Stochastic User Equilibrium with Equilibrated Choice Sets: Solving the Restricted SUE with Threshold and largescale

Transcription

1 Stochastic User Equilibrium with Equilibrated Choice Sets: Solving the Restricted SUE with Threshold and largescale application Thomas Kjær Rasmussen, David Paul Watling, Carlo Giacomo Watling & Otto Anker Nielsen heart 2015 Conference, Copenhagen September 11 th 2015

2 Outline Motivation Framework Algorithm Large-scale case-study Conclusions 2/23 DTU Transport, Technical University of Denmark

3 Motivation Solution algorithms exist for large-scale application of DUE efficient, e.g. by avoiding simulation convergence can be verified from gap measure Solution algorithms exist for large-scale application of SUE only approximate SUE explicit/implicit choice set generation (i.e. subset of all possible routes) often apply simulation in path generation and/or flow allocation Solution algorithm for large-scale application of RSUET efficient generic avoid simulation convergence should be easy to verify 3/23 DTU Transport, Technical University of Denmark

4 Solution algorithm framework Given Φ and Ω then the route flow x G is a RSUET(Φ,Ω) if and only if for all r R m and m = 1,2,,M: 0 Ω : 0 Φ : Solution algorithm which decomposes path generation and path loading RSUET model formulated with one condition concerning the distribution of flow and one posing a cost restriction on unused paths 4/23 DTU Transport, Technical University of Denmark

5 Solution algorithm framework Given Φ and Ω then the route flow x G is a RSUET(Φ,Ω) if and only if for all r R m and m = 1,2,,M: 0 Ω : 0 Φ : Path-based solution algorithm RSUET model formulated in space of paths path-based measures can be included in utility function for flow allocation memory issues related to path storage seem not a concern in modern computers 5/23 DTU Transport, Technical University of Denmark

6 Solution algorithm Step 0: Initialisation Step 1: Column generation phase. Systematically grow choice set to ensure fulfilment of second condition at convergence Step 2: Restricted master problem phase. Distribution of flow among used paths according to choice model to ensure fulfilment of first condition at convergence Step 3: Network loading phase. Perform network loading to get link flows, link costs and path costs resulting from the path flows Step 4: Threshold condition phase. Check if threshold condition is violated for any OD-pair. Remove violating paths, redistribute flow and perform network loading Step 5: Convergence evaluation phase. Apply consistent convergence criterion to evaluate whether both equilibrium conditions are fulfilled within a threshold. If not: Return to step 1. 6/23 DTU Transport, Technical University of Denmark

7 Solution algorithm Step 0: Initialisation Step 1: Column generation phase. Systematically grow choice set to ensure fulfilment of second condition at convergence Step 2: Restricted master problem phase. Distribution of flow among used paths according to choice model to ensure fulfilment of first condition at convergence Step 3: Network loading phase. Perform network loading to get link flows, link costs and path costs resulting from the path flows Step 4: Threshold condition phase. Check if threshold condition is violated for any OD-pair. Remove violating paths, redistribute flow and perform network loading Step 5: Convergence evaluation phase. Apply consistent convergence criterion to evaluate whether both equilibrium conditions are fulfilled within a threshold. If not: Return to step 1. 7/23 DTU Transport, Technical University of Denmark

8 Solution algorithm: Column generation phase Step 1 Column generation phase. Let k m,n-1 denote the current number of unique paths in the choice set of used paths for OD-pair m=1, 2,..., M in iteration n-1. For RSUET(min,Ω): For RSUET(Φ,Ω): For RSUET(max,Ω): For each origin, For each OD-pair mϵm, based on Perform k m, n-1 -shortest path perform a shortest actual link travel costs t, search for each OD-pair afn1 path search to all check for a new route to add to m=1, 2,..., M based on destinations based on the choice set R by applying actual link travel costs t. mn, a fn 1 actual link travel costs some path generation method If for any OD-pair m=1, t. If for any ODpair m=1, 2,..., M a which supports the fulfilment of afn1 2,..., M a new unique path i the Φ operator. If for any OD- pair is generated among the k new unique path i is m=1, 2,..., M a new unique path i generated paths, add it to generated, add it to is generated, add it to the choice the choice set R mn, with flow the choice set R mn, with flow x mi, n 1 0. set R with flow xmi, n1 0; mn, if several routes possible, add only the shortest one. Relates to the second RSUET condition xmi, n1 0; if several new unique paths are generated, add only the shortest one. 0 Φ : Choice set is systematically grown 8/23 DTU Transport, Technical University of Denmark

9 Solution algorithm Step 0: Initialisation Step 1: Column generation phase. Systematically grow choice set to ensure fulfilment of second condition at convergence Step 2: Restricted master problem phase. Distribution of flow among used paths according to choice model to ensure fulfilment of first condition at convergence Step 3: Network loading phase. Perform network loading to get link flows, link costs and path costs resulting from the path flows Step 4: Threshold condition phase. Check if threshold condition is violated for any OD-pair. Remove violating paths, redistribute flow and perform network loading Step 5: Convergence evaluation phase. Apply consistent convergence criterion to evaluate whether both equilibrium conditions are fulfilled within a threshold. If not: Return to step 1. 9/23 DTU Transport, Technical University of Denmark

