MODELS AND METHODS FOR THE OPTIMAL LOCATION OF TRAFFIC SENSORS AND VMSs A. Sforza

1. MODELS AND METHODS FOR THE OPTIMAL LOCATION OF TRAFFIC SENSORS AND VMSs A. Sforza DIS - Università di Napoli “Federico II “ Corso di Ottimizzazione su Rete A.A. 2010/11

2. Outline of presentation • Context • Flow intercepting facility location problems • Applications in Traffic Management and Control • Optimization models proposed in literature • Computational experience • Proposals of new constraints • A simple heuristic and some improving modifications • Application to Traffic Network in Naples

3. Facility Location Problems - Flow generating and/or attracting facilities vertex – point – path - Flow intercepting facilities in vertices – on links

4. Flow generating and/or attracting facilitiesvertex location

5. Flow generating facilitiesService reaches the clients or vice-versa

6. Flow intercepting facilities

7. Flow intercepting facilities

8. Flow intercepting facilitiesO-D demand flows

9. Flow intercepting facilitiesin the vertices (two facilities)

10. Flow intercepting facilitieson the links (three facilities)

11. The flow intercepting facility location problem is a problem of path covering

12. Applications in Traffic Management and Control • Location of: • Traffic counting sensors (for o-d matrix estimation) • To know a set of link flows or all the link flows • Variable message systems • Fixed • Mobile • Traffic checkpoints

13. Applications – Service Facilities A classification scheme • Voluntary service facilities • Car service stations, automatic teller machine • Unconscious service facilities • Traffic counting sensors • Unvoluntary service facilities • Variable message signs • Compulsory service facilities • Traffic check points • Inspection Stations

14. Traffic Management and Control Applications Traffic counting sensors No need of double counting Variable message systems There could be the need of double (or more) intercepting

15. m = 2 facilities No double counting Double counting for path p2 p1 p2 p3 p1 p2 p3

16. Available information - Information on path flows • Information on link flows • Assumption The flow pattern is not modified by facility location This is surely true for traffic sensors It could be not true for VMS

17. yj = 1 0 if there is a facility located at node j otherwise, j  N xp= 1 0 if at least one of the facilities is located on path p otherwise, p  P Information on path flows- Problem variables G= (N, A) N, set of vertices; A, set of links p path, P set of paths

18. Model P1: Maximization of the intercepted flow with a fixed number of facilities

19. Model P2: Minimization of the facility number to intercept a fixed % of the total demand

20. Intercepting all the demanded flows  pPfpxp  C* If we want to intercept all the demanded flows that is if C* =  pPfp  pPfpxp   pPfp  xp = 1  pP The second constraint disappears The first set of constraints becames  j pyj  1

21. Model P3: Minimization of the facility number to intercept the total demand (i.e. to cover all the paths)

22. Model Output Solving the model P1 produces the location of the m facilities giving the maximization of the intecepted flows, but it does not always give the exact values of the yp variables Solving the model P2 produces the number and the location of the facilities needed to intercept a fixed percent of the total demand and the list of the covered paths (i.e. exact values of yp variables) Solving the model P3 produces the number and the location of the facilities needed to intercept the total demand (i.e. all the paths)

23. Location in vertices Location on links Location in vertices is powerful for sensor location to counting the flows of all the junction movement It is possible from the technological viewpoint using cameras and virtual sensors for each lane and so for each movement in the junction. Unfortunatly its result can be affected by errors, sometimes relevant as we will see after. For VMS location vertex location is not practicable, because users have to be informed in the middle of the link

24. Transform a vertex model in a link modelthrough a dummy vertex In any case a vertex model is much more manageable, because the number of variables is more tractable with respect to the number of variables of a link model. Really it is possible to adopt a vertex model as a link model using a dummy vertex for each link • For a single direction • For both directions

25. Computational tests problem P1Nnodes_o/dpairs_pathsforodpairs_nodesforpaths

26. Computational tests problem P2 (60%)Nnodes_o/dpairs_pathsforodpairs_nodesforpaths

27. Computational tests problem P3Nnodes_o/dpairs_pathsforodpairs_nodesforpaths

28. Modification 1 of P2 model for traffic sensors location The constraint (2) can be referred to a single o/d pair:  pPodfpxp  C* for each o/d pair of a given set of o/d pair where Pod is the set of paths used to serve this o/d pair

29. Modification 2 of P2 model for traffic sensors and VMS location To ensure that at least k paths of an od pair are interceptedthe model can be integrated with the constraint:  pPodxp  K for each o/d pair of a given set of o/d pair where Pod is the set of paths used to serve this o/d pair

