined as follows to indicate which of the
preceding conditions triggered p(aLC
i,j,τ,t). That is,
The representation of function Gin terms of a decision tree is pre-
sented in Figure 2. The preceding statistical testing is conducted based
on the sample observations of the travel time mt or the travel distance
md for both CDTA- and DDTA-guided vehicles in the network at
each time instant. Namely, at each time instant t, the relationship of
µ•,L
i,j,τ,tand µ•,C
i,j,τ,tagainst the following hypotheses is tested:
Because the sample sizes of the two distributions nLand nCmay
not be large (namely, >30 depending on the observation resolu-
tion; normally 1 to 5 min), the tdistribution is used for testing the
statistic tt:
tt
snn
L
ij t C
ij t
p
L
ij t C
ij t
=−
(
)
+
ˆ
ˆ
()
•,
,,, •,
,,,
•,,,, •,,,,
µµ
ττ
ττ
11 5
HL
ij t C
ij t
1:•,
,,, •,
,,,
µµ
ττ
≠
HL
ij t C
ij t
0:•,
,,, •,
,,,
µµ
ττ
=
pa pa pa
NS
ij t CL
ij t LC
ij t
,,,
,,, ,,,
τ
ττ
(
)
=
(
)
+
(
)
=
10
0
if
otherwise
πτ
pa
LC
ij t,,,
(
)
=
[]
=
10
1
if condition (3) is met
if condition (4) is met
Condition 2: is statistically tested by using the
estimators and
and or and is statistically
tested by using the estimators
and
µµ
µµ
µµ ξµµ
µ
τ
mt C mt L
mt L mt L mt L mt C mt C mt C
mt C mt L i j md C md L
md L md L md L
sn sn
t
sn
,,
,, , ,, ,
,,,, ,,
,, ,
ˆ,ˆ,ˆ,ˆ,;
ˆˆ ˜;
ˆ,ˆ,
ˆ
=
(
)
(
)
(
)
<>
(
)
µµmd C md C md C
sn
,, ,
,ˆ,()
(
)
4
If C and L are
statistically
different in mt
if ?
If C and L are
statistically
different in md
YES
NO
if ?
YES
NO
tji Lmt
tji Cmt hh ,,, ,
,,, ,
τ
τ
≤
YES
NO
1)( ,,, =
tji
LC
ap
τ
1)( ,,, =
tji
CL
ap
τ
YES
NO
1)( ,,, =
tji
CL
ap
τ
1)( ,,, =
tji
LC
ap
τ
1)( ,,, =
tji
NS
ap
τ
tji Lmd
tji Cmd hh ,,, ,
,,, ,
τ τ
≤
FIGURE 2 Decision tree representation of Function G.
where
The criteria for determining which vehicle is eligible for a switch
are discussed in the next section.
Switching Criteria C
While the function Gupdates the action probabilities, the switching
decisions for each CDTA- and DDTA-guided vehicle are made based
on the criteria set Cand the associated rule-based guidance switching
schemes. For example, if p(aLC
i,j,τ,t) =1, the DDTA class vehicles under-
perform the CDTA class vehicles, and some vehicles in the popula-
tion that satisfy certain criteria will be designated to receive CDTA
guidance for the remaining portion of the trip. The general idea is to
select those that are in the lower tier of the population. The formal
definitions of the control switching criteria Care provided hereafter:
First, define the travel time and travel distance for vehicle vat time
tas CT t
vand CDt
v, respectively. When a DDTA class vehicle vwith
origin i, destination j, and departure time τreaches the downstream
agent node nof the current link, the following rules are applied:
1. If p(aNS
i,j,τ,t) =1, no switch will be performed.
2. If p(aLC
i,j,τ,t) =1, check π[p(aLC
i,j,τ,t) =1] and the prevailing
travel time and distance of vehicle v. If one of the following con-
ditions is satisfied, then vehicle vin the DDTA class will be pro-
vided with a new path selected from the CDTA routing policies.
Vehicle v’s current routing class is changed from DDTA to CDTA.
Condition 1 indicates that the DDTA class vehicles are detected
to underperform CDTA class vehicles by using the MoE mt. If the
current travel time of vehicle vis greater than µmt,L +ρ
mt,L, then this
vehicle is considered a low performer, and the quality of the guid-
ance it receives will be improved by assigning it to the counterpart
class. Along the same line, Condition 2 indicates that the DDTA
class vehicles underperform the other class. If the current travel dis-
tance for vehicle vis less than µmd,L +ρ
md,L, this vehicle is considered
a low performer and will be reassigned to the counterpart class.
