01707402.pdf
TRANSCRIPT
Path-based System Optimal Dynamic Traffic Assignment Models:Formulations and Solution Methods
Wei Shen, Yu Nie, and H. Michael Zhang∗
Abstract— The evaluation of path marginal cost, i.e., thegradient of the objective function with respect to path flows,lies in the kernel of solution algorithms for path-based SO-DTA models. We identify a common critical deficiency inexisting path marginal cost evaluation methods, develop anew path marginal cost evaluation method by tracing pathflow perturbation propagations and design the correspondingsolution algorithm for path-based SO-DTA models in networksin mono-centric cities. Our numerical experiments indicatethat this algorithm can generate numerical solutions close toanalytical solutions while the solution scheme based on theexisting path marginal cost evaluation method cannot.
I. INTRODUCTION
The system optimal dynamic traffic assignment (SO-DTA)
problem determines the time-dependent traffic flow pattern
that minimizes the total system costs. The problem is of great
importance to evaluating the efficiency of real-time traffic
management strategies, such as dynamic congestion pricing,
incident management and emergency evacuation plans.
The SO-DTA problem is traditionally formulated and
solved as a mathematical program which minimizes the
sum of travel costs over a feasible set defined mainly by
link-based traffic propagation and demand conservation con-
straints. (e.g., Merchant & Nemhauser 1978 [1], Carey 1987
[2], Wie 1998 [3], Ziliaskopoulos [4], etc.) . However, to
represent realistic traffic propagation rules, non-convexity in
the feasible set is often inevitable, because explicitly ensuring
the first-in-first-out (FIFO) rule for multiple commodities at
the link level and analytically describing traffic dynamics
both require non-linear equality constraints, making the
model difficult to solve.
Path-based SO-DTA models, which encapsulate traffic
propagation into a path cost mapping and hence may bypass
the non-convexity issue associated with link-based SO-DTA
models. However, research along this line is rather limited.
One major reason is that solving path-based SO-DTA models
usually requires gradients of the total system cost, i.e., the
change in the total system cost with respect to the unit change
in the path flow, which we call path marginal cost (PMC)
hereafter. Since the path cost mapping usually does not have
a closed form, the PMC evaluation is not straightforward.
This paper is motivated to make a thorough study on
path-based SO-DTA models, including model formulations
and solution procedures. In particular, we emphasize the
most critical part in the solution procedure, i.e., the PMC
evaluation. We try to clarify existing misconceptions about
the PMC evaluation, identify the associated difficulties, and
∗corresponding author, Tel: 530-754-9203, Email: [email protected]
propose an improved PMC evaluation scheme for networks
with a special type of topology.
The remainder of this paper is organized as follows:
Section II introduces the formulation of the path-based SO-
DTA model. The optimal conditions that resemble Wardrop’s
second principle [5] are provided and the importance of
the PMC evaluation in solving path-based SO-DTA mod-
els is emphasized. The PMC evaluation is then discussed
thoroughly in Section III. Section IV presents the solution
procedure for the path-based SO-DTA models in networks
in mono-centric cities. Computational results and discussions
are reported in Section V, and Section VI presents conclu-
sions and future research directions.
II. THE PATH-BASED SO-DTA MODEL
We consider a general transportation network with multi-
ple origin-destination (OD) flows. The whole study horizon
Td is discretized into N intervals of length δ. We assume that
Td is long enough for all the traffic flows to clear the network.
The goal of the model is to find the optimal departure time
choice and route choice path flow pattern such that the total
system travel cost, including travel time cost and schedule
delay cost, is minimized.
