交通规划第七章交通分配Lecture 07

2 3x2 8x2 38 0
x1 2.033
x2 2.467
n
t1 14.13, t2 10.08
S x ti xi 53,592 veh.min
i 1
Compare UE and SO Solutions • User equilibrium
Beckmann Transformation Model
如何证明上述Beckmann模型的解符合Wardrop第一平衡原理？
应用KKT条件求解上述模型，其结果即Wardrop第一 平衡原理！
系统最优 SO

Example 1
Example (UE)
Two routes connect a city and a suburb. During the peak-hour morning commute, a total of 4,500 vehicles travel from the suburb to the city. Route 1 has a 60-mph speed limit and is 6 miles long. Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are as follows: •Route 1 HPF increases at the rate of 4 minutes for every additional 1,000 vehicles per hour. t1 = 6 + 4x1 •Route 2 HPF increases as the square of volume of vehicles in thousands per hour. t2 = 4 + x22
BPR function
t t0* 1 * V C



revised BPR function
Davidson function
Davidson, K. B. A flow–travel time relationship for use in transportation planning. Proc. 3rd ARRB Conf. 3 (1), pp.183-194, 1966.
Lecture 07
路径选择
Route Choice
内容：

路网节点的描述 无约束和有约束优化方法 最短路径

关联矩阵
Route Choice the route choice (traffic assignment) problem
is to find the flow (and traffic time) on each of

有约束非线性规划问题
有约束非线性规划问题
有约束非线性规划问题
Incidence Matrix （关联矩阵）
Incidence Matrix （关联矩阵）
1-5
2-5
Incidence Matrix （关联矩阵）
Link Performance Function
Route 1
City
Route 2
Suburb
Apply Wardrop 1st principle 6 + 4x1 = 4 + x22 x1 + x2 = 4.5 Substituting and solving: 6 + 4x1 = 4 + (4.5 – x1)2 6 + 4x1 = 4 + 20.25 – 9x1 + x12 x12 – 13x1 + 18.25 = 0 x1 = 1.6 or 11.4 (total is 4.5 so x1 = 1.6 or 1,600 vehicles) x2 = 4.5 – 1.6 = 2.9 or 2,900 vehicles Check answer: t1 = 6 + 4(1.6) = 12.4 minutes t2 = 4 + (2.9)2 = 12.4 minutes
x1 x2 4.5
x13 4.53 2 2 S x 6 x1 2 x 18 4 x1 4.5 x1 4.5 x1 3 3
2 1
for a minimum :
dS 6 4 x1 4 20.25 9 x1 x12 0 dx
simplifyin : x12 13x1 18.25 0 Same equation as before g
Principles of Route Choice
0-1分配法
增量分配法
二次加权分配法
均衡分配方法
用户最优 UO__review & redefine
UE, user equilibrium No traveler can improve his/her travel time by unilaterally changing routes. It was formally proposed by J. G. Wardrop in 1952
Example (SO)
Two routes connect a city and a suburb. During the peak-hour morning commute, a total of 4,500 vehicles travel from the suburb to the city. Route 1 has a 60-mph speed limit and is 6 miles long. Route 2 is half as long with a 45-mph speed limit. The HPFs for the route 1 & 2 are as follows: •Route 1 HPF increases at the rate of 4 minutes for every additional 1,000 vehicles per hour. t1 = 6 + 4x1 •Route 2 HPF increases as the square of volume of vehicles in thousands per hour. t2 = 4 + x22
Route 1
City
Route 2
Suburb
Apply Beckmann Model
1. Formulate the SO equation
2 S x ti xi 6 4 x1 x1 4 x2 x2 n i 1 3 6(4.5 x2 ) 4(4.5 x2 ) 2 4 x2 x2
Akcelik function
Akcelik, R. Travel time functions for transport planning purposes: Davidson’s function, its time dependent form and an alternative travel time function”, Australian Road Research, 21(3), pp. 49-59, 1991.
无约束非线性规划问题
有约束非线性规划问题
Second Berkeley Symposium on Mathematical Statistics and Probability Date: July 31-August 12, 1950
有约束非线性规划问题
Harold William Kuhn (born 1925) is an American mathematician. A Professor-Emeritus of Mathematics at Princeton University. Albert William Tucker (28 Nov. 1905 – 25 Jan. 1995) was a Canadian mathematician. former chair of Princeton University's Mathematics Department. William Karush (1 Mar. 1917 – 22 Feb. 1997) was a professor emeritus of California State University at Northridge and is a mathematician best known for his contribution to Karush– Kuhn–Tucker conditions.
Lloyd Stowell Shapley (born June 2, 1923) is a distinguished American mathematician and economist. He is a Professor Emeritus UCLA, affiliated with departments of Mathematics and Economics.
2.
Minimize the SO function
3 S ( x) 6(4.5 x2 ) 4(4.5 x2 ) 2 4 x2 x2 3 27 2 x2 4(4.5 x2 ) 2 x2
3.
Solve the minimized function dS 2 4 2 4.5 x2 1 3x22 0 dx
John Forbes Nash, Jr. (born June 13, 1928) is an American mathematician. Serving as a Senior Research Mathematician at Princeton University during the latter part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences.