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 3 1 2 8 7 11 6 4 10 5 9 12 13

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

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

10. 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]

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, Xi,j ≤ bi,j

12. Creating the Linear Program • Minimize CT*X, where C=[-1;-1;…;-1] Subject to • A*X=0 • Xi,j ≤ bi,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 Xi,j

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

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?