Routing Algorithms in Traffic Assignment Modeling Tuan Nam Nguyen - PowerPoint PPT Presentation

Routing Algorithms in Traffic Assignment Modeling Tuan Nam Nguyen and Gerhard Reinelt University of Heidelberg Institute of Computer Science nam.nguyen@informatik.uni-heidelberg.de gerhard.reinelt@informatik.uni-heidelberg.de Workshop on Traffic Optimization Heidelberg, October 8th, 2015

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Motivation Figure: Traffic in Hanoi on October 8th, 2015. Source. Vnexpress.net Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 2 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Outline 1 Introduction Traffic Assignment Modeling (TAM) Routing Problems in TAM 2 K Shortest Paths (KSP) Some Exising Algorithms New Heuristic Method HELF Computational Results 3 K Dissimilar Shortest Paths (KDSP) 4 Application in TAM A Case Study: Hanoi, Vietnam Computational Results Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 3 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Traffic Assignment Modeling Traffic assignment modeling Traffic assignment modeling (TAM) aims at forecasting the number of trips on different links (road sections) of the network given the travel demand between different pairs of zones (or areas). Figure: An example of traffic assignment. Source: TranOpt Plus Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 4 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Traffic Assignment Modeling Traffic assignment modeling Traffic assignment modeling (TAM) aims at forecasting the number of trips on different links (road sections) of the network given the travel demand between different pairs of zones (or areas). Figure: An example of traffic assignment. Source: TranOpt Plus Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 4 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Traffic Assignment Modeling Traffic assignment modeling Traffic assignment modeling (TAM) aims at forecasting the number of trips on different links (road sections) of the network given the travel demand between different pairs of zones (or areas). Figure: An example of traffic assignment. Source: TranOpt Plus Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 4 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Traffic Assignment Models Each TA model is made based on a number of assumptions on traffic behaviors, traffic networks, etc. All or Nothing (AON) model: Drivers choose the shortest path (in terms of length) to travel without consideration of traffic flow on the path. Incremental (INC) model: repeat the AON model for travel time, i.e. the travel time on links are updated regularly according to the traffic flow and drivers choose the shortest path (in terms of travel time) to travel. System Equilibrium model: drivers are cooperated to each other to minimize the total travel time on the whole system. This assumption based on the second principle of Wandrop. User Equilibrium (UE) model: Travel time on all used paths between two nodes are equal and less than those of un-used paths. This is based on the first principle of Wandrop. Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 5 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix The User Equilibrium Models t 1 ( x 1 ) = 50(1 + 0 . 15 � x 1 / 200) 4 � D st = 1000 s t t 2 ( x 2 ) = 100(1 + 0 . 15 � x 2 / 300) 4 � t 3 ( x 2 ) = 300 + x 3 500 450 Traveling time (s) 400 350 300 Equilibrium point 250 200 150 t 2 ( x ) 100 t 1 ( x ) 50 0 0 100 200 300 400 500 600 700 800 900 1000 Traffic flow x 1 = 1000 − x 2 , x 3 = 0 (vehicles) Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 6 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix The User Equilibrium Models t ij ( x ij ): travel time function on link ( i , j ) where x ij is the total traffic flow on the link ( i , j ), P rs : the set of potential paths from r ∈ P to s ∈ P, � 1 if link (i,j) is on path p δ p ij = , 0 otherwise x p is the traffic flow on the potential path p ∈ P rs . rs : We have a generalized user equilibrium (GUE) model as following: x ij � � Minimize Z = t ij ( ω ) d ω, (GUE) ( i , j ) ∈ E 0 � � δ p ij x p ∀ ( i , j ) ∈ E Subject to x ij = (1) rs p ∈ P rs r , s ∈ P � x p ∀ r , s ∈ P rs = d rs (2) p ∈ P rs x p rs ≥ 0 ∀ r , s ∈ P , p ∈ P rs . (3) Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 7 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix The User Equilibrium Models For each pair of zones, the potential paths are possible paths between the nodes that satisfy some of routing behaviors, such as: Short in length Loop-less Dissimilar to each other Short time in congestion. Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 8 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Routing Problems and Applications 5 Given a graph G ( V , E ), two nodes s , t ∈ V . D 3 3 Shortest path (SP) problem 2 K shortest paths (KSP) problem 1 A E 3 B Loop-less paths: KSLP problem Non-loop-less paths: KSNLP problem 1 3 2 Dissimilar shortest paths (DSP) problem C K shortest loop-less paths D ( p 1 , p 2 ) ≥ α Figure: A example of graph. Multi-objectvie shortest paths (MOSP) problem Popular algorithms: Floyd-Warshall, Bellman-Ford, Dijkstra, Yen, Martin, Eppstein, etc. Figure: A part of map of NewYork. Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 9 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Dissimilar paths with minDifference = 30%. N = 1 Figure: Dissimilar shortest paths finding. Source: TranOpt Plus Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 10 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Dissimilar paths with minDifference = 30%. N = 2 Figure: Dissimilar shortest paths finding. Source: TranOpt Plus Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 10 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Dissimilar paths with minDifference = 30%. N = 3 Figure: Dissimilar shortest paths finding. Source: TranOpt Plus Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 10 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Dissimilar paths with minDifference = 30%. N = 10 Figure: Dissimilar shortest paths finding. Source: TranOpt Plus Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 10 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Applications of routing GPS navigation Logistic planning Games Routing services Army Traffic planning Figure: Routing service. Source: Google. Note: Many paths routing problems are more and more importatnt in real applications. Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 11 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix K Shortest Paths Table: Existing algorithms for KSP problem. Year Author The Title 1971 J. Y. Yen Finding the k shortest loopless paths Enumerating elementary paths and cutsets by 1975 K. Aihara Gaussian elimination method An algorithm for generating all the paths between 1976 T. D. Am et al. two vertices in a digraph and its application An algorithm in paths removing approach. Next 1984 E. Q. V. Martins shortest path is found after removing previous shortest paths from the graph Finding a minimum weight K-link path in graphs 1993 A. Aggarwal et al. with Monge property and applications 1997 D. Eppstein Finding the k Shortest Paths Labeling approach as the extension of Dijkstra to 1999 Martin and Santos find k shortest paths K ∗ : A heuristic search algorithm for finding the 2011 H. Aljazzar et al. k shortest paths Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 12 / 38

Introduction K Shortest Paths K Dissimilar Shortest Paths Application in TAM Appendix Paths Removing Approach G(V,E),k,s,t initialize step NO | D | < k Principle YES Given a graph G ( V , E ) and a pair of p = ShortestPath ( G , s , t ) nodes ( s , t ). If there are at least k possible different paths from s to t then the k th shortest paths from s to t NO p � = is the shortest path in the graph after NULL removing only k − 1 previous shortest YES paths from s to t . D ← D ∪ { p } RemovePath(G,p) Stop Tuan Nam Nguyen Routing Algorithms in Traffic Assignment Modeling Heidelberg, Oct. 8, 2015 13 / 38