utsp seminar

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

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