30. Modification 3 of P2 or P3 modelsfor VMS Location To ensure that at least h plants intercept a path p the model can be integrated with the constraint:  jpyj h for each path p of a given set of relevant paths

31. Computational Times (sec)

32. CT Modification 1 of P2 Model

33. CT Modification 2 of P2 Model

34. Need of heuristic For real networks with medium-large size an heuristic approach seems unavoidable

35. A small network 1 4 3 2 5 6 7 Path 1: 1- 2 - 5 Path 2: 1 - 2 – 4 Path 3: 1 – 3 – 4 Path 4: 1 – 3 – 7 Path 5: 2 - 5 Path 6: 2 – 4 - 6 Path 7: 3 – 4 - 6 Path 8: 3 – 7

36. O/D paths 1 4 3 2 5 6 7 Path 1: 1- 2 – 5 (1) Path 2: 1 - 2 – 4 (2) Path 3: 1 – 3 – 4 (2) Path 4: 1 – 3 – 7(1) Path 5: 2 – 5 (1) Path 6: 2 – 4 - 6 (1) Path 7: 3 – 4 – 6 (1) Path 8: 3 – 7 (1)

37. A greedy heuristic[Berman et al. (1992), Yang and Zhou (1998)] Coverage matrix B (path/link incidence matrix) The rows correspond to the paths p p P The columns correspond to the links aa A Each element bpa = 1 if link a belongs to the path p = 0 otherwise The coverage matrix can be obtained with an assignment model

38. The coverage matrix B

39. A greedy heuristic[Berman et al. (1992), Yang and Zhou (1998)] Scheme of the heuristic Step 0: set k=0. Let B(k) be the coverage matrix Step 1: Compute fa(k)= f a(k), a A Step 2: Find aj: fJ (k)= max a A{ fa(k) } and locate a facility in link aj (if more than one choose the link with lowest index,or better, choose the link belonging to the greatest number of paths) Step 3: Update the coverage matrix and generate B(k+1) deleting the column corresponding to link aj (bpj(k+1)=0 p P) deleting the rows corresponding to the paths intercepted from aj) (bpa(k+1)=0 a A, for each p such that bpj(k)=1 Step 4: if bpa=0 p P, a A , then STOP. otherwise, set k=k+1 and return to step 1

40. First step of the heuristic

41. Second step

42. Third step

43. Forth step

44. Fifth and last step

45. Comparison between heuristic and exact approach This heuristic produces very fast solution, but the result can be much far from the exact solution

46. Model P3 exact solution 4 facilities on links 2-4, 3-4, 2-5, 3-7 1 4 3 2 5 6 7 Path 1: 1- 2 – 5 (1) Path 2: 1 - 2 – 4 (2) Path 3: 1 – 3 – 4 (2) Path 4: 1 – 3 – 7(1) Path 5: 2 – 5 (1) Path 6: 2 – 4 - 6 (1) Path 7: 3 – 4 – 6 (1) Path 8: 3 – 7 (1)

47. Greedy solution 5 facilities on links 1-2, 1-3, 2-5, 4-6, 3-7 1 4 3 2 5 6 7 Path 1: 1- 2 – 5 (1) Path 2: 1 - 2 – 4 (2) Path 3: 1 – 3 – 4 (2) Path 4: 1 – 3 – 7(1) Path 5: 2 – 5 (1) Path 6: 2 – 4 - 6 (1) Path 7: 3 – 4 – 6 (1) Path 8: 3 – 7 (1)

48. A simple improvement of the heuristic The heuristic can be improved in the step 2 Step 2: Find aj: fJ (k)= max a A{ fa(k) } and locate a facility in link a (if more than one choose the link with lowest index) Alternative 1. Choose the link belonging to the greatest number of paths 2. Modify the selection criterion of the links

49. A simple network 3 6 10 8 2 5 7 9 1 4 O/D pair1 – 9 2 – 9 2 – 10 3 – 10 Path 1: 1-4-7-9 Path 2: 2-5-7-9 Path 3: 2-5-8-10 Path 4: 3-6-8-10

50. Possible solution 1 (sub-optimal) 3 6 10 8 2 5 7 9 1 4 O/D pair1 – 9 2 – 9 2 – 10 3 – 10 Path 1: 1-4-7-9 Path 2: 2-5-7-9 Path 3: 2-5-8-10 Path 4: 3-6-8-10