1. Modeling Traffic in St. Louis By Julia Greenberger

2. Goals To 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

3. St. Louis Map with Construction

4. Creating the Model Use 13 nodes to keep model manageable Use 18 links between these nodes to have 18 unknown variables

5. Map with Routing 1 2 3 4 5 6 7 8 11 9 10 13 12

6. Simplified Routing Map 1 2 7 4 3 5 9 13 10 12 6 8 11

7. Creating the Model (cont.) Find the maximum capacity of cars on the streets used in the model using b i,j = # of cars ≈ (# of lanes)*(speed limit)*(c), Where b i,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, green c=.75; medium traffic, yellow c=.5; heavy traffic, red

8. Map of Traffic Flow Use map to find c

9. Routing Map with Maximum Road Capacities 1 2 7 4 3 5 9 13 10 12 6 8 11 30.5 45 240 18.7 48 240 25.5 68 25 48 25.5

10. Creating the Linear Program Let X i,j = the number of cars traveling from node i to node j, where i,j: 1-13 We want to maximize X 1,2 + X 2,3 + … + X 12,13 Let X=[X 1,2 ; X 2,3 ;… ; X 12,13 ] To maximize the sum of the entries in X, we can maximize C T *X, where C=[1;1;…;1] or we can minimize C T *X, where C=[-1;-1;…;-1]

11. 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, X i,j ≤ b i,j

12. Creating the Linear Program Minimize C T *X, where C=[-1;-1;…;-1] Subject to  A*X=0  X i,j ≤ b i,j Solve using linprog in MATLAB

13. Results from Linear Program Maximum flow in total system is 30 cars Flow is limited by some streets with very small X i,j

14. Modifying Linear Program 1 2 7 4 3 5 9 13 10 12 6 8 11 30.5 45 240 18.7 48 240 25.5 68 25 48 25.5 240

15. Results The 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

16. Limitations We only used 13 nodes In reality, there are hundreds of nodes from Creve Coeur to downtown St. Louis Uncertainty in traffic coefficients

17. Questions?