When a CDTA class vehicle vwith origin i, destination j, and
departure time τreaches the downstream node nof the current link,
the following rules are applied:
3. If p(aNS
i,j,τ,t) =1, no switch will be performed.
4. If p(aCL
i,j,τ,t) =1, check π[p(aCL
i,j,τ,t) =1] and the prevailing travel
time and distance of vehicle v. If one of the following conditions is
satisfied, then vehicle vwill be provided with a new path selected
from the DDTA routing policies. Vehicle v’s current routing class
is changed from CDTA to DDTA.
Condition and210:,,, ,,
πµρ
τ
pa CD
CL
ij t v
tmd C md C
(
)
=
[]
=≤+
Condition and110:,,, ,,
πµρ
τ
pa CT
CL
ij t vtmt C mt C
(
)
=
[]
=≥+
Condition and210:,,, ,,
πµρ
τ
pa CD
CL
ij t v
tmd L md L
(
)
=
[]
=≤+
Condition and110:,,, ,,
πµρ
τ
pa CT
CL
ij t vtmt L mt L
(
)
=
[]
=≥+
D.F.: ˆ•,,,, •,,,,
sn n
pL
ij t C
ij t
=+−
ττ
2
ˆˆˆ
•,,,, •,,,, •,,,, •,,, ,
•,,,, •,,,,
snsns
nn
pL
ij t L
ij t C
ij t C
ij t
L
ij t C
ij t
=−
(
)
(
)
+−
(
)
(
)
+−
ττττ
ττ
11
2
22
Chiu and Mahmassani Paper No. 03-2627 43
The criteria for switching vehicle vfrom the CDTA class to the
DDTA class are similar to those defined for the reverse switch.
Furthermore, ρmt,C, ρmd,C and ρmt,L, ρmd,L are the thresholds that trig-
ger the guidance class switch for vehicle vunder Condition 1 or 2. The
values of ρmt,C, ρmd,C, ρmt,L, ρmd,L are set to minimize the gap of the
respective means of the two distributions of the travel time and
the travel distance measures for CDTA- and DDTA-guided vehi-
cles. The formulae for computing ρmt,C, ρmt,L, ρmd,C, and ρmd,L are
described as follows.
The switching decisions under the situation in which µ•,Cis statisti-
cally tested to be greater than µ•,Lsolve following problem.
Find a value ρ•such that when switching all CDTA vehicles that
satisfy CT t
v≥µ
ˆ•,C+ρ
•to receive the DDTA guidance, one has the
new distribution estimators h•,C
i,j,τ,t(µ
ˆ
~•,C, s
ˆ
~•,C, n
~•,C) and h•,L
i,j,τ,t(µ
ˆ
~•,L, s
ˆ
~•,L,
n
~•,L), with µ
ˆ
~•,C=µ
ˆ
~•,L.
To solve this problem, first compute
The next step is to make µ
ˆ
~•=µ
ˆ
~•,Cby removing the largest elements
in h•,C
i,j,τ,t(µ
ˆ
~•,C, s
ˆ
~•,C, n
~•,C) that satisfy Equation 6:
where max[v] is a nonincreasing sequence of all the CTt
vin h•,C
i,j,τ,t(µ
ˆ
~•,C,
s
ˆ
~•,C, n
~•,C) starting from the largest CTt
vthat satisfies Equation 6. Once
max[v] is solved, one obtains ρ•=arg min {CTt
vCTt
v∈max[v]}.
The switching decisions under the situation in which µ•,Cis sta-
tistically tested to be less than µ•,Lare obtained by solving the fol-
lowing problem, which is very similar to that described previously.
The problem and solutions are stated as follows:
Find a value ρ•such that, when switching all DDTA vehicles that
satisfy CTt
v≥µ
ˆ•,L+ρ
•to receive the CDTA guidance, one has the
new distribution estimators h•,C
i,j,τ,t(µ
ˆ
~•,C, s
ˆ
~•,C, n
~•,C) and h•,L
i,j,τ,t(µ
ˆ
~•,L, s
ˆ
~•,L,
n
~•,L), with µ
ˆ
~•,C=µ
ˆ
~•,L.