The following notations are used throughout this paper:
a) Set notationsRS set of OD pairs
P rs set of routes connecting OD pair rsTd the whole departure time horizon, Td =
{1, 2, ..., N}b) Indicesrs OD pair, rs ∈ RSp route between OD pair rs, p ∈ P rs
t index for departure time, t ∈ Td
c) Variables to be determinedfrs
pt flow entering route p ∈ P rs at time tf path flow vector, f = {frs
pt } with dimen-
sion n = N∑
rs∈RS |P rs|d) Functions of path flow fcrspt(f) actual path travel time for flow entering
path p ∈ P rs at time t, which is a unique
mapping with respect to fφrs
pt(f) generalized cost incurred by travelers en-
tering path p ∈ P rs at time t, which is a
unique mapping with respect to fqrst demand between OD pair rs at time t
e) Parameters given
Proceedings of the IEEE ITSC 20062006 IEEE Intelligent Transportation Systems ConferenceToronto, Canada, September 17-20, 2006
WA2.4
1-4244-0094-5/06/$20.00 ©2006 IEEE 1298
Qrs total demand for OD pair rs during the
study horizon
cs(t) schedule delay cost for travelers arriving
at destination at time tt̃s desired arrival time for travelers going to
destination s, t̃s ∈ Td
Δs arrival time flexibility for travelers going
to destination s, Δs ≥ 0α cost of one unit of travel time for travelers,
α > 0βs unit cost of schedule delay caused by the
early arrival of travelers at destination s,
βs > 0γs unit cost of schedule delay caused by the
late arrival of travelers at destination s,
γs > 0.
Note that destination-based parameters (Δs, βs, γs) are
used to reflect the difference of value-of-time among trav-
elers associated with different destinations. According to
empirical data, γ > α > β, and we have the following
relationship:
φrspt(f) = αcrs
pt(f) + cs[t + crspt(f)] (1)
where cs(t) is piecewise linear and can be represented by:
cs(t) =
⎧⎨⎩
βs[(t̃s − Δs) − t] if t < t̃s − Δs
0 if t̃s − Δs ≤ t ≤ t̃s + Δs
γs[t − (t̃s + Δs)] if t > t̃s + Δs
(2)Using the defined path variables and functions, the SO-
DTA problem optimizing both departure time and route
choices can be formulated as the following minimization
problem :
minf∈Ω
TC(f) =∑t∈Td
∑rs∈RS
∑p∈P rs
frspt · φrs
pt(f) (3)
subject to ∑p∈P rs
frspt = qrs
t ,∀rs ∈ RS, t ∈ Td (4)
∑t∈Td
qrst = Qrs(given),∀rs ∈ RS (5)
frspt ≥ 0,∀rs ∈ RS, k ∈ Krs, t ∈ Td (6)
According to Karush-Kuhn-Tucker (KKT) conditions, the
first-order necessary conditions of optimality are constraints
(4) - (6) plus frspt
∂L(f ,u)∂frs
pt= 0,∀r, s, p, t and
∂L(f ,u)∂frs
pt≥ 0.
Namely,
frspt
(∂TC(f)∂frs
pt
− μrs
)= 0,∀rs ∈ RS, p ∈ P rs, t ∈ Td
(7)
∂TC(f)∂frs
pt
− μrs ≥ 0,∀rs ∈ RS, p ∈ P rs, t ∈ Td (8)∑t∈Td
∑p∈P rs
frspt − Qrs = 0,∀rs ∈ RS (9)
frspt ≥ 0,∀rs ∈ RS, k ∈ Krs, t ∈ Td (10)
To facilitate further discussion, we provide the definition
of PMC below explicitly.
Definition 1 (Path marginal cost PMCrspt (f)): Given a
specific path flow pattern f = {frspt ,∀p, t, rs}, the path
marginal cost for path p at time t represents the increase in
the total system cost when the path inflow on p is increased
by one unit. Namely,
PMCrspt (f) =
∂TC(f)∂frs
pt
=
∑τ∈Td
∑rs∈RS
∑p∈RS
frspτ φrs
pτ (f)
∂frspt
(11)
It is obvious that (7) and (8) convey the Wardrop second
principle in terms of time-dependent path marginal cost, i.e.,
at dynamic system optimum, the time-dependent marginal
cost on all the paths actually used are equal and less than the
marginal cost on any unused path. Consequently, if we can
efficiently evaluate path marginal cost PMCrspt ,∀r, s, p, t,
algorithms for solving equilibrium problems may be applied
to solve this SO-DTA problem, at least approximately1.
