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INFORMS, November 2005 1
Network Equilibrium Models:Varied and Ambitious
Michael Florian
Center for Research on Transportation
University of Montreal
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INFORMS, November 2005 2
The applications of network equilibriummodels are varied:
They range from simple to the very complex
- single mode, single class equilibrium assignment - multi-class equilibrium assignment - generalized cost on road and transit networks - equilibrium assignment on congested transit networks - path analyses - complex multi-modal equilibration - combined mode trips - dynamic network equilibrium models
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INFORMS, November 2005 3
The single mode single class networkequilibrium model
( )0
minav
aa A
s x dx∈∑ ∫
subject to , ,
0, ,
( , )
ci
i
k ik K
k i
a ak kk K
h g i I
h k K i I
v h a Aδ
∈
∈
= ∈
≥ ∈ ∈
= ∈
∑
∑
kh
Numerous algorithms have been developped for its solution;As is well known the arc flows , are unique, but the pathflows , are not unique.
av
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INFORMS, November 2005 4
Some selected applications
• The presentation includes examples of the applicationof various network equilibrium models carried outaround the world
• The first set of applications was carried out with staticmodels of increasing complexity both for road andtransit networks
• The second set of applications presents new resultswith a dynamic network equilibrium model
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INFORMS, November 2005 5
Some Straightforward Applications
• Madrid, Spain• Pretoria, South Africa• Auckland, New Zealand
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INFORMS, November 2005 6
Madrid- AM Peak Flows and Speeds
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INFORMS, November 2005 7
Regional Study of the Province ofGauteng, South Africa
Study carried out byVela VKE – Pretoria, South Africa,
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INFORMS, November 2005 8
Scenario Without New facility
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INFORMS, November 2005 9
Scenario with New facility
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INFORMS, November 2005 10
Scenario comparison
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INFORMS, November 2005 11
Planning Transport in Auckland
Study carried out by AucklandRegional Council, Auckland, NZ
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INFORMS, November 2005 12
AM Vehicle Flows, 2001
36%
59%
5%MotorwaysArterialsLocal
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Complex Variable Demand Network EqulibriumModels
INRO
Demand Model
EquilibriumTrip Assignment
Impedance
Step sizeMechanism
• The “Step Size Mechanism” maytake different forms depending on theknowledge that one has of theunderlying model.
• Does the model have an equivalentconvex cost optimizationformulation?
• Can the model be formulated as avariational inequality?
• The model is very complicated andone carries out “ad-hoc” feedback bysome averaging scheme.
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INFORMS, November 2005 14
• Some well known variants of such models are the Combined Distribution-Assignment Model , Combined Distribution-Assignment-Mode ChoiceModel, Equilibrium Assignment Model with Variable Demand,...
• One can use adaptations of nonlinear programming algorithms to obtain thesolution of such models
• The “Step Size Mechanism” may be trivially stated to be the result of a linesearch on the objective function
where xk is a current solution and dk is a direction of descent
Complex Variable Demand Network EquilibriumModels:
Equivalent Convex Cost Optimization Formulations
INRO
1
0 1min ( ( )), ( )k k k k k k kF x d x x x d x
λλ λ+
≤ ≤+ − = + −
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INFORMS, November 2005 15
It is well known that models are with asymmetric cost functions, such asintersection delays, transit travel time depending on auto travel times in multi-mode models can be formulated as :
Network EquilibriumModels:Variational Inequality Formulations
* *( )( ) 0a a as v v v− ≥
subject to , ,
0, ,
( , )
ci
i
k ik K
k i
a ak kk K
h g i I
h k K i I
v h a Aδ
∈
∈
= ∈
≥ ∈ ∈
= ∈
∑
∑
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INFORMS, November 2005 16
•Such models may be solved by a variety of algorithms. Often the sufficientconditions for convergence are impossible to verify
• A common heuristic method used in practice is the Method of SuccessiveAverages
•The “Step Size Mechanism” may be related to the averaging of the link costs orthe averaging of link flows
where T(xk) is the computed procedure that is used to obtain the next iterate
Network EquilibriumModels:Variational Inequality Formulations
x x x x x T xk kk
k k k k
kk
k
+ + +
=
∞
= + − =
< < − = +∞∑
1 1 1
1
0 1 1
α
α α α
( ) ( )
( )
;
;
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INFORMS, November 2005 17
A Complex Model:Rigorous Formulation-Heuristic Solution Algorithm
• Santiago, Chile• Complex demand model• Road Network Equilibrium• Transit Network Equilibrium• Combined modes: road-transit; transit-transit
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Santiago, Chile Strategic Planning Model
• Base network
• 409 centroids including 49 parking locations
• 1808 nodes, 11,331 directional links
• 1116 transit lines and 52468 line segments
• 11 modes, including 4 combined modes
(bus-metro, txc-metro, auto-metro and auto passenger-metro)
• The demand
• subdivided into 13 socio-economic classes
• 3 trip purposes ( work, study, other )
• driving license holders can access to 11 modes
• no license holders can access to 9 modes
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Base Network of Santiago, Chile
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Variational Inequality Formulation
Find ( h T* *, ) ∈Ω such that
pn ij g G m ggp
rpnm
r rr
pn ijpng
ijpng
ijpng
gp
m gijpnm
ijpng
ijpnm
ijpnm
p
C h T h h T T T
T T T T h T
∑ ∑ ∑ ∑ ∑
∑⊆ ∈
∈
− + − +
− ≥ ∀ ∈
( )
* * * * *
* * *
[ ( , )( ) ln ( ))
ln( / )( ) , ( , ) .
