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 manageableUse 18 links between these nodes to have 18

unknown variables

Map with Routing

1 23

45

67 8 11

910

1312

Simplified Routing Map

1 2

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

1 2

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

4825.5 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

maximizeCT*X, where C=[1;1;…;1]

or we can minimizeCT*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 carsFlow is limited by some streets with very small

Xi,j

Modifying Linear Program

1 2

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

4825.5 25.5

240

ResultsThe maximum flow in total system did not

changeThe flow on Forest Park Parkway decreased from

15 to 12.3 carsModel supports the hypothesis that the opening

of Highway-40 will decrease traffic flow on local streets

LimitationsWe only used 13 nodesIn reality, there are hundreds of nodes from

Creve Coeur to downtown St. LouisUncertainty in traffic coefficients

Questions?