1. Traffic Assignment Part II CE 573 Transportation Planning Lecture 17

2. Objectives • Understand user equilibrium • Execute a link cost update • Communicate capacity restrained traffic assignment concept • Execute incremental assignment Michael Dixon

3. User Equilibrium Principles • Principle A: Under equilibrium conditions traffic arranges itself in congested networks in such a way that no individual trip maker can reduce his path costs by switching routes. • Principle B: Under equilibrium conditions traffic arranges itself in congested networks such that all used routes between an O-D pair have equal and minimum costs while all unused routes have greater or equal costs (to the used path costs). Michael Dixon

4. User Equilibrium Mathematical Form Michael Dixon

5. Updating Link Costs • Traffic assignment: assign trips to links • Gives volume on links • Input link volumes into link cost functions • Update link costs Michael Dixon

6. Updating Link Costs (Example) • Using equations of this form for link cost for each link (travel time): • Where • ta link travel cost per unit distance on link a, • toa free flow travel time per unit distance on link a, • Va flow on link a, • Qpa practical capacity of link a, and • α and β calibration parameters for link cost (α=1.0 and β=3). • Given the volume estimated from All-Or-Nothing (AON) assignment (last class example) • Update the link costs Michael Dixon

7. Updating Link Costs (Example) Michael Dixon

8. Update P-Matrix and Compare to Previous Matrix • Which links are, and are not, used? • If you were to assign traffic with the updated trees, would it be the same as it was before? • Which traffic assignment is correct? • Should you combine the two traffic assignments (before and after the update)? If so then how? Michael Dixon

9. Capacity Restrained Traffic Assignment Algorithms • Incremental Assignment • Method of Successive Averages • Frank-Wolfe Algorithm Michael Dixon

10. General Traffic Assignment Approach • n = n + 1 • Get previous solution (tan and Van-1) • Update minimum cost path trees • Do AON to get Fanlink volumes given previous link costs • Search direction: (Fan - Van-1) • Step size: (Φ) • Update link volumes (Van) • Update link costs (tan) Michael Dixon

11. Incremental Assignment Concept • The trip matrix T is assigned in proportions. • Properties of the proportions z • Each proportion 0 ≤ zn ≤ 1 • All proportions • After a proportion of T is assigned to the network • link volumes are updated • link costs are updated • Assignment is complete when all proportions (1 through N) are assigned Michael Dixon

12. Incremental Assignment Algorithm • Initialization of network database: • 1) get set of link costs (most likely freeflow) • 2) set all link volumes to 0, • 3) select set of proportions • 4) begin at first proportions(n = 0). • Set n = n + 1 • Update the minimum cost trees • All-or-nothing assignment (Fan) • Update the link flowsVan = Van-1 + φ Fan, where φ = zn. • Update link costs (tan) based on Van and go to step 2. Michael Dixon

13. Incremental Assignment Application • Proportions: Say there are two • z1 = 0.6 • z2 = 0.4 • Use previous example network • See Excel file “Incremental Assignment Example.xls” • What differences do you see between the AON file and the Incremental file? • Why did these differences occur? • Did we reach equilibrium? Michael Dixon

14. Method of Successive Averages Concept • Find previous solution • Find current AON solution • Establish a search direction • Establish a step size = 1/(number of iterations) • Calculate the current solution Michael Dixon

15. Method of Successive Averages Algorithm • Initialization of network database: • 1) get set of link costs (most likely freeflow) • 2) set all link volumes to 0, • 3) set iteration number (n = 0). • Set n = n + 1 • Update the minimum cost trees • All-or-nothing assignment (Fan) • Update the link flows, where φ = 1/n • Update link costs (tan) given Van. • Check for convergence; if no convergence then go to step 2. Michael Dixon

16. Method of Successive Averages Application • Use previous example network • See Excel file “MSA Assignment Example.xls” • What differences do you see between the AON file and the MSA file? • Why did these differences occur? • Did you reach equilibrium? Michael Dixon