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: hmzhang@ucdavis.edu

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.

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