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UNIVERSITY TRANSPORTATION RESEARCH CENTER
Dynamic User Equilibrium Model for Combined
Activity-Travel Choices Using Activity-Travel
Supernetwork Representation
Prepared by
Gitakrishnan Ramadurai
Ph.D. Candidate
Satish Ukkusuri, Ph.D.
Assistant Professor and Blitman Career Development Chair
Department of Civil and Environmental Engineering
Rensselaer Polytechnic Institute
Department of Civil and Environmental Engineering
Rensselaer Polytechnic Institute
110 8th Street
Troy, NY 12180
518-276-6033
518-2764833 (fax)
www.rpi.edu
August 2008
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Rensselaer Polytechnic Institute, 110 8th Street
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Integrated urban transportation models have several benefits over sequential models including consistent solutions, quicker
convergence, and more realistic representation of behavior. Static models have been integrated using the concept of
Supernetworks. However integrated dynamic transport models are less common. In this paper, activity location, time of
participation, duration, and route choice decisions are jointly modeled in a single unified dynamic framework referred to as
Activity-Travel Networks (ATNs). ATNs is a type of Supernetwork where virtual links representing activity choices are added to
augment the travel network to represent additional choice dimensions. Each route in the augmented network represents a set
of travel and activity arcs. Therefore, choosing a route is analogous to choosing an activity location, duration, time of
participation, and travel route. A cell-based transmission model (CTM) is embedded to capture the traffic flow dynamics. The
dynamic user equilibrium (DUE) behavior requires that all used routes (activity-travel sequences) provide equal and greater
utility compared to unused routes. An equivalent variational inequality problem is obtained. A solution method based on route-
swapping algorithm is tested on a hypothetical network under different demand levels and parameter assumptions.
Integrated urban transport model, Activity-Travel Networks,
Dynamic user equilibrium, Route-Swapping Algorithm
Unclassified Unclassified
19
Gitakrishnan Ramadurai, PhD Candidate,
Satish Ukkusuri, Assistant Professor, Department of Civil and Environmental Engineering
Dynamic User Equilibrium Model for Combined
Activity-Travel Choices Using Activity-Travel Supernetwork presentation
Noname manuscript No.
(will be inserted by the editor)
Dynamic User Equilibrium Model for Combined
Activity-Travel Choices Using Activity-Travel
Supernetwork Representation
Gitakrishnan Ramadurai ·Satish Ukkusuri
the date of receipt and acceptance should be inserted later
Abstract Integrated urban transportation models have several benefits over sequen-
tial models including consistent solutions, quicker convergence, and more realistic rep-
resentation of behavior. Static models have been integrated using the concept of Su-
pernetworks. However integrated dynamic transport models are less common. In this
paper, activity location, time of participation, duration, and route choice decisions are
jointly modeled in a single unified dynamic framework referred to as Activity-Travel
Networks (ATNs). ATNs is a type of Supernetwork where virtual links representing
activity choices are added to augment the travel network to represent additional choice
dimensions. Each route in the augmented network represents a set of travel and activity
arcs. Therefore, choosing a route is analogous to choosing an activity location, dura-
tion, time of participation, and travel route. A cell-based transmission model (CTM) is
embedded to capture the traffic flow dynamics. The dynamic user equilibrium (DUE)
behavior requires that all used routes (activity-travel sequences) provide equal and
greater utility compared to unused routes. An equivalent variational inequality prob-
lem is obtained. A solution method based on route-swapping algorithm is tested on a
hypothetical network under different demand levels and parameter assumptions.
Keywords Integrated urban transport model ·Activity-Travel Networks ·Dynamic
user equilibrium ·Route-Swapping Algorithm
G. Ramadurai
PhD Candidate
Department of Civil and Environmental Engineering
Rensselaer Polytechnic Institute
Tel.: +1-518-2768306
Fax: +1-518-2764833
E-mail: ramadg@rpi.edu
S. Ukkusuri
Assistant Professor and Blitman Career Development Chair
Department of Civil and Environmental Engineering
Rensselaer Polytechnic Institute
Tel.: +1-518-2766033
Fax: +1-518-2764833
E-mail: ukkuss@rpi.edu
2
1 Introduction
Urban transport modeling involves several dimensions of individual choice including
activity participation, location, time of participation, duration, choice of mode, and
route. Often the choice models are sequentially applied with feedback: initially, the
choice environment is assumed fixed and the individual choices are determined. Sub-
sequently, given the individual choices the choice environment is adjusted. If feedback
is involved, the two steps are repeated until the individual choices and the resulting
choice environment are in equilibrium. We also refer to this state as converged solution.
This process of iteratively solving a sequence of models forms the basis of the four-step
urban transportation modeling paradigm.