III. PATH MARGINAL COST EVALUATION
In the static case, the PMC is the sum of the link marginal
cost(LMC). In the dynamic case, the PMC evaluation is
much more complicated since path flows are not assigned
to links on the path simultaneously. However, for traffic
dynamics models which do not consider link interactions,
such as the point queue model, the exit flow function,
the link performance function and so on, a decomposition
scheme from path marginal cost to link marginal cost is
still possible. To illustrate this, we further introduce the
following additional notations:
Link variables and functionsuat flow entering link a at time tua link inflow vector, ua = {uat,∀t ∈ Td}cat link travel time for flow entering link a at time t
ust flow arriving at destination s at time t
For dynamic traffic models not considering link interac-
tions, cat is uniquely determined by the inflow pattern ua
on link a. Hence, we can treat cat as a function of ua,
i.e., cat = cat(ua). Then the total travel cost TC(f) can
be written as follows:
TC(f) = α∑t∈Td
∑a∈A
uatcat(ua) +∑t∈Td
∑s∈S
ustc
s(t) (12)
Substituting (12) into (11) and using the chain rule, the
following relationship can be easily derived:
PMCrspt (f) = α
∑a∈A
∑k∈Td
LMCak(ua)Indakptrs(f)
+∑s∈S
∑k∈Td
cs(k)Indskptrs(f) (13)
1Note that the solution based on this method may only be approximatesince the KKT conditions may not be totally sufficient due to the non-closedform of φrs
pt(f) in the objective function.
1299
where LMCak(ua) and Indakptrs(f) are link marginal
costs and path flow perturbation propagation index, whose
definitions are provided below.
Definition 2 (Link marginal cost): Given a specific link
inflow pattern ua = {uat, t ∈ Td} for link a, the link
marginal cost for link a at time t represents the change in
the total link cost when the link inflow at time t is increased
by one unit. Namely
LMCat(ua) :=∂
∑τ∈Td
uaτ caτ (ua)
∂uat,∀a ∈ A, t ∈ Td (14)
Definition 3 (Path flow perturbation propagation index):Given a specific path flow pattern f and the corresponding
link inflow pattern u. the path flow perturbation propagation
index Indakptrs(f) represents the change in the inflow of link
a at time k when the path flow at time t is increased by
one unit. Namely,
Indakptrs(f) =
∂uak
∂frspt
(15)
According to (13), PMCs for traffic models not consider-
ing link interactions can still be regarded as additive as long
as path flow perturbation propagations are correctly captured.
Ghali & Smith (1995) [6] provides a sound analytical
formulation for LMCs based on the link cumulative curves
for the point queue model. It is shown that the link marginal
cost is equal to the time difference between the time when
the vehicle enters the link and the earliest time after that
when the queue on the link vanishes. We utilize this LMC
evaluation method in our discussion.
The problem of evaluating the path flow perturbation
propagation seems to be neglected in most existing path-
based SO-DTA studies. Most researchers (e.g., Ghali &
Smith 1995 [6], Peeta 1994 [7], etc. ) simply assume that
the path flow perturbation travels along the path at the same
speed as that of the additional flow unit. In other words, if
we denote za(t) as the entering time at link a for a vehicle
departing from the origin at time t and following path p,
then
Indakptrs =
{1 if a ∈ p and k = za(t)0 otherwise
(16)
Based on this assumption on path flow perturbation prop-
agations, PMCs in the dynamic case are additive according
to link traversal times (we assume that a path p consists of
a series of links a1, a2, . . . , am). Namely,
PMCrspt (f) =
m∑i=1
LMCai,zai(t)(uai) + cs(zs(t)) (17)
For narrative convenience, this PMC evaluation method is
referred to as the link traversal time (LTT) method hereafter.
Unfortunately, as we shall see in the following discussion,
this assumption on path flow perturbation propagation is
actually NOT true due to link bottleneck restrictions. We
q c1 c21 2 3
1
ft2
ft
link 1 link 2
[0, T]
> c1>c
2q
Fig. 1. An illustration network with two sequential links
demonstrate this claim by showing in a simple example
network involving two sequential links how the path flow
perturbation propagates along the path.
The illustration network in Fig. 1 contains two links 1
and 2 and each link has a bottleneck at its downstream end.