To solve this problem, first compute
The next step is to make µ
ˆ
~•=µ
ˆ
~•,Lby removing the largest elements
in h•,L
i,j,τ,t(µ
ˆ
~•,L, s
ˆ
~•,L, n•,L) that satisfy Equation 7:
ˆ
˜ˆ˜
˜max
ˆ
˜˜max ˆ˜
max ˆ
˜ˆ
˜˜ˆ
˜˜
•
•,•,max
•,
•,•,•,•,max
max ••,•,••,
µµ
µµ
µµ µ
=−
−
[]
⇒−
[]
(
)
=−
⇒−
[]
=−
⇒
∈
[]
∈
[]
∈
[]
∑
∑
∑
LL v
t
vv
L
LL LL v
t
vv
vt
vv LL L
nCT
nv
nvn CT
CT v n n
CT
vvt
vv LL
n−
(
)
−−
(
)
=
∈
[]
∑ˆ
˜ˆ
˜ˆ
˜˜()
•
max •,••,
µµµ07
ˆ
˜ˆ
˜ˆ
˜ˆ˜ˆ˜
˜˜
••,•,•,•,•,•,
•,•,
µµ µ µµ
=== +
+
CLCC L L
CL
nn
nn
ˆ
˜ˆ˜
˜max
ˆ
˜˜max ˆ˜
max ˆ
˜ˆ
˜˜ˆ
˜˜
•
•,•,max
•,
••,•,•,max
max ••,•,••,
µµ
µµ
µµ µ
=−
−
[]
⇒−
[]
(
)
=−
⇒−
[]
=−
⇒
∈
[]
∈
[]
∈
[]
∑
∑
∑
CC v
t
vv
C
CCCv
t
vv
vt
vv CC C
v
nCT
nv
nvn CT
CT v n n
CT
tt
vv CC
n−
(
)
−−
(
)
=
∈
[]
∑ˆ
˜ˆ
˜ˆ
˜˜()
•
max •,••,
µµµ06
ˆ
˜ˆ
˜ˆ
˜ˆ˜ˆ˜
˜˜
••,•,•,•,•,•,
•,•,
µµ µ µµ
=== +
+
CLCC L L
CL
nn
nn
where max[v] is a nonincreasing sequence of all the CTt
vin h•,L
i,j,τ,t(µ
ˆ
~•,L,
s
ˆ
~•,L, ñ•,L) starting from the largest CTt
vthat satisfies Equation 7. Once
max[v] is solved, one obtains ρ•=arg min{CTt
vCTt
v∈max[v]}.
The online switching decision process is presented in Figure 3.
ORPUA UPDATING PROCEDURE
ORPUA is updated based on prespecified updating intervals. At each
updating instant, the following steps are executed:
1. For each origin i, destination j, and departure time τ, compute
the distribution estimator hmt,C
i,j,τ,t(µ
ˆmt,C, s
ˆmt,C, nmt,C), hmt,L
i,j,τ,t(µ
ˆmt,L, s
ˆmt,L,
nmt,L), hmd,C
i,j,τ,t(µ
ˆmd,C, s
ˆmd,C, nmd,C), and hmd,L
i,j,τ,t(µ
ˆmd,L, s
ˆmd,L, nmd,L) over all the
guided vehicles present in the network at time t.
2. For each i, j, τ, test the null hypotheses: µL=µCusing both Ht
mt
and Ht
md.
3. Update p(aCL
i,j,τ,t), π(aCL
i,j,τ,t) p(aLC
i,j,τ,t), π(aLC
i,j,τ,t) and p(aNS
i,j,τ,t),
π(aNS
i,j,τ,t) based on the updating function G.
4. If p(aCL
i,j,τ,t) =1, then for each vehicle vthat satisfies Equation 6,
switch its guidance class from CDTA to DDTA. If p(aLC
i,j,τ,t) =1, then
for each vehicle vthat satisfies Equation 7, switch its guidance class
from DDTA to CDTA. If p(aNS
i,j,τ,t) =1, then there is no switching.
5. Repeat Steps 1 to 4 until the end of the operating horizon.
PERFORMANCE EVALUATION OF ORPUA
This section presents results of simulation experiments to highlight the
performance of ORPUA under various IRP and supply–demand sce-
narios. The objective of these experiments is to investigate, under var-
44 Paper No. 03-2627 Transportation Research Record 1857
ious levels of a priori IRP, how ORPUA updates vehicles’ routing poli-
cies in response to the unfolding traffic dynamics and disturbances.
