1. TRB Planning Applications Conference May 2009, Houston,TX A Caveat on O-D Matrix Estimation/Adjustment: Deviations from a seed matrix and Simultaneous multi-class adjustments Michael Florian and Yolanda Noriega CIRRELT and INRO

2. TRB Planning Applications Conference May 2009, Houston,TX Contents of presentation 1. Motivation 2. The gradient method for adjusting O-D matrices 3. Deviations from the matrix to be adjusted 4. Using deviations from the reference O-D matrix 5. Extension of the method to multi-class adjustments of O-D matrices 6. Conclusions

3. TRB Planning Applications Conference May 2009, Houston,TX Motivation –The use of counts to adjust out of date origin- destination matrices is a commonly used method; –Practically all the transportation planning software packages offer a way to adjust matrices; –The benefits are obvious: the counts are reliable and far less expensive than an O-D survey;.

4. TRB Planning Applications Conference May 2009, Houston,TX Motivation –The analysis of sub-areas that are then used for dynamic traffic assignment or micro-simulation applications benefit from an adjusted O-D matrix that better replicates counts; –Other traffic operations applications benefit from more reliable turning movements that can be obtained as a by-product of the O-D adjustment.

5. TRB Planning Applications Conference May 2009, Houston,TX More an Art than a Science –The adjustment of an O-D matrix by using counts requires good judgment in addition to the method used for the adjustment; –The quality and consistency of the counts must be analyzed and verified; –The network coding and the volume/delay functions used should be free of errors, as much as possible;

6. TRB Planning Applications Conference May 2009, Houston,TX More an Art than a Science –Finally, the adjustment process should not distort the structure of the O-D matrix that is being adjusted; –A balance must be achieved between the fit to the counts and the changes of the O-D matrix that result from the adjustment.

7. TRB Planning Applications Conference May 2009, Houston,TX The Gradient Method for O-D Adjustment –Let’s recall what is being done in this O-D adjustment method: –The inputs are the link counts and the existing O- D matrix; –The aim is to obtain an assignment that fits the counts as best as possible; –The objective then is to obtain the smallest value of the difference between the assigned flows and the counts:.

8. TRB Planning Applications Conference May 2009, Houston,TX The Gradient Method for O-D Adjustment Find new O-D matrix g such that the objective function SUM (assigned flows(g) – counts)^2 is as small as possible. SUM is over all links that have counts –The adjusted matrix g is computed with a method that is based on computing the rate of decrease of the objective function. –Constraints can include screen lines, cordon counts, productions, attractions…

9. TRB Planning Applications Conference May 2009, Houston,TX The Gradient Method for O-D Adjustment Step 0. Initialization; Step 1. Equilibrium assignment to obtain the link volumes; Step 2. Computation of the link derivatives and the objective function. Step 3 Computation the gradient matrix; Step 4. Computation of the link derivatives.; Step 5. Compute the maximal gradient; Compute the optimal step length; Update the demand matrix; Step 6. Update iteration counter; If the maximum number of iterations is reached STOP. otherwise go to Step 1;

10. TRB Planning Applications Conference May 2009, Houston,TX But considering only the link counts may not be enough –An example of a matrix adjustment for the City of Montreal illustrates the issue: –The heavy links indicate the location of counts

11. TRB Planning Applications Conference May 2009, Houston,TX Montreal flow comparison for SOV’s without adjustment Link R^2=.89

12. TRB Planning Applications Conference May 2009, Houston,TX Montreal flow and O-D matrixcomparison for SOV’s after adjustment Link R^2=.99 O-D R^2=.89

13. TRB Planning Applications Conference May 2009, Houston,TX How can one control the deviations of the O-D matrix? –It is possible to introduce a demand term in the O-D matrix adjustment objective: Find new O-D matrix g such that the a weighted sum (weight) [SUM (Adj.O-D – Initial O-D)^2] +(1-weight) [SUM (assigned flows(g) – counts)^2] where weight is less than 1, is as small as possible. SUM is over all links that have counts –The adjusted matrix g is computed with a similar method that is based on computing the rate of decrease of the objective function by using the gradient method.

