Transportation Engineering
2010 Fall Semester
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Transportation Engineering-Lecture 12 Slides-Engineering, Slides of Transportation Engineering
Route Choice, Finding Shortest Path, Dijkstra’s Algorithm, All-or-Nothing Assignment, Link Performance Function, User Equilibrium, Wardrop’s First Principal, UE Formulation, SO Assignment, Wardrop’s Second Principal, System Optimal, Transportation Engineering, Seungmo Kang, Engineering, Korea University, South Korea.
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Transportation Engineering
2010 Fall Semester
Route Choice
-^ 100 trips •^ From here •^ To Sinseol-DongStation •^ Using Car •^ Which Route?
Route 1 Route 2
Route 3
Source: http://map.google.com
Route Choice
•^ All-or-Nothing Assignment •^ User Equilibrium (UE) Assignment •^ System Optimal (SO) Assignment
Dijkstra’s Algorithm^1
9 8 1 : Node 1^10 : Link with Travel Time 10
9 8 4 11
14 6 12
10 11 13 15 13
-^ Find Shortest Path fromZone 5 to Zone 10
Dijkstra’s Algorithm 1 3
25(3)^
40(8) 19(4)^
11(1) 25(4)^ 35(7)13(5) 26(4)
-^ Result 25(3) : Travel time from node 5 is 25 and the previous node is 3
Shortest Path from Node 5 to Node 10
Link Performance Function • Link travel time vs. Flow demand • Travel time increases with flow demand • Grows exponentially if the demand is larger than the capacity • BPR (Bureau of Public Road) Function
ఉݔ ݐ ൌ ݐሼ1 ߙ ሽ ܥݐ: Link Travel Time ݐ: Free Flow Travel Time 4 ݔ: Link Flow Demand ܥ: Capacity (Max. flow Rate)3.5 ߙ: Parameter to be calibrated 3 (default: 0.15)2.5 ߚ: Parameter to be calibrated(default: 4) 2 1.5 1 0.5 0 0 1000
2000 3000 4000
(^5000) ሻ݄/݄݁ݒሺݔ ሻݎ݄ሺݐ^ Capacity^ ݐ
Link Performance Function • Can flow exceed “capacity”? • On a link, the capacity is thought of as a maximum“outflow.” • Demand is inflow. • If inflow > outflow for a period of time, there isqueueing (and delay). • For Example: For a 1 hour period, if 2100 cars arriveand 2000 depart (=capacity), 100 are still there. • This equation tries to represent that phenomenon in asimple way.
UE Example O/D Trip = 5 The link perfomance functions t=2+xand t=1+2x. 11 22 ^
Flow conservation:^ ^ x+x= 5^12 UE status:^ ^ Travel time for path 1 =Travel time for path 2^ ^ t= t^1 2 ^ 2+x= 1+2x^1 2 Solve:^ ^ 2+5-x=1+2x^22 ^ x= 2, x= 3^2
Route 2t=1+2x^22 Route 1t=2+x^11
UE Formulation
SO Assignment This can be formulated as amathematical programMin Subject to
, for all k, and O-D pairs, for all k, and O-D pairs Where,^ a^ a for all links
Z =^ X ta f^ =^ q^ k k^ f^0 k^ X^ =^ f^ δ^ a^ k^ ak^ OD^ K
Example O/D Trip = 5 The link performance functions t=2+xand t=1+2x. 11 22 ^ Flow conservation:^ ^ x
+x= 5 12
Route 2t=1+2x^22 Route 1t=2+x^11
-^ Result^ XXT^1
TTotal 1 2 Travel Time AoN^0