As opposed to the sequential procedure, several studies have explored integrated
choice models particularly with respect to static transport models. Integrated urban
transportation models have several benefits over sequential models including consistent
solutions, quicker convergence, and more realistic representation of behavior. Static ur-
ban transport models have been integrated using the concept of Supernetworks ((Sheffi,
1985); also referred to as Hypernetworks, (Sheffi and Daganzo, 1979, 1980)). However
integrated dynamic transport models are less common.
Dynamic traffic assignment models (Peeta and Ziliaskopoulos, 2001) have been
developed over the past two decades and have addressed several of the short-comings
of the static traffic assignment procedures. In particular, DTA models have increased
traffic flow and behavior realism and the explicit modeling of time-varying flows. These
advantages allow DTA to be applied to real-time traffic management, ATIS, and other
ITS measures (Mahmassani, 2001; Ben-Akiva et al, 2001). While traditionally DTA
models were restricted to determining route choices given an exogenous time-sliced
demand matrix, more recently DTA models that capture two choice dimensions - route
and departure time choice - have been developed (Friesz et al, 1993; Ran et al, 1996;
Huang and Lam, 2002; Wie et al, 2002; Szeto and Lo, 2004; Zhang and Zhang, 2007).
To capture behavioral realism better there is a need to consider additional choice
dimensions within a dynamic traffic assignment framework.
Initial work toward integrating additional choice dimensions in DTA models are
(Abdelghany et al, 2001, 2003). Abdelghany et al (2001) develop dynamic spatial mi-
croassignment procedures when the unit of analysis is trip chains instead of trips. How-
ever, they do not model additional choice dimensions such as departure time, activity
location and duration endogenously. Abdelghany et al (2003) addresses a more general
choice problem. They determine the departure time, route choice and the sequence of
activities simultaneously.
More recent studies in integrated dynamic models include Lam and Huang (2003);
Zhang et al (2005); Kim et al (2006); Rieser et al (2007). Lam and Huang (2003) de-
velop a dynamic equilibrium model considering activity location, route, and departure
time dimensions. Their framework, however, assumes the duration of activity partici-
pation as exogenous. Capturing activity duration is essential to understand the effect
of activity scheduling on traffic congestion. An integrated work activity scheduling and
departure time choice model in a network with bottleneck congestion is developed by
Zhang et al (2005). However, they consider single activity participation only. A logi-
cal extension is to consider multiple activities and activity chaining decisions. This is
the focus of the paper by Kim et al (2006). They present an activity chaining model
formulated from the perspective of a time use problem with budget constraints. Their
model includes a dynamic traffic assignment simulation model to obtain network travel
3
times and an iterative day-to-day dynamic process where activity chains are updated
based on the network travel times computed in previous iteration. Whether such an
iterative procedure results in consistent solutions and the performance of the solutions
compared to more holistic frameworks are interesting research questions that merit
attention. Rieser et al (2007) describe a multi-agent simulation (MATSim) that takes
individuals complete activity sequence as input. Individual’s behavior in terms of their
route choice and departure time choice are determined iteratively with a traffic flow
simulator. They describe a conceptual framework to extend the MATSim to incorporate
activity rescheduling and participation decisions.
In this paper, activity location, time of participation, duration, and route choice
decisions are jointly modeled in a single unified dynamic framework referred to as
Activity-Travel Networks (ATNs). The proposed integrated framework is motivated
by the following considerations: (a) to capture activity demand-supply dynamics in
addition to transportation demand-supply dynamics, and (b) to obtain a consistent
equilibrium solution across all dimensions of choice. ATNs is a type of Supernetwork
where virtual links representing activity choices are added to augment the travel net-
work to represent additional choice dimensions. Each route in the augmented network
represents a set of travel and activity arcs. Therefore, choosing a route is analogous to
choosing an activity location, duration, time of participation and travel route.
2 ATN Representation and Motivation
ATNs use a network representation where nodes are activity centers that are joined
by travel links. Activities are represented by arcs that both originate and terminate
in the same node (activity centers). Each activity arc is characterized by a unique
activity type and a set of durations. An activity-travel sequence for an individual can
be represented as a ‘route’ that includes both travel and activity arcs. All individuals at
the beginning of the model start from ‘home’ and must participate in a predefined set of
activities. All activity-travel sequences that traverses the set of activity arcs in which an
individual participates in are considered feasible sequences. The model time frame may
be set arbitrarily and is presented in a discrete-time setting. Durations of arc-traversal
for travel arcs is always assumed to be a function of flow, while for activity arcs it is
assumed fixed. Consistent with rational behavior assumption, each individual chooses
the activity-travel sequence that provides the maximum generalized utility. However,
modeling the network dynamics at an individual level is computationally intensive.
