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Lecture 13-1 4-step Model – Trip Assignment CVEN672 1 Trip Assignment Definition Assign Tij onto alternative routes on the network to predict the Link flows and to evaluate the network performance. CVEN672 2 Trip Assignment Generalized Cost = function of time, distance, $$ + possibly others. From the user’s perspective; shortest path (stochastic vs. deterministic)UE From the planner’s perspective: minimize the network wide travel cost SO j i CVEN672 3 Trip Assignment: Link Performance Fucntion Each path has an associated Link performance function (LPF): Typical forms: t = t0eV/C t = t0ab(V/C) Most common from the Bureau of Public Roads (BPR) t = t0(1 + a(V/C)b) which becomes when linear (b = 1, a = aC/t0): t = t0 + aV CVEN672 4 Trip Assignment Four classes of TA methods: 1. 2. 3. 4. ALL-or-NOTHING Assignment for uncongested network (constant cost) [Dijkstra: Shortest path algorithm] find the shortest path and assign all travels on the SP between a pair of OD. User Equilibrium (Wardrop principles) Every traveler is on the shortest path to the destination. No user can be better off by unilaterally changing to a different route. Stochastic TA (due to “perceived” costs, lack of info for users) Users do not have perfect real time information about cost. Their route choice decisions have uncertain factor. Dynamic assignment (how about time variable???) Traffic do not happen on the roads at one single instance of time. Early travel decisions have impact on later traffic congestions. The interactions over time are considered. CVEN672 5 Shortest Path (Dijkstra) Assuming a known network with a known deterministic link performance, the objective is to find a path from an origin to a destination at the least cost. Shortest path calculations are idealistic, but are most fundamental to the network flow problem and to the network traffic assignment problem. Many sophisticated algorithms are built with the shortest path algorithm. There are many variations (Dijkstra, Moore), but all get to the same results. The complexities are different slightly. CVEN672 6 Notations • • • • • • N Set of nodes M set of unlabeled notes Bar(M) set of unlabeled nodes ni ith node in consideration C(ni,nj) cost of travel from node i to node j. C(nj) cost of travel from the home node to node j. CVEN672 7 Shortest Path (Dijkstra) CVEN672 8 Label L(ni) = [predecessor, cost] CVEN672 This is the algorithm to summarize based on the example 9 Shortest Path (Dijkstra) CVEN672 10 DIJKSTRA’S ALGORITHM • Greedy Algorithm – Greedy algorithms use problem solving methods based on actions to see if there’s a better long term strategy. Dijkstra’s algorithm uses the greedy approach to solve the single source shortest problem. It repeatedly selects from the unselected vertices, vertex v nearest to source s and declares the distance to be the actual shortest distance from s to v. The edges of v are then checked to see if their destination can be reached by v followed by the relevant outgoing edges. • Disadvantages – The major disadvantage of the algorithm is the fact that it does a blind search there by consuming a lot of time waste of necessary resources. – Another disadvantage is that it cannot handle negative edges. This leads to acyclic graphs and most often cannot obtain the right shortest path. CVEN672 11 Exercise: find the shortest path from AE CVEN672 12 All-or-nothing Assignment, use Dijkstra to find S.P. for each O/D pair and then assign all flow to it A-C = 400; A-D = 200; B-C = 300; B-D = 100 CVEN672 13 Some math background to the Dijkastra’s Algorithm (optional) • Linear programming min cij xij d_o ij x – LP to inset later oj 1, j x x • Dual program ij i x is jk 0,j N \{o, s} k 1. i max d s d 0 s.t. d j d o coj , j N \{o} d j di cij , i, j N \{o} CVEN672 14