14. TRB Planning Applications Conference May 2009, Houston,TX Results obtained by varying the weight O-D R^2=.91Link R^2=.96

15. TRB Planning Applications Conference May 2009, Houston,TX Results obtained by varying the weight O-D R^2=.99Link R^2=.93

16. TRB Planning Applications Conference May 2009, Houston,TX Original link fitConsidering the demand term

17. TRB Planning Applications Conference May 2009, Houston,TX Best link fitConsidering the demand term

18. TRB Planning Applications Conference May 2009, Houston,TX Which is the right adjustment? The analyst is the judge! The ability to inspect the structure of the adjusted O-D matrix is important. (It is also possible to give weights for particular link counts and elements of the O-D matrix to be adjusted).

19. TRB Planning Applications Conference May 2009, Houston,TX Considering only the link counts may not be enough: another example

20. TRB Planning Applications Conference May 2009, Houston,TX O-D matrix comparisons weight=0weight=0.05

21. TRB Planning Applications Conference May 2009, Houston,TX Link flow comparisons weight=0 weight=0.05

22. TRB Planning Applications Conference May 2009, Houston,TX O-D matrix comparisons weight =0.1weight=0.2

23. TRB Planning Applications Conference May 2009, Houston,TX Multi-Class OD Matrix adjustment Over the past 15 years, the use of multi-class equlibrium assignments has become quite common; So there is an interest to extend the gradient method for simultaneous multi-class OD matrix adjustment; The gradient method was extended for multi-class OD matrix adjustment It is implemented and sample results using data from the Montreal region network will be shown.

24. TRB Planning Applications Conference May 2009, Houston,TX Extension of the method to multi-class adjustments of O-D matrices Find new O-D matrices g(c) for each class such that the a weighted sum (weight) [SUM (Adj.O-D – Initial O-D) 2 ] +(1-weight) [SUM (assigned flows(g) – counts) 2 ] where weight is less than 1, is as small as possible. SUM is over classes and links that have counts

25. TRB Planning Applications Conference May 2009, Houston,TX The Multi-Class Method Step 0. Initialization; Step 1. Multi-class assignment to obtain the link volumes; Step 2. Computation of the link derivatives and the objective function; Step 3. Compute the gradient matrices for each class; Step 4. Obtain the link flow derivatives; Step 5. For each class Compute the maximal gradient; Compute the optimal step length; Update of the demand matrices; Step 6. Update the iteration counter If the maximum number of iterations is reached STOP; Otherwise go to Step 1.

26. TRB Planning Applications Conference May 2009, Houston,TX We tried three approaches : The first approach is the multi-class adjustment where the demand for every class is adjusted iteration by iteration (MC Adjustment); The second approach consists on adjusting the demand for one class at the time, leaving however the flows of all classes variable during the assignments (SEQ Adjustment); Similarly, in the third approach the demand of one class is adjusted at the time, but here the volumes of the other classes are considered as fixed (FIX Adjustment). Three Approaches for Multi-Class Adjustments

27. TRB Planning Applications Conference May 2009, Houston,TX Computational results

28. TRB Planning Applications Conference May 2009, Houston,TX Computational results

29. TRB Planning Applications Conference May 2009, Houston,TX Computational results

30. TRB Planning Applications Conference May 2009, Houston,TX Auto Demand to be Adjusted

31. TRB Planning Applications Conference May 2009, Houston,TX MC Adjustment of Auto Demand

32. TRB Planning Applications Conference May 2009, Houston,TX Regular Trucks to be Adjusted

33. TRB Planning Applications Conference May 2009, Houston,TX MC Adjustment of Regular Trucks Demand

34. TRB Planning Applications Conference May 2009, Houston,TX Demand to be Adjusted Heavy Trucks

35. TRB Planning Applications Conference May 2009, Houston,TX MC Adjustment of Heavy Trucks Demand

36. TRB Planning Applications Conference May 2009, Houston,TX MC Adjustment of Heavy Trucks Demand 100% on flows, cars 99.99% on flows, cars

37. TRB Planning Applications Conference May 2009, Houston,TX MC Adjustment of Heavy Trucks Demand 99.97% on flows, cars 100% on flows, regular trucks

38. TRB Planning Applications Conference May 2009, Houston,TX MC Adjustment of Heavy Trucks Demand 99.97% on flows, regular trucks 99.99% on flows, regular trucks

39. TRB Planning Applications Conference May 2009, Houston,TX Conclusion The adjustment of O-D matrices is a process that should be done carefully with inspection of the adjusted matrix; comparison with the matrix to be adjusted is important; One can carry out the simultaneous adjustment of the O-D matrices for several classes.

40. TRB Planning Applications Conference May 2009, Houston,TX THE END