Therefore, we treat all individuals residing in the same ‘home’ node, who participate in
the same set of activities as similar. We accordingly modify the behavioral framework to
be consistent with Wardrop’s (Wardrop, 1952) equilibrium framework. The behavioral
rule adopted is ‘all used routes (activity-travel sequences) provide equal and greater
utility compared to unused routes’. In other words, at equilibrium no individual can
improve her utility by unilaterally changing her travel choice decisions.
A primary motivation of the ATNs representation is to capture the effect of activity
and transportation demand-supply dynamics in travel choice decisions. Consider a
hypothetical scenario in the double-diamond network shown in figure 1. The network
consists of eight nodes: Home node (H), Work node (W), four Non-work activity centers
(N1-N4), and two intermediate nodes (I1 and I2). The nodes are connected by twelve
arcs: 3, 4, 10, and 11 are the activity arcs and the rest are travel arcs. Let us call the
diamond with the home node as the residential neighborhood diamond (R-diamond)
4
and the other as business neighborhood diamond (B-diamond). The total demand for
travel from home to work is 100 individuals; all individuals drive alone to work. Further,
50 individuals drive directly from home to work while 50 individuals make a stop to
participate in a non-work activity en route to work. All individuals have to arrive at
work at the same time, (say) T. All travel arcs have a capacity of 50 vehicles per
time unit and free-flow traversal time of one time unit, while the duration of non-work
activity participation (which is also the time for traversal of activity arc) is two time
units. The utility of participating in the non-work activity is 100 utils (let utils be
the unit of measuring utility) while the utility of travel on an arc is -5*(travel time)
utils. As mentioned earlier, the travel arcs have fixed capacities: at free-flow a travel
arc traversal would fetch -5 utils, while a queuing delay by one time unit would result
in a payoff of -10 utils.
There exist two possible activity-chain sequence in the double-diamond network:
i) Home to Work, and ii) Home to Non-work activity to Work. The former can be
accessed via four different paths while the latter has eight paths - four paths each
that visit a non-work activity center in R-diamond and B-diamond. Since the utility of
participating in the non-work activity in all four nodes is the same, based on traditional
models of utility maximization, they attract equal amount of traffic. Therefore such
an assignment model would result in each of the eight paths that pass through the
non-work activity having a flow of 50/8. The corresponding total free-flow traversal
time is 7 time units (therefore start time is (T−7)th time unit). For the individuals
who drive straight to work, the traversal time is 5 time units; the corresponding flow
is divided among the four paths (50/4). However, link 7, with a capacity of 50 vehicles
per time unit, has an upstream demand of 75 vehicles at the start of (T−3)th time
unit. This leads to delay by one time unit for 25 individuals and a loss in overall utility
of 125 utils (assuming there is no late arrival penalty).
On the other hand, if traffic dynamics is incorporated in the assignment model,
we would obtain a solution where none of the individuals visit the non-work activity
center in the R-diamond. In this case, there is no delay for any of the individuals and
the total overall utility is 125 utils more than the previous case. The reason for the
difference in utilities is the limited capacity of link 7. Ignoring the traffic flow dynamics,
could lead to sub-optimal assignment patterns. Therefore, it is important to consider
transportation demand-supply dynamics.
5
Activity demand-supply dynamics also play a similar important role in individ-
ual decisions. Examples include activity centers with access time restrictions, social
interaction activities that provide greater utility with increased participation and ca-
pacity restrictions in shopping mall check-out counters. Consider the example of ca-
pacity restrictions in shopping mall check-out counters: current models that ignore
such an activity supply capacity restriction could over-estimate trip-chaining of shop-
ping activity by commuters or under-estimate non-peak hour shopping trips. If in the
double-diamond network example above, the non-work activity centers located in the
B-diamond had the following modified utility specification: 100 utils if flow on arc is
less than or equal to 15 individuals, 75 otherwise; then, the corresponding destination
choice and traffic assignment model would result in 15 individuals choosing to partici-
pate in the non-work activity in B-diamond while 10-individuals choose the R-diamond.
ATNs can model the above described as well as several other activity demand-supply
dynamics.
3 Conceptual Framework
We present the overall conceptual framework in this section. A similar conceptual
framework for the general transportation planning problem was presented by Florian
et al (1988). The framework presented here builds on the work by Florian et al (1988)
and includes activity characteristics in addition to travel characteristics.
3.1 Definitions and Notation
h: index for household.
ih: index for individuals in household h.
ih∈1,2,...,Ih.
G={ν, α}is the activity-travel network, where νis the set of nodes and αis the set
of arcs.
α3 {αT, αA}correspond to the set of travel and activity arcs.
Aih: Set of activities individual ihparticipates in.
Atrav
ih: Set of travel activities for individual ih.
A•: Set of activities for all individuals residing in node.
The elements of the above sets are characterized by attributes that denote their
‘state’. We represent the set of characteristics as Ω[.].
Ων[Xν]: Representing accessibility measures for different activities at node ν.