The capacities of the bottlenecks at link 1 and link 2 are c1
and c2, respectively, and free flow travel times of link 1 and
link 2 are t1f and t2f . During time [0, T ], vehicles enter the
network from link 1 at a constant flow rate q. We assume
that q > c1 > c2. Obviously, queues will develop at both of
the bottlenecks. The cumulative curves for these two links
are illustrated in Fig. 2, where t1e and t2e represent the times
that the queues on link 1 and link 2 vanish, and N is the
total number of vehicles released in [0, T ].According to the cumulative curves in Fig. 2, the vehicle
entering link 1 at time t1 ∈ [0, Td] will enter link 2 at time
t2. Suppose we want to evaluate PMCt1(f). To simplified
the discussion, no schedule delay cost is considered.
Based on the definition, PMCt1(f) can be evaluated by
constructing the new cumulative curves for link 1 and 2 with
an additional flow unit entering link 1 at time t1 (Fig. 2).
Fig. 2 shows that the additional costs incurred in link 1 and
link 2 by the additional unit path flow are t1e−t1 and t2e−t2,
respectively. Hence,
PMCt1(f) = LMC1,t1(f) + LMC2,t1e(f)
= (t1e − t1) + (t2e − t1e)= t2e − t1 (18)
However, the LTT method predicts the additional costs
in link 1 and link as link marginal cost LMC1,t1(u1) and
LMC2,t2(u2) ( Fig. 3). Namely,
PMC ′t1(f) = LMC1,t1(u1) + LMC2,t2(u2)
= t1e − t1 + t2e − t2
> PMCt1(f) (19)
Consequently, PMCt1(f) is larger than LMC1,t1(u1) +LMC2,t2(u2) by (t1e − t2). In other words, the LTT method
tends to overestimate the PMC for two sequential links in this
simple network. The reason of the overestimation is that the
path flow perturbation actually travels more slowly than the
additional flow unit because of the link bottleneck capacity
restriction. More specifically, the perturbation caused by an
additional unit flow entering link 1 at time t1 will not
propagate onto link 2 so long there is a queue present on
link 1. Namely,
Ind2kptrs =
{1 if k = t1e0 otherwise
(20)
1300
0 t 0 t1
ft
q c1 c1c2
NN
1
ft 2 1
f ft t+
1 unit vehicle
1
et 2
etTt1 1
et
1 unit vehicle
t2 t2
# #Additional cost incurred by
the additional unit vehicle
Additional cost incurred by
the additional unit vehicle
original cum. curves
new cum. curves
Fig. 2. Path marginal costs for the illustration network
0 t 0 t1
ft
q c1 c1 c2
NN
1
ft2 1
f ft t+
1 unit vehicle
t1
1 unit vehicle
T 1
et1
et 2
ett2t2
# #1 1( )aMC t
2 2( )aMC tL L
Fig. 3. Link marginal costs for the illustration network
In fact, this propagation rule is also applied to two sequen-
tial links in a merge. For more general networks involving
diverges, evaluating path flow perturbation propagation is
much complicated since the path flow perturbation will also
affects the inflows of links not on the path as well.
In view of the deficiency in the existing PMC evaluation
method, i.e., the problematic assumption on the path flow
perturbation propagation, we present a new path marginal
cost evaluation method for networks in mono-centric cities,
i.e., networks without diverges2.
To evaluate PMCs, we need to keep track of the path flow
perturbation propagation among links. For networks without
a diverge, this is quite easy to achieve based on the dynamic
network loading results. If we denote dai(t) as the actual
time that the perturbation of the path flow departing at time
t reaches link ai, we have the following relationship:
Indaikptrs =
{1 if k = dai
(t)0 otherwise
∀i = 1, . . . , m (21)
Substituting (21) into (13), we get:
PMCrspt =
m∑i=1
LMCai,dai(t)(uai) + cs[ds(t)] (22)
Based on the DNL (Dynamic Network Loading) results,
dai(t) can be derived by the following recursion relation-
ships:
da1(t) = t (23)
dai(t) = wai−1 [dai−1(t)], i = 2, . . . , m (24)
ds(t) = wam[dam
(t)] (25)
where wai(t), referred to as the path flow perturbation
propagation lag hereafter, is the earliest time after t + cai(t)
2A thorough discussion for general networks will be reported elsewhere.
when the queue on link ai vanishes and can be read directly
from the cumulative curves.
