lec24_2_04_studen

lec24_2_04_studen - Outline Effective Route Guidance in...

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c ± Outline Route Guidance; User Optimum; Capacities. Constrained c ± 1 G =( ) k demands ( o i ,d i ) r i f P . t a ( · ) and L C ( f ):= a A t a ( f a ) f a 2 x x +1 2 2 2 2 1 0 0 0 0 2 2 2 · 4+ 2 · c ± 2 UE unique UE UE / SO α ( L ) UE / SO UE / SO α ( L ) BUE / SO α ( L ) c ± 3 Effective Route Guidance in Traffic Networks Lectures developed by Andreas S. Schulz and Nicol´ as Stier May 13, 2004 2004 Massachusetts Institute of Technology Lecture 1 Equilibrium; System User Equilibria in Networks with Lecture 2 System Optimum; Dantzig-Wolfe Decomposition; Constrained Shortest Paths; Computational Results. 2004 Massachusetts Institute of Technology Review of Traffic Model Directed graph V,A with capacities, with rate Flows on paths Can be non-integral. Traversal times: latency functions continuous nondecreasing belong to a given set (e.g. linear) The total travel time of a flow is: 1=10 2004 Massachusetts Institute of Technology Review of First Lecture No capacities With capacities Set of may be non-convex unbounded 2004 Massachusetts Institute of Technology
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optimize optimize system c ± 4 SO 51 51 1 1 1 2 2 100 c a =1 c a =1 c ± 5 SO SO C ( SO ) = . Unfair! 51 51 1 1 1 2 2 100 c a =1 c a =1 c ± 6 SO C ( f ) = 104 but fair! 51 51 1 1 1 2 2 100 c a =1 c a =1 c ± 7 selfish users central planner the goal own travel time welfare fair, not efficient efficient, not fair fair, efficient 2004 Massachusetts Institute of Technology Long Detours in Instance with constant latencies: 2004 Massachusetts Institute of Technology Long Detours in routes 1 unit along each path: 100 + 3 2004 Massachusetts Institute of Technology Long Detours in Compare to routing 1 unit along the other paths: 2004 Massachusetts Institute of Technology
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Constrained c ± Constrained 8 SO cannot due to unfairness UE Use instead! CSO = min total c ± Constrained 9 c ± 10 UE Notation: ± a ± P = a P ± a c ± Constrained 11 System Optimum 2004 Massachusetts Institute of Technology System Optimum Route Guidance be implemented in practice does not take into account the global welfare constrained SO travel time s.t. demand satisfied users are assigned to “fair” routes capacity constraints 2004 Massachusetts Institute of Technology System Optimum Technological Requirements exact knowledge of the current position 2-way communication to a main server 2004 Massachusetts Institute of Technology Constrained System Optimum Constrained SO : Normal Lengths Normal lengths : a-p r io r ibe l ie fo fnetwo rk Geographic distances Free-flow travel times (times in empty network) Travel times under normal
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Constrained SO : Definition Fix a tolerance ε 0 Apath P ∈P i is valid if ± P (1 + ε ) × min Q ∈P i ± Q Definition: CSO ε = min total travel time s.t. P
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lec24_2_04_studen - Outline Effective Route Guidance in...

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