ΩαT[(m, n),(f, T T (f))]: Representing source and sink node, flow, and travel-time of
travel arc αT. The source and sink node are shown together because they represent
known characteristics while flow and travel-time have to be solved for.
ΩαA[(n, δ),(U, f )]: Activity-center node, duration, utility of traversing the arc, and
flow in arc.
ΩAih[ts, δ, n]: Activity start time, duration, and location node.
6
ΩAtrav
ih
[ts, δ, o, d, µ, ρ]: Travel start time, duration, origin, destination, mode, and route.
Ωh[.]: Characteristics of the household hsuch as type of household, number of vehicles.
Ωih[.]: Characteristics of individual such as age, gender, employment status.
3.2 Relationships: ATN Framework
We use two types of functional relationships, Φand Ψ, to capture the various complex
relationships between the above variables. Φfunctions are direct functional maps from
<mn to <m(for example, regression equations), while Ψfunctions represent more
complex relationships such as a fixed-point mapping.
Several different frameworks arise based on the relationship assumptions among
the above sets and their characteristics. The set of relationships below represent the
framework adopted in this paper.
Aih=Φ(Ωh, Ωih, Ων,ˆ
Ωα)∀ih(1)
{ΩA•, ΩAtrav
•}=Ψ1(ΩαT, ΩαA, Ων)∀A(2)
{ΩαT[f, T T (f)], ΩαA[U, f ]}=Ψ2(ΩA•, ΩAtrav
•)∀αTand αA(3)
The reader may note that the Φfunction is at an individual level while the Ψ
functions are at a network or zonal level. Also, the Φfunction is similar to disaggregate
demand models while Ψ2is similar to an aggregate network assignment model. Given
the complexity of the Ψfunctions they are not modeled at an individual or disaggregate
level.
Φdetermines the set of activities that an individual participates in. Among other
factors, this could depend on household and individual characteristics, activity center
location and accessibility characteristics, transportation and activity supply charac-
teristics, and also the set of fixed activities the individual participates in. The reader
may note that the Φfunction includes estimates of transportation and activity supply
characteristics denoted as ˆ
Ωα. In this paper, we assume Φ, the set of activities that an
individual participates in, as known; the focus of this paper is on the two Ψ-functions
only.
The Ψfunctions represent complex relationships between arc (both travel and activ-
ity arc) characteristics and characteristics of activities. Ψ1determines the characteris-
tics such as start time, duration and location for the set of activities and corresponding
travel of all individuals in every node. They are assumed to depend on activity and
travel supply characteristics represented by duration/traversal time, flow, location, and
on activity-center accessibility characteristics Ων.Ψ2, on the other hand, maps a given
set of activity and travel characteristics to a set of flows and corresponding travel times
and utilities on arcs. The two relationships Ψ1and Ψ2together represent the fixed-point
problem shown below.
{ΩαT[f, T T (f)], ΩαA[U, f ]}=Ψ2(Ψ1(ΩαT, ΩαA, Ων)) (4)
7
4 Operational Framework
Two critical issues to operationalize the ATN framework are flow propagation dynamics
and utility function specification. We discuss their implementation details below.
4.1 Dynamics of Flow Propagation
Traffic flow has been modeled at different levels in the past. The most realistic models
are disaggregate microsimulation models (Gartner et al, 2001) where behavior of each
vehicle on the network is modeled explicitly. On the other hand, macroscopic mod-
els (Gartner et al, 2001) describe traffic flow based on relationships between speed,
flow, and density. Though microscopic models are more accurate they require greater
computation time and lack analytical solutions.
Macroscopic models, on the other hand, can be modeled as side constraints to pro-
vide approximate, quick solutions and are more suitable for analytical DTA models.
Macroscopic models can be further divided into exit flow models, point queue models,
and physical queue models. In this study, we use a network level simulation adaptation
of the cell transmission model (CTM) (Daganzo, 1994, 1995; Ziliaskopoulos, 2000).
The CTM is capable of capturing the effect of spillbacks (physical queue) and shock
wave propagation (two-regime flow). Also, the fixed-point problem in equation 4 is
formulated as a variational inequality (VI) problem. Existing VI solution techniques
are based on heuristic searches and require several iterations of network loading step.
Therefore, embedding a microsimulation model would require multiple runs of a com-
putationally intensive model and could significantly increase the running time of any
algorithm. We present the details of the CTM below.
We assume the activity-travel network to be divided into a series of inter-linked
cells. Cells represent a segment of a travel link or an activity location. Unlike (Da-
8
ganzo (1994), Daganzo (1995)) we assume variable cell lengths. As mentioned in Da-
ganzo (1995) this implies a trade-off between computational resource requirements and
the level of accuracy of the CTM model to the classic LWR model - shorter cells can
more closely replicate the LWR model but may demand more computational resources.