Consequently, the PMC can still be regarded as ”additive”
except that we replace the original time zai(t) which is the
time that the additional flow unit reaches link ai by the time
dai(t) which is the actual time that the path flow perturbation
reaches link ai. For narrative convenience, we will refer
to this new PMC evaluation method as the perturbation
propagation time (PPT) method.
IV. SOLUTION PROCEDURE
A. The heuristic method of successive average (MSA) algo-rithm
Once PMCs are available, we can transform the path-based
SO-DTA model into an equilibrium problem and solve it
using variational inequality methods. It is well know that
equilibrium conditions like the first order optimality condi-
tions of the path-based SO-DTA model can be transformed
into the following variational inequality (VI) problem:∑t∈Td
∑rs∈RS
∑p∈P rs
PMCrspt (f∗)[frs
pt − frs∗pt ] ≥ 0,∀f ∈ Ω
(26)
where Ω is a polyhedron defined by (9) and (10).
Ever since Friesz et al. [8] and Smith [9] proposed the VI
formulation of the predictive user equilibrium dynamic traffic
assignment problem, the solution algorithms to dynamic
equilibrium problems in transportation have been studied
extensively. Since the comparison of the performance of
different algorithms is beyond the scope of this paper, in
this study, we simply adopted the heuristic MSA algorithm
to solve this path marginal cost equilibrium problem.
We describe the complete steps of the MSA algorithm
for solving the path-based SO-DTA problem in networks in
mono-centric cities as follows:
MSA algorithm for solving the path-based SO-DTA model:
Step 0. Select an initial path flow pattern f0 and set k = 0.
Step 1. Load fk into the network.
Step 2. For all rs ∈ RS, search for the time-dependent
path [p∗, t∗] with the least marginal cost, i.e., [p∗, t∗] =argminp∈P rs,t∈Td
PMCrspt (f).
Step 3. Obtain the auxiliary path flow pattern g(fk) by
assigning all the demands Qrs,∀rs ∈ RS onto [p∗, t∗];Step 4. Set λ = 1/k and update the solution by setting
fk+1 = (1 − λ)fk + λg(fk);Step 5. Check if ||fk+1 − fk||/||fk|| < ε (a predetermined
parameter). If yes, stop; otherwise, set k = k + 1 and return
to step 1.
B. Algorithm for time-dependent least marginal cost pathsearching
The only unresolved part of the above heuristic MSA
algorithm is to search for the time-dependent least marginal
cost path. Our time-dependent least marginal cost path
searching algorithm is designed based on the DOT algorithm
by Chabini (1998) [10] for time-dependent minimal cost path
1301
(TDMCP) searching, which has been shown to have the
minimal computational complexity among all the existing
TDMCP algorithms.
If we denote Di(t), i ∈ N as the label for node i at time t,i.e., the temporal minimal cost, pi(t) as the pointer denoting
the predecessor link on the temporal time-dependent shortest
path, and cij(t) as the time-dependent cost for link (ij) at
time t, the DOT algorithm for time-dependent minimal cost
path searching can be described as follows:
DOT algorithm for TDMCP searching:
Step 0: Initialization: set Di(t) = ∞,∀i �= s and Ds(t) =0,∀t < N . Set ps(t) := 0,∀t.Step 1: Set Di(N) := the static shortest path tree rooted at
s with all costs defined by cij(N). Furthermore, note that
Di(t) = Di(N),∀t ≥ N .
Step 2: For t = N − 1 down to 0:
for (i, j) ∈ Aif Di(t) > cij(t) + Dj(t + τij(t))
Di(t) := cij(t) + Dj [t + τij(t)];pj [t + τij(t)] := [i, t];
endif
endfor
In our case, the path marginal cost is actually not additive
according to link traversal times but according to path flow
perturbation propagation times along the path. Hence, t +τij(t) in the original algorithm should be replaced by wij(t)to represent the correct path flow perturbation propagation
relationships in a compacted time-space expansion network.
After this revision, the DOT algorithm can be applied to
search for the time-dependent least marginal cost path.