φβ
φ
1
0 Ω
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Trip Ends and Conservation of FlowConstraints
jij
pngipn
g
T O i p n∑∑ = ∀, , , ( )α ipn
inij
pngjp
g
T D j p∑∑∑ = ∀, , ( )ξ ip
T T j p n gijpnm
ijpng
m g
− = ∀∈
∑ 0, , , , ( )Lijpng
φ gp ( ) , , , ,h T j p n gr
pnmijpnm
r R m
− = ∀∈∑ 0 ( )µ ij
pnm
h r p n mrpnm ≥ ∀0, , , , ( )γ r
pnm
T ij p n mijpnm > ∀0, , , ,
T ij p n gijpng > ∀0, , , ,
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INFORMS, November 2005 22
Network Equilibrium Models: car and transit
• Multi-class network equlibrium model• Multi-class transit network equlibrium model• Heuristic equilibration that resorts to averaging of
flows and travel impedances
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Solution Procedure
Multiple Transit Assignments
Standard Transit Assignment
for buses
Trip Distribution and Mode Choice
MetroMSA Impedance
Equil. AutoAssignment
Congested time
Equilibrium Transit Assignment
for metro
BusImpedance
AutoImpedance
Park-and-Ride Model for auto-metro and bus-metro
Convergence?
end All Impedances
Start and Initialize
MSA Auto Volume
Transit Autoequivalent flow
Auto Skims
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INFORMS, November 2005 24
Santiago, Chile Strategic Planning Model
• The next slide shows the convergence of the MSA algorithmthat uses link flow averaging for the car network and traveltime averaging for the transit network.
• The convergence of both the car demand and link flows aregiven for two variants: uncongested transit assignment andequilibrium transit assignment.
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Convergence of equilibration
Normalized Gap vs. Iterationscongested vs. non-congested metro assignment with metro capacity
00.5
11.5
22.5
33.5
44.5
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Iteration
Nor
mal
ized
gap
(%
)
cong. Link vol. cong. demand non-cong. Link vol. non-cong. demand
Auto demand
Auto link volume
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Metro Volume (non-congested vs. congestedversion)
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Metro Volume Changes
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Auto-metro volume (non-congested vs.congested version)
Congested
Non-congested
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Metro Volume - metro 5A, non-congested version
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Metro Volum2 - metro 5A, congested version
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INFORMS, November 2005 31
Some Complex Applications
• Los Angeles, California• Toll Highways Poznan, Poland• Toll Bridge, Montreal,
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INFORMS, November 2005 32
The SCAG Regional TransportationPlanning Model
• A complex and very large scale model
• Lack of rigorous formulation; networkequilibrium sub-models
• A multi-class multi-mode network equilibriummodel with asymmetric cost function is part ofthe model
• Heuristic solution algorithm based on an outeraveraging scheme
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INFORMS, November 2005 33
SCAG MODELFLOW CHART:
START
auto skims forPK
auto skims forOP
transit skims forPK
transit skims forOP
HBW Logsums for PK (mc) HBW Logsums for OP (mc)
trip distribution for PK (gravity) trip distribution for OP (gravity)
mode choice model for PK mode choice model for OP
demand computations (time–of-day model)
auto-truck assignments for AM auto-truck assignments for MD
is the number of outer loops satisfied?