Allowing for variable cell lengths is straight-forward in simulation adaptations of the
CTM. Activity cells do not have a physical length; traversal time of activity cells are
determined based on the duration of activity participation information contained in
the route chosen for travel. The links between cells do not have any physical signifi-
cance. An example of the cell transmission model representation of activities at a node
is shown in Figure 2.
Notation:
Let,
xi,r
tis the number of vehicles following route rin cell ithat entered the cell at (t−ti
f)th
time interval or earlier. In the case of activity cells, this represents the number of ve-
hicles that have ‘resided’ in cell ifor a time period equal to the duration of activity
participation.
xi
t=P
∀r
xi,r
t
Xi
t: the total number of vehicles in cell iat time t.
yi,r
t: flow on route r, out of cell iat time t.
yi
t=P
∀r
yi,r
t
Ni: Number of vehicles that can be accommodated at jam density for cell i.
Qi: Maximum flow capacity out of cell i.
We assume Nand Qare time invariant and therefore drop the time subscripts in
their representation.
In the discussion below, cells jand kare assumed to be immediately downstream
of cell iand cells gand hare immediate upstream of cell i.
Finally, Pgrepresents the fraction of flow from cell gthat enters the downstream
merge cell, i. The sum over all such fractions is one (in our illustration we assume two
cells - gand h- merge into cell i. Therefore Pg+Ph= 1).
Flow propagation is achieved by repeatedly solving three sets of equations - first set
of equations determine the outflow from a cell (y) between time-step t-1 to t, second
set of equations determine the individual route break-ups and the final set determine
the current cell occupancy (xi,r
t) based on past occupancy, inflows, and outflows.
yi
tis determined from the following equations:
For activity cells: yi
t=xi
t.
9
For ordinary travel cells: yi
t= min(xi
t−1, Qi, N j−Xj
t−1).
For cells that merge into a single cell, several cases arise. We deal with each below:
Case 1: If min(xg
t−1, Qg) + min(xh
t−1, Qh)>(Ni−Xi
t−1)
Case 1a: If min(xg
t−1, Qg)> P g(Ni−Xi
t−1) and min(xh
t−1, Qh)> P h(Ni−Xi
t−1), then
yg
t=Pg(Ni−Xi
t−1) and yh
t=Ph(Ni−Xi
t−1).
Case 1b: Else If min(xg
t−1, Qg)≤Pg(Ni−Xi
t−1), then yg
t= min(xg
t−1, Qg) and yh
t=
(Ni−Xi
t−1)−yg
t.
Case 1c: Else, yh
t= min(xh
t−1, Qh) and yg
t= (Ni−Xi
t−1)−yh
t.
Case 2: Else, yg
t= min(xg
t−1, Qg) and yh
t= min(xh
t−1, Qh).
For diverge cells, xi
tis split into two parts xi,rj
tand xi,rk
tsuch that xi,rj
t(xi,rk
t)
contains all vehicles that take cell j(k) next. This is determined based on the next cell
in route r. The outflow into each of the two diverge links may be determined similar
to an ordinary cell with route specific cell occupancies xi,rj
tand xi,rk
tinstead of xi
t.
The second set of equations determine the flow on each route r:yi,r
t=yi
t
xi
t−1
xi,r
t−1.
The third step of determining current occupancy follows from xi,r
t=xi,r
t−1+yj,r
t−
yi,r
t.
The reader is referred to Lo and Szeto (2002) for a detailed discussion on obtaining
average travel times from the CTM simulation. An additional step required in the
current model is to deduct activity participation durations from the computed travel
times.
4.2 Utility Function Specification
The next critical step in the ATN framework is the utility function specification. The
focus of the present paper is not on estimating utility function form or parameters.
We assume reasonable functional forms and parameter values to illustrate the ATN
framework. However, accurate estimation of utility function form and parameters is an
important issue that needs further investigation in the future.
Let,
Acbe the set of all possible activity combinations.
Ra
od : Set of routes from origin oto destination dcontaining activity arcs αAsuch
that they traverse all activities in activity combination a∈Ac.ris a route that be-
longs to the set Ra
od. Each route rrepresents a set of travel and activity arcs. Therefore
choosing a route r, results in the choice of activity location, duration, time of partici-
pation and travel route.
10
Ua,r
od denotes utility derived by individuals departing from oand reaching d, partici-
pating in activity chain a∈Acusing route r.
ha,r
od : Path flow from oto d, participating in activity chain combination a∈Acusing
route r.
The temporal dimension in dynamic traffic assignment models (such as departure
or arrival time index) is not associated with the above definitions since all individuals
are always traveling on the network or participating in an activity.
Similar to Lam and Huang (2003), we assume an additive specification for the
above utility expression.
Ua,r
od =Ua(r)−Utrav (r) (5)
where, Ua(r) is the utility derived from participating in activity combination a∈Ac
and is a function of route r.Ua(r) can be represented as the sum of utilities derived
from traversing each activity arc αain route r.