Several factors are considered in the experiment design, including dif-
ferent spatial patterns and loading and various demand prediction
errors in demand matrix in the CDTA as well as incident occurrence
situations. These experiments are conducted on the Fort Worth net-
work with 445 links and 179 nodes, the commonly used midsized net-
work in much of the literature (4, 11–13, 15). The experimental design
and experimental results are described in the following sections.
Impacts of Various Levels of CDTA Demand
Prediction Errors
In this experiment, the errors for the CDTA predicted demand are
assumed to be the only error source in generating the CDTA routing
policies. The actual demand pattern, represented as an O-D matrix, is
assumed to be either pattern P-1 or pattern P-2, presented in Figure 4,
and the number of vehicles generated is assumed to be about 17,200,
representing heavy traffic conditions. The demand prediction errors
are characterized in terms of the structural pattern of the demand as
well as the number of generated vehicles.
The first experiment contains the following sets of scenarios:
1. Scenario 1: Specify actual demand patterns and loading level
to be 17,000 vehicles.
2. Scenario 2: Solve the a priori HDTA experiments based on var-
ious predicted demand patterns and levels. Compute and store opti-
mal routing policies as the “a priori routing policies” for predicted
demand patterns and levels.
3. For each predicted demand, conduct a run with IRP Λ=(1, 0)
(i.e., all the vehicles are CDTA-guided). This run is specified as
(100C).
4. For each predicted demand, conduct a run with IRP Λ=(0, 1).
This run is denoted as (100LL).
5. For each predicted demand, conduct several runs with IRPs
ranging from Λ=(1, 0) to Λ=(0, 1) without ORPUA. These runs are
specified as (WoSw), in which vehicles are first randomly assigned
with a class (CDTA or DDTA) based on IRP, and then the initial
paths are assigned based on the associated class. Once a vehicle is
assigned to a path, it stays on that path for the entire trip; namely,
there will be no switching.
6. For each predicted demand, conduct several runs with IRPs
ranging from Λ=(1, 0) to Λ=(0, 1) in increments of 0.1 with
ORPUA. These runs are specified as (WSw), in which vehicles
receive initial paths in the same manner as WoSw. A vehicle’s class
and path will be updated by ORPUA along the trip.
7. With the hindsight of the actual demand, conduct a run that
solves the system optimal for this demand. This run is denoted as
(100CSO) and will serve as the benchmark for comparison.
The experiment results are presented in Figure 5, where the pre-
dicted demand is assumed to be either 12,000 vehicles (30% lower
than actual) or 19,000 vehicles (11.6% higher than actual). Figures
5aand 5bindicate that the average travel time for the benchmark
scenario (100CSO) is 15.02 min. When CDTA underpredicts actual
demand by 30.0%, if 100% of the total vehicles receive a priori rout-
ing policies (denoted as 100C), the average travel time increases to
22.88 min, which is equivalent to a 52.3% increase above the bench-
mark. If 100% of the vehicles receive local guidance (denoted as
100LL), the average travel time is 19.60 min, which is equivalent to
a 30.5% increase above the benchmark. Comparing the results of
Vehicle j reaches the
downstream agent node
Does the affiliated
class underperform the
counterpart class?
Is vehicle
j’s current travel
time greater than the
Switch Threshold?
Vehicle j is switched to
counterpart class
No action
performed
No action
performed
YES
YES
NO
NO
FIGURE 3 Online guidance switching decision for vehicle j.
these two scenarios indicates that 100C underperforms 100LL be-
cause of the significant underprediction of the actual demand. The
generally convex shape of the performance curve of the WoSw runs
implies that system performance could improve by splitting the
vehicles into two classes. Thus, there is an advantage to hybridizing
the central and local DTA routing policies. When the CDTA over-
predicts the actual demand, the inflated demand leads to better solu-
tion quality than when the actual demand is underpredicted. As also
indicated in Figures 5aand 5b, the average travel time for the 100C
run is 17.2 min, which outperforms the 19.02 min of 100LL. The
decreasing trend is rather clear compared with the 30.0% overpredic-
tion scenario. The apparent optimal IRP, at around Λ=(0.8, 0.2) to Λ
=(0.9, 0.1), suggests that assigning more vehicles to receive CDTA
guidance is more desirable. This finding suggests that, by simply
inflating the predicted demand in generating CDTA policies, the
system could maintain a reasonable performance level.