V. NUMERICAL RESULTS
In this section, we give numerical results to demonstrate
how the proposed algorithm for path-based SO-DTA models
based on the PPT PMC evaluation method performs. For
comparison purpose, a similar solution procedure based on
the LTT PMC evaluation method is also implemented. All the
algorithms are coded in MS-VC++ and run on a Windows-
XP PC (Intel Pentium M 1.60 GHz, 768 MB of RAM).
A. Numerical example I
To demonstrate how the prediction of path flow pertur-
bation propagations affects the accuracy of the final system
optimum solution, an example network with two routes in
parallel is constructed. To simplify the discussion, we only
focus on the system optimal route choice, and the time-
dependent departure rates are assumed to be given. The
free flow travel times of route 1 and route 2 are 60min
and 12min respectively. Vehicles depart from the origin at
a constant departure rate q = 3000veh/hr for one hour.
Route 1 does not have any bottlenecks. Three scenarios
which differ from each other in the number of bottlenecks
on route 2 are designed. The capacity characteristics of three
scenarios are summarized in Table I. We expect that the more
bottlenecks on a route, the more errors might be incurred by
the inaccurate prediction of path flow propagations.
TABLE I
NETWORK CHARACTERISTICS IN ALL THE SCENARIOS
tf : MIN, s: VEH/HR
Scenario I route 1 no bottleneck, tf = 60route 2 bottleneck I: tf = 12, s = 1500
Scenario II route 1 no bottleneck, tf = 60route 2 bottleneck I: tf = 6, s = 2000;
bottleneck II: tf = 12, s = 1500
Scenario III route 1 no bottleneck, tf = 60route 2 bottleneck I: tf = 4.8, s = 2000;
bottleneck II: tf = 8.4, s = 1800;
bottleneck III: tf = 12, s = 1500
0
2
4
6
8
10
12
14
0 20 40 60 80 100 120Assignment interval (30s)
De
pa
rtu
re r
ate
(ve
h/in
terv
al)
scenario I
scenario II
scenario III
analytical solution
0
5
10
15
20
25
30
0 20 40 60 80 100 120Assignment interval (30s)
De
pa
rtu
re r
ate
(ve
h/in
terv
al)
scenario I
scenario IIscenario III
analytical solution
(a) Route 1
(b) Route 2
scenario III
scenario II
scenario I Analytical solution
Analytical solution
scenario I
scenario IIscenario III
Fig. 4. Numerical solutions based on the PPT method
Note that tf is measured from the origin to the bottleneck
or destination. A quick calculation reveals that the analytical
solutions for all the three scenarios are the same as follows:
For [0, 36min] : d1(t) = 1500veh/hr, d2(t) = 1500veh/hr
For [36min, 60min] : d1(t) = 0, d2(t) = 3000veh/hr
We now apply both the PPT and LTT methods and combine
them with the heuristic MSA algorithm to solve the SO-DTA
problem. The numerical solutions of route choice patterns
based on these two methods, in comparison to the analytical
solution, are depicts in Fig. 4 and Fig. 5.
In scenario I, the numerical solutions based on both the
PPT and LTT methods are identical and very close to the
analytical solution. This is not a surprise because when there
is only 1 bottleneck on route 2, the PMCs are actually
LMCs and no path flow perturbation propagation indices are
required to obtain PMCs. In scenario II and III, the PPT
method can still achieve very good accuracy compared to the
analytical solution, while the numerical solutions based on
the LTT method show distinct deviations from the analytical
solution.
1302
0
2
4
6
8
10
12
14
0 20 40 60 80 100 120
Assignment interval (30s)
Depart
ure
rate
(veh/inte
rval)
scenario Iscenario IIscenario IIIanalytical solution
0
5
10
15
20
25
30
0 20 40 60 80 100 120Assignment interval (30s)
Depart
ure
rate
(veh/inte
rval)
scenario I
scenario II
scenario III
analytical solution
(a) Route 1
(b) Route 2
scenario III
scenario II
scenario I
Analytical solution
scenario III
scenario II
scenario I
Analytical solution
Fig. 5. Numerical solution based on the LTT method
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100 120 140 160 180 200 220 240
Assignment interval (30s)
De
pa
rtu
re r
ate
(ve
h/in
terv
al)
PPT
LTT
Analytical solution
Fig. 6. Numerical solutions based on both PPT and LTT methods
B. Numerical example II
The second numerical example is designed to test whether
the heuristic MSA method based on the PPT PMC evaluation
method can generate an accurate system optimal departure
time choice pattern. The testing network contains only 1 link
and we aim at deriving the system optimal departure time
choice pattern. The link free flow travel time is tf = 10min,
and there is a bottleneck with capacity s = 1800veh/hr at the
downstream end of the link. The total demand is 1500veh.