Ua(r) = X
∀αA∈r
Uα(r, f) (6)
where, fis the flow in activity link. In general, utility derived from activity participation
may be assumed to be a function of type and duration of activity, time of participation,
location of activity with respect to the origin/destination of flow on route r, and the
total flow on activity link αA.
Utrav (r) = β∗T T (r) is the disutility from travel on route r. where, βis a parameter
to convert travel-time into utility units and TT (r) is the total travel time on route r.
5 Mathematical Formulation of ATNs
5.1 Dynamic User Equilibrium Conditions
We can now express the DUE conditions as follows:
Ua,r
od =
=Ua
od if ha,r
od >0
≤Ua
od if ha,r
od = 0
∀o,d,a∈Ac,and r(: r∈Ra
od) (7)
Subject to the condition that flow on network should satisfy demand. This is ex-
pressed as:
X
∀r∈Ra
od
ha,r
od =X
∀ih∈(o,d)
ζa
ih∀a∈Ac, o, d (8)
where, ζa
ih=1. . . if activity combination a∈Aih
0. . . otherwise
Ua
od is the maximum utility derived by individuals departing from oand reaching
d, participating in activity combination a∈Acusing route r.
DUE conditions, however, are not always satisfied in capacitated networks (Szeto
and Lo, 2006). Discontinuities in travel time or utility functions could result in non-
existence of solutions. These discontinuities could arise from time discretization or
due to capacity restrictions in the network. In capacitated networks it is possible that
packets of flow are broken because of the lack of available capacity downstream. Any
11
discrete-time model in capacitated networks exposes itself to the above drawback. Fur-
ther study is required to understand the properties of DUE in discrete-time capacitated
network models.
5.2 Equivalent variational inequality formulation
The above DUE conditions can now be formulated as an equivalent VI problem.
X
∀a∈Ac
(ha−ˆ
ha)TUa(h)≥0∀ha∈Haand ∀a∈Ac(9)
where,
Hais the set of feasible route flows traversing all activities in activity combination a,
given by (7),
hais the vector of route flows ∈Ha,
ˆ
hais the vector of route flows that satisfy the DUE condition in equation 6, and
Uais a vector whose each element is given by Ua,r
od −Ua
od.
5.3 Solution Approach
The utility derived from traversing the activity-travel sequence represented by route r,
expressed as the sum of utility derived from participating in activities and the disutility
from travel, is assumed to be a monotone decreasing function of flow on route r.
Therefore, a route-swapping algorithm (Lam and Huang, 2003; Szeto and Lo, 2006;
Nagurney and Zhang, 1997) is adopted to obtain solutions to the VI problem shown
in (8). The detailed algorithm is presented below:
Step 0: Initialize. Set iteration counter i= 0.
Choose an initial feasible vector of flows h(i).
Step 1: Computation. Load flow h(i) and compute travel times TT(r) using the Cell-based
transmission model.
Compute utilities Ua,r
od using (5) ∀r, a, o, d .
Set Ua
od = max
∀r∈Ra
od
Ua,r
od ∀a, o, d.
Step 2: Update flows. Set ˆ
Ra
od =r∈Ra
od :Ua,r
od =Ua
od.
For ever activity combination a∈Ac,
ha,r
od (i+ 1) = max(0, ha,r
od (i) + ρha,r
od (i)(Ua,r
od −Ua
od)) ∀r∈Ra
od\ˆ
Ra
od
Σa
od =P
∀r∈Ra
od\ˆ
Ra
od
(ha,r
od (i)−ha,r
od (i+ 1)) ∀a, o, d.
ha,r
od (i+ 1) = ha,r
od (i) + Σa
od
|ˆ
Ra
od|∀r∈ˆ
Ra
od ∀r, a, o, d.
ρis a scale parameter.
12
Step 3: Check for convergence. Compute π=P
∀r,a,o,d
(Ua,r
od −Ua
od)ha,r
od and ˆπ=P
∀r,a,o,d
Ua
od ha,r
od .
If π
ˆπ< then terminate. is a convergence tolerance value.
else, i = i + 1; Go to Step 1.
Nagurney and Zhang (1997) use the route-swapping algorithm (referred to as Eu-
ler’s method) for the static traffic assignment problem. They show that the Euler’s
method converges only when the link costs are strictly monotone increasing. However
when implementing the algorithm for path based formulations they reported that the
algorithm did not converge in their limited trials. Other studies (Lam and Huang,
2003; Szeto and Lo, 2006) also report lack of smooth convergence in their implementa-
tions of the algorithm. Therefore, the route swapping algorithm has convergence issues
when implemented in path-based formulations. In the numerical trials we test different
scenarios and report under what conditions the route swapping algorithm appears to
provide consistent DUE solutions to a test problem.