auto-truck assignments for PM auto-truck assignments for NT
transit assignments for AM transit assignments for MD
END
successive average link volume for outerloop of AM
successive average link volume for outerloop of MD
Trip generation
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INFORMS, November 2005 34
Network Overview - Highway Networkby facility type
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INFORMS, November 2005 35
Network Overview - Highway Networkwith parking lots
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INFORMS, November 2005 36
Equilibration Algorithm Convergence Results
SCAG Model Convergence (AM peak)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 1 5 9 13 17 21 25
loops
link
rela
tive
diffe
renc
e
inner-loop outer-loop
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INFORMS, November 2005 37
Assignment ResultsAM peak volume
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INFORMS, November 2005 38
Assignment ResultsAM link speed
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INFORMS, November 2005 39
Assignment ResultsAM truck volume by class
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INFORMS, November 2005 40
Assignment ResultsAM HOV VMT Grid Map
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INFORMS, November 2005 41
Assignment ResultsHCM Level of Service
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INFORMS, November 2005 42
Scenario ComparisonVMT changes
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INFORMS, November 2005 43
Toll highway analysis – Poznan, Poland
It involves the following models:
-Multi-class equilibrium assignment with generalizedcosts
-Demand models for toll-no toll choice and future yeardemands
-Equilibration of the demand for toll highway andnetwork performance
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INFORMS, November 2005 44
The multi-class equilibrium model withtolls
( )0
minav
c c ca a a
a A c C a As x dx v tθ
∈ ∈ ∈
+∑ ∑∑∫subject to , ,
0, ,
( , , )
ci
ci
ck i
k K
ck i
ca ak k
k K
h g i I c C
h k K i I
v h a A c Cδ
∈
∈
= ∈ ∈
≥ ∈ ∈
= ∈ ∈
∑
∑
khav
The numerical solution of this model is well known;It is worthwhile to point out that the flows by class, are not unique,nor are the path flows , but the arc flows are unique.
cav
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INFORMS, November 2005 45
Base year – No Toll
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INFORMS, November 2005 46
Base year – Medium Cost Toll
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INFORMS, November 2005 47
Base year – High Toll
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INFORMS, November 2005 48
Year 2010 – High Cost Toll
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INFORMS, November 2005 49
Year 2020 – Medium Cost Toll
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INFORMS, November 2005 50
Year 2010 – Low Cost Toll
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INFORMS, November 2005 51
Toll Bridge Study – Montreal, Canada
• Study carried out by the Ministry of Transportationof Quebec
• The analysis relied heavily on the analysis of pathsgenerated by the assignment algorithm
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INFORMS, November 2005 52
The Montreal Region
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INFORMS, November 2005 53
The new proposed bridge
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INFORMS, November 2005 54
Zones around Bridge
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INFORMS, November 2005 55
Demand for Current and Future Year
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INFORMS, November 2005 56
The Current Bridge Flows
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INFORMS, November 2005 57
Bridge Flows with New facility
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INFORMS, November 2005 58
Toll Income with Various Toll Levels
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INFORMS, November 2005 59
National Models
• These are very large multi-modal multi-class models• The underlying demand models are rather complex
and the running times are very high• An example of a national model is the PLANET
model developed for British Rail
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INFORMS, November 2005 60
The PLANET Model – British Rail
Zones Rail Road
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INFORMS, November 2005 61
Dynamic Network Equilibrium Model
• Solved by a hybrid optimization-simulation model adiscretized version of a variational inequality formulation of adynamic network equilibrium model
• The theoretical properties of the model are difficult to establish
• Wardrop’s user equilibrium in a temporal framework is a basisfor the model
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INFORMS, November 2005 62
( ) ( ) 0( )
( )
u t if h ti ks tk u ti
= >≥
otherwise ( ) ( )minu t s ti kk Ki
=∈
( ) ( )( )i
k ik K t
h t g t∀ ∈
=∑
Dynamic equilibrium
( ) 0kh t ≥
( )
( )( )
1,
i
k
t t T
i i I
g t i
s t t k
= ∈
= ∈
=
=
assignment interval
OD pair,
demand for OD pair
travel time ( ) on path
variables
constraints
equilibrium conditions
( )
( ) ( )
0,
i
k
T
I
K i
h t t k
=
=
=
=
demand period
set of OD pairs
set of paths for
flow on path
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INFORMS, November 2005 63
Dynamic assignment model
Networkdefinition
Time-dependentOD matrices
TrafficControl Data
STOP
calculate path flows (t)
run traffic simulation
choose initial paths
determine path traveltimes (t)
convergence achieved?YES NO
modify path sets:add a new path or
remove an existingpath (each O-D pair
& each interval)
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INFORMS, November 2005 64
Traffic simulation model
simplified model of vehicle interactions allows for anefficient event-based simulation
• car following• lane changing• gap acceptance
sophisticated lane selection heuristics• local lane selection rules• stochastic look-ahead strategy
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INFORMS, November 2005 65
Q
0
400
800
1200
1600
2000
0 90 K
flow
(veh
/hr/l
ane)
1
6030
-W
120
1
density (veh/km/lane)
V
free-flow speed
1
1
V
QL V R
K L
W L R
=
=
=
=
+
fundamental diagram
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INFORMS, November 2005 66
An application in Stockholm
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INFORMS, November 2005 67
An application in Stockholm
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INFORMS, November 2005 68
An application in Montreal
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INFORMS, November 2005 69
An application in Auckland,NZ
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INFORMS, November 2005 70
Ending Remarks
• The equilibrium model of route choice is hereto stay for both static and dynamic models
• We have a lot to thank to the landmarkcontribution of 1956