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8/3/2019 Utsp Seminar
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STOCHASTIC USEREQUILIBRIUM
SUBMITTED BY
SUDHA DAS KHAN 11ID60R18
JAYASHREE.A 11ID60R24
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TRAFFICASSIGNMENT
Traffic assignment, the selection of routes
(alternative called paths) between origins and
destinations in transportation networks.
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TRAFFIC ASSIGNMENT MODEL
Traffic assignment model has three submodels:
1. Supply model: simulates variation in network
performance due to user behaviour
2. Demand model: simulates variation in user
behaviour due to network performance
3. Supply-demand interaction model: simulates the
user network interaction
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TRAFFIC ASSIGNMENT BASED ON
1. Dynamic approach(within day or day to day)
2. Utility approach(deterministic or stochastic)
3. Service regularity(regular or irregular)
4. User information system(present or not)
5. User type(frequent or occasional)
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UTILITY THEORY
The utility to a individual traveler offered by given travel choice alternativedepends on:
speed
safety cost
travel time
translated into its monetary value of traveler.
The traveler wishes to maximize his/her utility
DETERMINISTIC UTILITY THEORY
o The utility of a traveler is the function of travel choice alternatives.
o It is assumed that the function remains constant for a particular route or
for a particular class of people.o Uij = bij cij tij
o
It is not possible to predict with certainty the alternative that the genericdecision maker will select
Uij utility received from trip made between i and jbij income of the travelercij cost of travel - parameter which converts time into monetary value
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STOCHASTIC UTILITY THEORY
The utility function takes into account the individual strength of preference for
route choice
It is not possible to predict the alternative of the traveler with certainty
Hence it is expressed in terms of probability
The perceived utility Uij, expressed as sum of systematic utility(Vij) and random
residual(ij)
Uij = Vij +
ij
Vji = E[U
ij]
2i,j = Var[Uij ]
And therefore,
E[Uij] = Vj
i Var[Vj
i
] = 0
E[ij] = 0 Var[ij] =
2i,j
The probability is given by,
Pi[j/Ii] = Pr[Vij +
ij > V
ik +
ik] = Pr[V
ij - V
ik >
ik -
ij]
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TRAFFIC ASSIGNMENT BASED ON UTILITYAPPROACH
DETERMINISTICAPPROACH
STOCHASTICAPPROACH
It does not take into account theoverlapping of the path
Simple structure and ease of use
It takes into account theoverlapping at the cost of pathenumeration
The drawbacks of modified logitmodel and taken careRequires Monte Carlo Simulation orof complete path enumeration andnumerical integration of the
multivariate Normal distribution
All or nothingassignment
Method of
Successive Average
Modified Algorithm
Capacity RestraintAssignment
IncrementalAssignment
Logit Model
Modified LogitModel
Probit Model
STOCHASTICAPPROACH
Logit Model
Modified LogitModel
Probit Model
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STOCHASTIC USER EQUILIBRIUM(LOGIT MODEL)
WHICH PATH IS CHEAPEST;
SHORTEST; LESSCONGESTED..
STOCHASTICLOADING METHOD CAPACITYRESTRAINT
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The proportion of traffic between O-D pair(r,s) choosingto use route kwhich has a mean cost of Crs
k is given by,
the cost term are independent Gumbel variate
is the spread parameter
The model has Markovian nature, then
B(j) set of before nodes for node j and cij is costof link
exp( )
exp( )
k
k rs
jrs
rs
j
CP
C
( )
exp( )rj ri ij
i B j
jW W C
Stochastic Loading Method
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Flow on link(i,j) from origin ris given by
Xrj outflow of traffic with origin rto nodej
Satisfaction function, the mean perceived cost of
travel between O-D pairs(r,s)
exp( ) / rjij
rj
WijX X C
1 1log( ) log exp( )
k
rs rs rsk
S W C
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CAPACITY RESTRAINT
Here capacity restraint in the form of link-cost flow
function
The link flow at iteration n+1 is
xn+1=xn+n[y(n)-x(n)]
n - step length (1/n+1) (Method of successive average)
y(n) all or nothing assignment solution
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STOCHASTIC USER EQUILIBRIUMALGORITHM
Step 1: Set up an initial flow pattern xij, typically by carrying out a
stochastic loading based on free flow link costs
Step 2:
Compute new link costs c(n) based on the current flow
pattern x(n)
Carry out a stochastic loading using these link costs,
producing an auxiliary flow pattern y(n)
Using the output values of Srs and the vector of partial
derivatives z/ xa, compute z0 and g0
0
( ) ( ) ( ) ( )ax
a a a a rs rs
a a rs
z x C x dx x c x q S x ( ) ( )( ) ( )
( ) ( )n na a
a a a a
a
dz x dc xg x y y x
d dx
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Step 3: Compute link cost based on auxiliary solution and
use these to determine z1 and g1
Step 4: using z0, z1 and g0, g1, estimate the optimal step
length n, n = -g0/(-g0+g1)
update the current solution,
xn+1=xn+n[y(n)-x(n)]
Step 5: at stochastic user equilibrium the auxiliary and
current flow pattern are the same
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CONCLUSION
The most widely used algorithm is Method of
Successive Average
Inspite of its drawbacks, the logit method is widelyused
Many algorithm are also developed to take care of
the drawbacks in logit model
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REFERENCE
ALGORITHMS FOR LOGIT-BASED STOCHASTIC USEREQUILIBRIUM ASSIGNMENT MIKE MAHER - Department ofCivil and Transportation Engineering, Napier University,Edinburgh EH10 5DT, U.K.
Transportation System Analysis: Models and Application byEennio Cascetta
Transportation System Engineering: theory and method byEennio Cascetta
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EXAMPLE:
Network consisting of three parallel links between asingle O-D pair. The link cost flow relations are:
c1=3+x1 c2=2+2x2 c3=2.5+1.5x3
and the demand is one.
n z(x) x1(n) x2
(n) x3(n)
0 -1.6055308 0.1863237 0.5064804 0.3071959
1 -1.6267193 0.2514581 0.4058328 0.3427091
2 -1.6268206 0.2588580 0.4062035 0.3349384
3 -1.6268212 0.2591917 0.4056907 0.3351176
4 -1.6268212 0.2592326 0.4056927 0.3350746
5 -1.6268212 0.2592345 0.4056899 0.3350756
6 -1.6268212 0.2592347 0.4056899 0.3350754