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Modeling Traffic in St. Louis By Julia Greenberger
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?