Two important components in the above algorithm are the convergence expres-
sion and the scale parameter ρ. Traditionally, the literature has utilized a convergence
check based on flow changes between two iterations (Nagurney and Zhang, 1997; Ab-
delghany et al, 2003). However, since path flows are not necessarily unique (assuming
the solution exists), using flow to determine convergence could lead to infinite loops in
the algorithm. A better convergence measure is the utility difference of used paths in
successive iterations. However, a direct comparison of utilities in used paths could lead
to the algorithm converging to a non-equilibrium solution. The expression used above
overcomes this problem by comparing the utility on all used paths to the maximum
possible utility. This ensures that the algorithm converges only when all used paths
have an utility that is close the maximum possible utility at equilibrium. If there are
no equilibrium solutions, the algorithm will not terminate. A reasonable upper limit
on the number of iterations is required to ensure the algorithm does not loop infinitely.
Consequently, if the algorithm terminates after reaching the maximum number of it-
erations, the solution must be checked to see if it has converged to an equilibrium
solution or not.
Nagurney and Zhang (1997) provide conditions for the scale parameter ρunder
which the algorithm converges to an equilibrium solution. Lam and Huang (2003)
show that these conditions allow for local stability if the cost functions are not strictly
monotonic. However, in discrete time capacitated networks such as the one dealt with
in this paper, the utility functions are likely to be discontinuous with sudden jumps
and falls. This adds to the complexity of the algorithm convergence and no theoretical
properties for the value of the scale parameter ρexist. This problem is also reported in
Szeto and Lo (2006). We test different values for the scale parameter and draw limited
insights.
6 Results from an Example Network
We demonstrate the application of the ATN framework and the proposed solution algo-
rithm on an example network. The example network considered is the double-diamond
network presented earlier. The equivalent cell-based representation of the network is
shown in figure 3. Free-flow traversal time, maximum flow capacity, and number of ve-
hicles at jam density for square (rectangular) cells are 1 minute, 1000 vehicles/minute,
and 3500 vehicles (3 minute, 1000 vehicles/minute, and 10500 vehicles).
13
There are two possible activity chains - home to work (H-W) and home to non-work
to work (H-NW-W). All individuals depart from home; departure times are set at every
fifth minute starting from (and including) 7:00 AM. The preferred work arrival time is
8:00 AM. The possible non-work activity durations are 5, 10, 15, 20, 25 and 30 minutes.
Since free-flow travel time is 25 minutes, 7:35 AM (7:30 AM) is assumed to be the latest
departure time for individuals participating in H-W (H-NW-W) activity chain. There
are 4 (8) travel route options available for activity chain H-W (H-NW-W). Therefore,
for the H-W activity chain there are (4x8=) 32 route options numbered from zero to
thirty one 1, while there are (8x27=) 216 route options for the H-NW-W activity chain
2numbered from thirty two to two hundred and forty seven. Three demand levels are
analyzed. Low, medium, and high demand representing 750, 3750, and 7500 individuals
participating in each activity chain combination are considered.
The utility profiles for the different activities are presented in Figure 4. Home stay
is rewarded with 100 utils/min. The utility derived from non-work activity partici-
pation are identical for all four locations; they depend on duration of participation
only. Starting at zero utils for zero minutes duration, the utility derived from every
additional minute increases linearly to a maximum of 125 utils/min for duration of 15
minutes and then drops linearly to 0 utils/min at 30 minutes of activity participation.
The utility is assumed to be independent of time of participation or flow on activity
arc. The preferred arrival time at work is 8:00 AM. Early arrivals are penalized at the
rate of -50 utils/min and late arrivals are penalized at -150 utils/min. Travel disutility
is -100 utils/min of travel.
Three different scale parameter ρvalues were tested. ρwas first assigned to an
initial value (1/n) and then progressively reduced to 1, 1/2, 1/3, 1/4... of the initial
value. Each value was held constant for niterations. For example when n= 10, the
initial value of ρis set to 0.1 for the first 10 iterations, then reduced to 0.05 (= 0.1/2)
1Includes departure time options 7:00, 7:05, ... 7:35 and 4 travel routes
2Including 4 locations, 2 travel routes, and 6 non-work activity duration options for individ-
uals departing at 7:00 AM and 7:05 AM, 5 for 7:10 AM departures, 4 for 7:15 AM departures
and so on - totaling to 27 activity duration combinations.
14
for iterations 11 through 20, then reduced to 0.033 (= 0.1/3) for iterations 21 through
30 and so on. Three different values of n(100, 1000, and 10000) were tested.
15
In terms of the overall results, the high demand case did not converge. It appears
that there is no equilibrium solution for the high demand case. The inverse of the initial
value of the scale parameter, n= 100 converged immediately (to a non-equilibrium
solution) and have not been included in the results presented below.