The performance curve of WSw runs not only exhibits a pattern
similar to that of the WoSw runs in both scenarios but also highlights
the effectiveness of ORPUA. For any IRP setting, the average travel
times of the WSw runs are consistently lower than those of the WoSw
scenarios. The results indicate that the online guidance updating
mechanism of ORPUA further improves the system performance for
all possible IRP. In addition, the advantages of ORPUA become
increasingly evident as IRP approaches Λ=(1, 0), under which more
vehicles are CDTA guided and the system is dominated by the poor-
quality CDTA guidance. Because the CDTA guidance is generated
Chiu and Mahmassani Paper No. 03-2627 45
by an inaccurate demand prediction, the system performance
degrades significantly with an increasing percentage of centrally
guided vehicles (as indicated in WoSw). However, the WSw runs
indicate that, with ORPUA, the system performance is less sensitive
to the quality of the CDTA routing policies and exhibits a higher
degree of system robustness.
The assumption that the demand prediction errors affect only the
overall loading level is further relaxed in the next set of experiments.
Figures 5cand 5dpresent the results of scenarios that have routing
schemes similar to those corresponding to Figures 5a5b. The only
difference is that the predicted demand follows a different spatial
pattern from the actual one so as to reflect more realistic demand
prediction errors.
With this new actual demand input, the average travel time for the
benchmark run is 10.0 min. Figures 5cand 5dconsistently present the
increasing trend of the average travel time with respect to increasing
the CDTA split percentage under different demand prediction error
levels. Also indicated in Figures 5c and 5d, when the CDTA under-
predicts the actual demand by 30%, the average travel time for the
100LL run is 11.1 min, which is equivalent to an 11% increase above
the benchmark. The average travel time is 15.4 min for the 100C sce-
nario, equivalent to a 54% increase above the benchmark. Both
WoSw and WSw runs show a steady increasing performance curve
with respect to IRP. Similar to the previous experiment, WSw out-
performs WoSw over all IRPs and shows less degradation with
respect to increasing split in terms of CDTA guidance.
(a)
(b)
FIGURE 4 Demand pattern (a) P-1 and (b) P-2. (Dim 1 and 2 O-D zones.)
46 Paper No. 03-2627 Transportation Research Record 1857
12
14
16
18
20
22
24
(0,1.0)
(0.1,0.9)
(0.2,0.8)
(0.3,0.7)
(0.4,0.6)
(0.5,0.5)
(0.6,0.4)
(0.7,0.3)
(0.8,0.2)
(0.9,0.1)
(1.0,0.0)
% of Initial Central-Guided Vehicles
Avg. Travel Time (min)
100CSO
100C
WoSw
WSw
100LL
12
14
16
18
20
22
24
(0,1.0)
(0.1,0.9)
(0.2,0.8)
(0.3,0.7)
(0.4,0.6)
(0.5,0.5)
(0.6,0.4)
(0.7,0.3)
(0.8,0.2)
(0.9,0.1)
(1.0,0.0)
Initial Routing Profile
Avg. Travel Time (min)
100CSO
100C
WoSw
WSw
100LL
Demand Prediction Error -30.0% Demand Prediction Error +11.6%
(a) (b)
10
11
12
13
14
15
16
(0,1.0.0)
(0.1,0.9)
(0.2,0.8)
(0.3,0.7)
(0.4,0.6)
(0.5,0.5)
(0.6,0.4)
(0.7,0.3)
(0.8,0.2)
(0.9,0.1)
(1.0,0.0)
Initial Routing Profile
Avg. Travel Time (min)
100CSO
100C
WoSw
WSw
100LL
10
11
12
13
14
15
16
(0,1.0)
(0.1,0.9)
(0.2,0.8)
(0.3,0.7)
(0.4,0.6)
(0.5,0.5)
(0.6,0.4)
(0.7,0.3)
(0.8,0.2)
(0.9,0.1)
(1.0,0.0)
Initial Routing Profile
Avg. Travel Time (min)
100CSO
100C
WoSw
WSw
100LL
Demand Prediction Error -30.0% Demand Prediction Error +11.6%
(c) (d)
10
11
12
13
14
15
16
(0,1.0)
(0.1,0.9)
(0.2,0.8)
(0.3,0.7)
(0.4,0.6)
(0.5,0.5)
(0.6,0.4)
(0.7,0.3)
(0.8,0.2)
(0.9,0.1)
(1.0,0.0)
Initial Routing Profile
Avg. Travel Time (min)
100CSO
100C
WoSw
WSw
100LL
10
11
12
13
14
15
16
(0,1.0)
(0.1,0.9)
(0.2,0.8)
(0.3,0.7)
(0.4,0.6)
(0.5,0.5)
(0.6,0.4)
(0.7,0.3)
(0.8,0.2)
(0.9,0.1)
(1.0,0.0)
% of Initial Central-Guided Vehicles
Avg. Travel Time (min)
100CSO
100C
WoSw
WSw
100LL
Demand Prediction Error -30.0% Demand Prediction Error +11.6%
(e) (f)
FIGURE 5 ORPUA performance with respect to IRP under various erroneously predicted demand scenarios. (a, b) Actual demand pattern,
P-1; predicted demand pattern, P-2. (c, d) Actual demand pattern, P-1; predicted demand pattern, P-2. (e, f) Actual demand pattern, P-1;
predicted demand pattern, P-2, incident situation.