10 Solution algorithm: Restricted Master Problem phase Step 2 Restricted master problem phase. Given the choice sets R for all m=1, 2,..., M, mn, apply the selected inner assignment component and averaging scheme to find the new flow solution X n. Relates to the first RSUET condition 0 Ω : Inner assignment component ordinary path-based SUE flow allocation methods efficient path-based DUE methods using cost transformation function for closed-form Logit type choice models exp 10/23 DTU Transport, Technical University of Denmark

11 Solution algorithm: Restricted Master Problem phase Step 2 Restricted master problem phase. Given the choice sets R for all m=1, 2,..., M, mn, apply the selected inner assignment component and averaging scheme to find the new flow solution X n. Relates to the first RSUET condition 0 Ω : Inner assignment component ordinary path-based SUE flow allocation methods efficient path-based DUE methods using cost transformation function for closed-form Logit type choice models exp exp,, 11/23 DTU Transport, Technical University of Denmark 1 exp exp exp exp exp exp

12 Solution algorithm: Restricted Master Problem phase Step 2 Restricted master problem phase. Given the choice sets R for all m=1, 2,..., M, mn, apply the selected inner assignment component and averaging scheme to find the new flow solution X n. Relates to the first RSUET condition 0 Ω : Inner assignment component ordinary path-based SUE flow allocation methods efficient path-based DUE methods using cost transformation function for closed-form Logit type choice models exp Averaging scheme MSA, MSWA, Self-regulated averaging, Armijo, 12/23 DTU Transport, Technical University of Denmark

13 Solution algorithm Step 0: Initialisation Step 1: Column generation phase. Systematically grow choice set to ensure fulfilment of second condition at convergence Step 2: Restricted master problem phase. Distribution of flow among used paths according to choice model to ensure fulfilment of first condition at convergence Step 3: Network loading phase. Perform network loading to get link flows, link costs and path costs resulting from the path flows Step 4: Threshold condition phase. Check if threshold condition is violated for any OD-pair. Remove violating paths, redistribute flow and perform network loading Step 5: Convergence evaluation phase. Apply consistent convergence criterion to evaluate whether both equilibrium conditions are fulfilled within a threshold. If not: Return to step 1. 13/23 DTU Transport, Technical University of Denmark

14 Solution algorithm: Threshold condition phase Step 4 Set m=1 Step 4.1 For each route r in the choice set R, check whether the threshold condition c mr (X n ) ({c ms (x) :s Rs mn, }; m ) is violated. If any route r violates this condition, and if n K min, then flag the route that Step 4.2 Step 4.3 mn, violates the threshold condition the most. If no route is flagged by Step 4.1 and if m<m, setm=m+1 and return to Step 4.1. Ifnoroutesare flagged by Step 4.1 and if m=m, continue to Step 4.3. Ifarouter is flagged by Step 4.1, removethe route from the choice set and redistribute flow x mr,n among the remaining currently-used routes s xms, n according to the following: xms, n xms, n xmr, n. dm xmr, n If m<m, setm=m+1 and return to Step 4.1. Ifm=M, continue. If no routes have been removed for any of the M OD-pairs, continue. Else, perform the network loading, compute the link travel costs t a (f n ), the generalised path travel costs C(X n ) and (if relevant/included) the path-size factors. Relates to the first RSUET condition 0 Ω : Lower risk of destabilisation :drop max one route per iteration per OD-pair and only after K min iterations 14/23 DTU Transport, Technical University of Denmark

15 Solution algorithm Step 0: Initialisation Step 1: Column generation phase. Systematically grow choice set to ensure fulfilment of second condition at convergence Step 2: Restricted master problem phase. Distribution of flow among used paths according to choice model to ensure fulfilment of first condition at convergence Step 3: Network loading phase. Perform network loading to get link flows, link costs and path costs resulting from the path flows Step 4: Threshold condition phase. Check if threshold condition is violated for any OD-pair. Remove violating paths, redistribute flow and perform network loading Step 5: Convergence evaluation phase. Apply consistent convergence criterion to evaluate whether both equilibrium conditions are fulfilled within a threshold. If not: Return to step 1. 15/23 DTU Transport, Technical University of Denmark

16 Solution algorithm: Convergence evaluation phase Step 5 Convergence evaluation phase. If the gap measure consisting of the sum of Used Unused Rel.Gap n and Rel.Gap n is below a pre-specified threshold ξ, Stop. Else, set n=n+1 and return to Step 1. Utilise cost transformation function to consistently evaluate convergence first condition (distribution of flow) Rel.Gap Used n M m1 rr m M m1 x c ( x ) c ( x ) mr, n mr n m,min n rr m x c ( x ) mr, n mr n 16/23 DTU Transport, Technical University of Denmark