The convergence measure for n= 1000 and n= 10000 are presented in Figures 5
and 6. As can be seen from the figures, the low demand case converges to an equilibrium
very smoothly. There are a few spikes in the medium demand case (for both values of
n), and even more spikes for the high demand case. Further, the rate of convergence
16
is faster for the lower value of n(higher value of scale parameter ρ). This is consistent
with the result in Szeto and Lo (2006).
As mentioned earlier the high demand case did not have any equilibrium solution.
However, in general the algorithm progressed such that the maximum flow was loaded
on to the route with maximum utility. This result was observed only for n= 10000.
Even though the rate of convergence is faster for n= 1000, the solution obtained for
the high demand case was poor. That is the maximum flow was not among routes with
maximum utilities.
The equilibrium flows and corresponding average utilities for both H-W and H-NW-
W activity chains for all three demand levels are presented in Table 1. The solutions
correspond to converged values for the low and medium demand cases and 100,000
iterations for the high demand case. The value of nused here is 10000. The low and
medium demand case have converged to an equilibrium solution since almost 99.9%
of the flows are in routes that provide maximum utility. In the high demand case,
however, the utility at equilibrium for almost all used paths in the H-W activity chain
and for 98.3% of flow for the H-NW-W activity chain are equal. About 1.7% of the flows
in the H-NW-W activity combination have utilities lesser than the maximum possible
utility - this could be because of the capacity restrictions not allowing these flows
to shift to a more attractive path without adversely affecting the utilities of others.
The presence of unequal flows even among symmeteric paths - 31, 7, 23 and 15 for
example - (notwithstanding approximations arising from sequential flow propagation
steps) indicates the possibility of non-uniqueness of solutions in terms of path flows in
DUE.
An intuitive result obtained in the example in Section 2 is also corroborated here.
The intuition is that individuals would prefer participating in the non-work activity at
the location in B-diamond instead of R-diamond to avoid queuing at the bottleneck link
joining the two diamonds. We observe this in Table 1b. The flows on routes shown in
the table accounts for 98.3% of equilibrium flow and all individuals contributing to this
flow prefer the non-work activity center in B-diamond (cells 30 and 31 shown in Figure
3). In terms of durations, individuals in the H-W activity chain prefer departing as late
as possible (7:35 AM) while few depart at 7:30 AM. For H-NW-W activity chain, 7:30
AM departure and a non-work activity duration of 5 minutes is the most preferred.
In summary, the route-swapping algorithm performed reasonably well. Even for
high demand cases when there is no equilibrium solution the algorithm approaches a
‘good’ solution that ensures more flow on higher utility paths. However as demand
increases the convergence is not smooth - there are several spikes as seen in Figures 5
and 6. A reason for the spikes may be because of the discrete time capacitated network
considered here - a small shift in flow to some paths may result in substantial delays
and disutilities. Also, the convergence value was found to depend on the ρparameter
though clear patterns were not obtained. These remain important issues for future
studies.
17
7 Summary and Further Work
In this paper, an integrated formulation to obtain equilibrium solutions across multiple
dimensions of travel choice is presented. The formulation is based on a Supernetwork
representation referred to as Activity-Travel Network (ATN) representation. In ATN
representation, nodes are activity centers that are joined by travel links. Activities
are represented by arcs that both originate and terminate in the same node (activity
centers). An activity-travel sequence for an individual can be represented as a ‘route’
that includes both travel and activity arcs. The ‘route’ choice in an ATN results in
simultaneous determination of activity location, time of participation, duration, and
route choice decisions.
18
The proposed integrated framework allows (a) to capture activity demand-supply
dynamics in addition to transportation demand-supply dynamics, and (b) to obtain a
consistent equilibrium solution across all dimensions of choice. A rigorous mathemat-
ical and operational framework for ATNs based on dynamic user equilibrium behav-
ior with an embedded cell-based transmission traffic flow model was presented. The
equivalent variational inequality problem was obtained. A solution method based on
route-swapping algorithm is proposed and demonstrated on an example network.
Several open issues merit further investigation: first, we need to derive the prop-
erties such as solution existence and uniqueness of the variational inequality problem.
Second, numerical or analytical results on convergence properties of solution algorithms
need to be developed. This would depend on the utility function specification and the
traffic flow dynamic model among other factors. Third, more sophisticated representa-
tion for the utility function and the activity-travel choice mechanism can be explored.
Finally, the solution algorithm presented here did not converge smoothly for higher
demand values (more congested cases). While this can be improved by adopting a finer
resolution of time discretization, it leads to increase in the number of route alterna-
tives. Algorithms that obviate the need for route enumeration can solve the problem
significantly faster. Faster solution algorithms will allow the Activity-Travel Network
framework to be adopted in real-time traffic management applications even in large
scale networks.
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