When CDTA overpredicts actual demand by 11.6%, the average
travel time for the 100C scenario is 13.78 min, a 38% increase from
the benchmark. In this experiment, the system performance at a high
CDTA split is apparently better than that indicated in the 30.0%
underprediction case, reiterating that overpredicting demand yields
better solutions than underpredicting. However, the improvement is
rather limited and it still underperforms the performance under the
100LL scenario. It is noted that the increasing trend in the perfor-
mance curve is rather distinct from that in Figures 5aand 5b. Fig-
ures 5aand 5bsuggest that CDTA outperforms DDTA with inflated
predicted demand that has the same spatial pattern as the actual one.
Figures 5cand 5dsuggest otherwise, indicating that this strategy has
its limited advantage only under circumstances in which the pre-
dicted spatial pattern for the demand matrix is very close to the
actual one. Furthermore, they also suggest that the system perfor-
mance is more sensitive to spatial pattern than the overall loading
magnitude. Focusing on estimating the spatial pattern could be an
effective strategy for O-D estimation.
Performance of ORPUA Under
Incident Situations
The set of experiments presented in this section considers a scenario
in which an incident occurs on the freeway from minute 5 to 40 and
reduces 95% of the freeway capacity. The experimental results pre-
sented in Figures 5eand 5fare generated with the same parameters as
those in Figures 5cand 5d, that is, the predicted demand is assumed
to have a different spatial pattern from the actual one and subject to
varying levels of error in the overall loading level. The same incident
is applied to all the experiments.
The results presented in Figures 5eand 5fare consistent with those
in previous experiments in the sense that the smaller the prediction
error, the better the CDTA performance. Overpredicting demand in
CDTA is more effective than underpredicting, but generally CDTA
still underperforms DDTA. Comparing the WoSw with the WSw sce-
narios, both show similar performance at low IRP, but the advantage
of ORPUA becomes clear for a high CDTA percentage.
CONCLUDING REMARKS
This paper introduces an ORPUA approach as a mechanism for oper-
ating the hybrid DTA online. Virtually all the experiments performed
confirm that the quality of the CDTA assignment solutions is rather
sensitive to various sources of error, such as demand prediction errors,
or incidents that are difficult to model in the predictive framework.
The DDTA performance is also subject to several unforeseeable dis-
turbances such as failure of local agents or communication difficul-
ties between agents in spite of the advantage that DDTA does not
require predictive inputs. Under such uncertain online performance
characteristics of both DTA systems, maintaining robust and satis-
factory system performance throughout the operating horizon could
be challenging; however, ORPUA provides a practical mechanism for
achieving this objective.
The ORPUA presented in this paper aims to provide an effective
updating mechanism for online operation. It takes the a priori opti-
mal IRP and updates the guidance supplied to vehicles in a real-time
fashion according to the unfolding network conditions and relative
performance of the two classes of vehicles. It does not anticipate the
future network conditions; instead, it reacts to them and optimizes
Chiu and Mahmassani Paper No. 03-2627 47
the overall system performance by improving the performance of
the inferior class of vehicles. The experimental scenarios considered
here suggest that implementations with ORPUA outperform those
without ORPUA with the same IRP.
In conclusion, applying the HDTA strategy leads to better system
performance than the pure CDTA or DDTA strategy under imperfect
predicted input circumstances. ORPUA was designed to be a uni-
fied approach that continually maintains a reasonable quality of the
assignment solutions regardless of the events that may cause sudden
performance degradation of either CDTA or DDTA. The optimal
operating point of the system is subject to the quality of the CDTA
and the DDTA routing policies. ORPUA improves the performance
and robustness of the system under real-time operation.
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