traffic assignment telab

8/9/2019 Traffic Assignment TElab

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Indian Institute of Technology, Kharagpur

Urban Transportation Systems PlanningTraffic Assignment

Department of Civil EngineeringIndian Institute of Technology Kharagpur


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Different types of traffic assignment techniquesinclude:

All-or-nothing assignment

Incremental assignment

Capacity restraint assignment

Stochastic assignment

Stochastic user equilibrium assignment

System optimum assignmentDynamic assignment, etc.



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Link Cost Function As the flow increases towards the capacity of the

stream, the average stream speed reduces from

the free flow speed to the speed corresponding to

the maximum flow


The minimum path computed

prior to the trip assignment

will not be the minimum afterthe trips are assigned


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Network Properties and Algorithms

It is useful to understand some of the network

properties like network connectivity, minimum

spanning tree, shortest path, etc.



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Minimum Spanning Tree

Spanning Tree: A tree with required minimum

number of links to ensure connectivity of the

network i.e. (n-1) links to connect n nodes

Minimum spanning tree: A spanning tree withminimum total cost or length

In many system, such as highways, computer

network, telephone lines, television cables, etc, weneed to identify the minimum spanning tree

Example: Minimum road network which needs to

be developed and maintained throughout the year

to ensure connectivity of all nodes



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So many ways, 5 links can be selected to ensure

connectivity of all the 6 nodes

But, for the minimum spanning tree (shown by

yellow lines), the total cost is minimum


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Two important features of minimum spanningtree:

Possible multiplicity: There may be several

minimum spanning trees of the same weight

having minimum no. of edges; in particular, ifall weights are the same, every spanning tree is

minimum

Uniqueness: If each edge has a distinct weightthen there will only be one, unique minimum

spanning tree



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Kruskal's Algorithm This is a greedy algorithm

Take a graph with 'n' vertices

Keep adding the shortest (least cost) edge, whileavoiding the creation of cycles

until (n - 1) edges have been added



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Example

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Step 2


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Step 3


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Step 4


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Step 5


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Prims Algorithm

Start at any vertex in a graph (vertex A, for

example), and find the least cost vertex (vertex B,

for example) connected to the start vertex.

Now, from either 'A' or 'B', find the next leastcostly vertex connection, without creating a cycle

(vertex C, for example).

Now, from either 'A', 'B', or 'C', find the next least

costly vertex connection, without creating a cycle,

and so on

Eventually, all the vertices will be connected,

without any cycles, and an MST will be the result



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Step 2


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Step 3


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Step 4


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Step 5


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Shortest path Finding a path between two vertices (or nodes)

such that the sum of the weights of its constituent

edges is minimized

Formally, given a weighted graph (that is, a set Vof vertices, a set E of edges, and a real-valued

weight function f : E R), and one element v of V,

find a path P from v to a v' of V so that

f(p), p P, is minimal among all pathsconnecting v to v'



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Dijkstra's algorithm It solves the single-pair, single-source, and

single-destination shortest path problems



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Algorithm

Let's call the node we are starting as initial node

and from which Y be the distance to an other

node

Assign some initial distance values and try toimprove them step-by-step and assign to every

node a distance value; Steps are

Set it to zero for initial node and to infinity forall other nodes and mark all nodes as unvisited

Set initial node as current

For current node, consider all its unvisited

neighbours and calculate their distance



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For example, if current node (A) has distance of 6,

and an edge connecting it with another node (B) is

2, the distance to B through A will be 6+2=8

If this distance is less than the previously

recorded distance (infinity in the beginning, zerofor the initial node), overwrite the distance

When we are done considering all neighbours of

the current node, mark it as visited

A visited node will not be checked ever again; its

distance recorded now is final and minimal

Set the unvisited node with the smallest distance

(from the initial node) as the next "current node"



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Example


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Find out shortest path in this network using

Dijkstra's algorithm


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Solution

Step 1: set node 1 at zero and others at infinitive

Step 2: node 1 is now current node

Step 3: from node 1 distances of nodes 2, 3, 6 are

7, 9, 14 respectively and minimum is 7, so


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Now node 2 become the current node from which

the distance of node 3 is (7+10)= 17>9 and node 4is (7+15)= 22

The distance of node 3 from node 2 is more than

the distance from node 1, so


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(7+10)=17>922


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the distance of node 4 is (9+11)= 20


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the distance of node 5 is (11+9)= 20

And distance from node 5 to node 4 is (20+6)= 26

> 20, so


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(11+9)=20

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