Modeling Traffic in St. Louis

By Julia Greenberger

GoalsTo create a model of the traffic flow of cars

traveling from Creve Coeur to downtown St. Louis

To use this model to determine the maximum flow of cars from Creve Coeur to downtown St. Louis

To predict the change in traffic flow on Forest Park Parkway once Highway 40 (I-64) reopens

St. Louis Map with Construction

Creating the ModelUse 13 nodes to keep model manageable

Use 18 links between these nodes to have 18 unknown variables

Map with Routing

1 23

45

678

11

9

10

1312

Simplified Routing Map

12

7

4

3

5 9 13

10

126

8

11

Creating the Model (cont.)Find the maximum capacity of cars on the streets used

in the model using

bi,j = # of cars ≈ (# of lanes)*(speed limit)*(c),

Where bi,j is the maximum capacity of the street from node i to node j and i,j:1-13

and c=traffic coefficient.

c=1; no traffic, greenc=.75; medium traffic, yellowc=.5; heavy traffic, red

Map of Traffic FlowUse map to find c

Routing Map with Maximum Road Capacities

12

7

4

3

5 9 13

10

126

8

1130.5

45

240

240

240

18.7

48

240

240

25.568

68

68 25

48

25.5 25.5

Creating the Linear Program

Let Xi,j = the number of cars traveling from node i to node j, where i,j: 1-13

We want to maximize X1,2 + X2,3 + … + X12,13

Let X=[X1,2; X2,3;… ; X12,13 ]

To maximize the sum of the entries in X, we can maximize

CT*X, where C=[1;1;…;1]

or we can minimize

CT*X, where C=[-1;-1;…;-1]

Creating the Linear Program

Assume the number of cars entering a given node is equal to the number of cars exiting that node

Create a matrix A, with equations that balance the flow in and out of each node

A = [ …0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,1;

…]To balance flow in and out of node, A*X=0

Using the constraint vector, Xi,j ≤ bi,j

Creating the Linear Program

Minimize CT*X, where C=[-1;-1;…;-1]Subject toi) A*X=0ii) Xi,j ≤ bi,j

Solve using linprog in MATLAB

Results from Linear Program

Maximum flow in total system is 30 cars

Flow is limited by some streets with very small Xi,j

Modifying Linear Program

12

7

4

3

5 9 13

10

126

8

1130.5

45

240

240

240

18.7

48

240

240

25.568

68

68 25

48

25.5 25.5

240

ResultsThe maximum flow in total system did not

change

The flow on Forest Park Parkway decreased from 15 to 12.3 cars

Model supports the hypothesis that the opening of Highway-40 will decrease traffic flow on local streets

LimitationsWe only used 13 nodes

In reality, there are hundreds of nodes from Creve Coeur to downtown St. Louis

Uncertainty in traffic coefficients

Questions?