The desired arrival time is t̃ = 7 : 00am. The schedule delay
parameters are δ = 0, α = 1, β = 0.8, γ = 1.2. It is easy to
derive the analytical SO solution for this problem as follows:
Earliest departure time ts = 6 : 20am
Latest departure time te = 7 : 10am
Departure rate a(t) during [ts, te] = s = 1800veh/hr
We show the numerical optimal departure time choice
patterns based on both methods in Fig. 6 (t = 0 corresponds to
the time 6:00 am), in comparison with the analytical solution.
As we can see from the results, the heuristic MSA
algorithm based on the PPT method still can converge to
the analytical solution while the same algorithm based on
the LTT method cannot. This is understandable because
in the LTT method, the deficient prediction of path flow
perturbation propagation affects the accuracy of marginal
schedule delay at the destination.
VI. CONCLUSIONS
This paper studies the solution procedure for path-based
SO-DTA models. The most critical part in the solution
procedure, i.e., the evaluation of PMCs, is identified and
discussed thoroughly. A solution algorithm for path-based
SO-DTA models based on a new PMC evaluation method
is developed and tested on simple networks. Our limited
numerical examples indicate that the proposed heuristic
MSA algorithm based on the PPT PMC evaluation method
can generate numerical solutions very close to analytical
solutions, for both SO-DTA problems optimizing departure
time choices and route choices.
At present our proposed solution method can only be ap-
plied to networks without diverges and the embedded traffic
dynamics models are restricted to those not considering link
interactions. The relaxation of either aspects may bring in
additional challenges in predicting path flow perturbation
propagations and is worth further investigation.
ACKNOWLEDGEMENTS
This research is supported in part by a grant from the Na-
tional Science Foundation under the number CMS#9984239.
The views are those of the authors alone.
REFERENCES
[1] Merchant, D.K. and Nemhauser, G.L., ”A model and an algorithmfor the dynamic traffic assignment problems”, Trans. Sci., vol. 12(3),1978, pp 183-199.
[2] Carey, M., ”Optimal time-varying flows on congested networks”, Oper.Res., vol. 35(1), 1987, pp 58-69.
[3] Wie, B.-W., ”A convex control model of dynamic system-optimaltraffic assignment”, Contr. Engr. Prac., vol. 6, 1998, pp 745-753.
[4] Ziliaskopoulos, A.K., ”A linear programming model for the singledestination system optimum dynamic traffic assignment problem”,Trans. Sci., vol. 34(1), 2000, pp 37-49.
[5] Wardrop, J.G., ”Some theoretical aspects of road traffic research”, InProceedings of the Institute of Civil Engr., II, vol. 1, 1952, pp 325-378.
[6] Ghali, M. O. and Smith, M. J., ”A model for the dynamic systemoptimum traffic assignment problem”, Trans. Res., B, vol. 29(3), 1995,pp 155-170.
[7] Peeta, S., ”System optimal dynamic traffic assignment in congestednetworks with advanced information systems”, Ph.D. thesis, Theuniversity of Texas at Austin, 1994.
[8] Friesz, T.L. and Bernstein, D. and Smith, T.E. and Tobin, R.L. andWie, B.W., ”A variational inequality formulation of the dynamicnetwork user equilibrium problem”, Oper. Res., vol. 41(1), 1993, pp179-191.
[9] Smith, M.J., ”A new dynamic traffic model and the existence and cal-culation of dynamic user eqiubria on congested capacity-constrainedroad networks”, Trans. Res., B, vol. 26, 1993, pp 49-63.
[10] Chabini, I., ”Discrete dynamic shortest path problems in transportationapplications”, Trans. Res. Rec., vol. 1645, 1998, pp 170-175.
1303