17 Solution algorithm: Convergence evaluation phase Step 5 Convergence evaluation phase. If the gap measure consisting of the sum of Used Unused Rel.Gap n and Rel.Gap n is below a pre-specified threshold ξ, Stop. Else, set n=n+1 and return to Step 1. Utilise cost transformation function to consistently evaluate convergence first condition (distribution of flow) second condition (composition of choice set) RSUET(min,Ω): Rel.Gap Used n Rel.Gap Unused n M m1 M m1 rr m M m1 x c ( x ) c ( x ) mr, n mr n m,min n rr m x c ( x ) mr, n mr n x x d min c ( ) min c ( ) m rrm, xmr 0 mr n rrm mr n M m1 d min c ( x ) m rrm, xmr 0 mr n RSUET(max,Ω): Rel. Gap 17/23 DTU Transport, Technical University of Denmark Unused n M m1 d max c ( x ) c ( x ) m rrm, xmr 0 mr n mr, k n M m1 d max c ( x ) m rrm, xmr 0 mr n

18 Zealand application of RSUET(min, min) Network and demand 3.2 million trips, 19 user classes, 3 vehicle types 18,706 one-directional links, 429 zones Restricted master problem phase cost function considering free-flow travel time, congested travel time and length MultiNomial Logit (MNL) and Path-Size Logit (PSL) inner assignment component DUE Path Swap Inner Logit averaging scheme Method of Successive Weighted Averages d n (MSWA) n n d d d 18/23 DTU Transport, Technical University of Denmark

19 Zealand application of RSUET(min, min) Threshold condition phase Ω : 1.2 min : 16,618 observed paths from GPS traces 1,169 observed link flows on network 19/23 DTU Transport, Technical University of Denmark

20 Results Fast convergence for MNL and PSL RSUET(min,1.2 min) Step-size parameter d influences convergence speed Robust towards congestion level slightly slower convergence larger choice sets increased effect of threshold condition 20/23 DTU Transport, Technical University of Denmark

21 Results Fast convergence and very reasonable choice set size final choice sets generated within first few iterations flow allocation converges very fast Comparison to GPS data: observed paths are generally reproduced Comparison to link flows: very successful reproduction Choice set size Coverage, λ=0.8 Efficiency Excluded R 2 link Min. Avg. Max. Ite 25 Ite 100 index paths flow θ= θ= θ= θ= θ= /23 DTU Transport, Technical University of Denmark

22 Results MultiNomial Probit (MNP) from Danish National Transport Model Increased coverage at cost of slow convergence choice sets continue to grow not stable link flows Choice set size Coverage, λ=0.8 Efficiency Excluded R 2 link Min. Avg. Max. Ite 25 Ite 100 index paths flow θ= θ= θ= θ= θ= MNP SUE mixed MNP SUE /23 DTU Transport, Technical University of Denmark

23 Conclusions Generic path-based RSUET(Φ,Ω) solution algorithm allows use of existing SUE flow allocation methods among used paths allows use of efficient DUE methods for certain Logit-type choice models (cost transformation function) allows consistent evaluation of convergence Application to large-scale case study very fast convergence to equilibrated solution (across scale parameters, demand levels and step-size strategies) reproduction of observed routes and link counts 23/23 DTU Transport, Technical University of Denmark

24 References Watling, D.P., Rasmussen, T.K., Prato, C.G., Nielsen, O.A., Stochastic user equilibrium with equilibrated choice sets: Part I Model formulations under alternative distributions and restrictions. Transportation Research Part B: Methodological, 77, Rasmussen, T.K., Watling, D.P., Prato, C.G., Nielsen, O.A., Stochastic user equilibrium with equilibrated choice sets: Part II Solving the restricted SUE for the logit family. Transportation Research Part B: Methodological, 77, Watling, D.P., Rasmussen, T.K., Prato, C.G., Nielsen, O.A., Bounded Rationality, Thresholds and Equilibrated Choice Sets: A Unified Approach for Deterministic and Stochastic User Equilibrium. Working Paper. Rasmussen, T.K., Watling, D.P., Prato, C.G., Nielsen, O.A., Largescale application and validation of the Restricted Stochastic User Equilibrium with Threshold. Working Paper. 24/23 DTU Transport, Technical University of Denmark

25 Thank you for your attention 25/23 DTU Transport, Technical University of Denmark

26 Results Effect of threshold condition, MNL RSUET(min, 1.2min) Path Category ID l 1r [km] t FreeTT, 1r (x) [min] t CongTT, 1r (x) [min] c 1r (x) c 1r (x)/c 1,min (x) Flow [%] /23 DTU Transport, Technical University of Denmark

27 Results Increase due to recalculation of correction terms in PSL Increase with increasing demand/congestion level 90/105/130/145/180 seconds per iteration for scale-parameter 1.0/1.25/1.5/1.75/2.0 27/23 DTU Transport, Technical University of Denmark