Decreasing Traffic . . . Traffic Assignment: . . . Towards a More . . . How to Estimate, Take Into A Seemingly Natural . . . A More Realistic . . . Account, and Improve Travel Taking Uncertainty . . . Logit Discrete Choice . . . Towards an Optimal . . . Time Reliability in Exponential Disutility . . . Transportation Networks Title Page ◭◭ ◮◮ Ruey L. Cheu, Vladik Kreinovich, Fran¸ cois Modave, Gang Xiang, Tao Li, and Tanja Magoc ◭ ◮ Page 1 of 14 University of Texas, El Paso, TX 79968, USA contact email vladik@utep.edu Go Back Full Screen Close Quit
Decreasing Traffic . . . 1. Decreasing Traffic Congestion: Formulation of the Traffic Assignment: . . . Problem Towards a More . . . A Seemingly Natural . . . • Practical problem: decreasing traffic congestion. A More Realistic . . . • Important difficulty: a new road can worsen traffic con- Taking Uncertainty . . . gestion. Logit Discrete Choice . . . • Conclusion: importance of the preliminary analysis of Towards an Optimal . . . the results of road expansion. Exponential Disutility . . . Title Page • Traditional approach assumes that we know: ◭◭ ◮◮ – the exact amount of traffic going from zone A to zone B ( OD-matrix ), and ◭ ◮ – the exact capacity of each road segment. Page 2 of 14 • Limitations: in practice, we only know all this with Go Back uncertainty. Full Screen • What we do: we show how to take this uncertainty into Close account in traffic simulations. Quit
Decreasing Traffic . . . 2. Traffic Assignment: Brief Reminder Traffic Assignment: . . . Towards a More . . . • Traffic demand: # of drivers d ij who need to go from A Seemingly Natural . . . zone i to zone j – origin-to-destination (O-D) matrix. A More Realistic . . . • Capacity of a road link – the number c of cars per hour Taking Uncertainty . . . which can pass through this link. Logit Discrete Choice . . . � � β � � v • Travel time along a link : t = t f · Towards an Optimal . . . 1 + a · , where: c Exponential Disutility . . . • t f = L/s is a free-flow time ( s is the speed limit), Title Page • a ≈ 0 . 15 and β ≈ 4 are empirical constants. ◭◭ ◮◮ • Equilibrium: when ◭ ◮ • the travel time along all used alternative routes is Page 3 of 14 exactly the same, and Go Back • the travel times along other un-used routes is higher. Full Screen • Algorithms: there exist efficient algorithms for finding Close the equilibrium. Quit
Decreasing Traffic . . . 3. How We Can Use the Existing Traffic Assignment Traffic Assignment: . . . Algorithms to Solve Our Problem: Analysis Towards a More . . . A Seemingly Natural . . . • Main objective: predict how different road project af- A More Realistic . . . fect future traffic congestion. Taking Uncertainty . . . • Future traffic demands – what is known: there exist Logit Discrete Choice . . . techniques for predicting daily O-D matrices. Towards an Optimal . . . • What is lacking: we need to “decompose” the daily Exponential Disutility . . . O-D matrix into 1 hour (or 15 minute) intervals. Title Page • 1st approximation: assume that the proportion of drivers ◭◭ ◮◮ starting at, say 6 to 7 am is the same as now. ◭ ◮ • Need for a more accurate approximation: Page 4 of 14 – drivers may start early because of congestion; Go Back – if a new road is built, they will start later; Full Screen – the % of those who start 6–7 am will decrease. Close • We cover: both approximations. Quit
Decreasing Traffic . . . 4. Towards a More Accurate Approximation to O-D Traffic Assignment: . . . Matrices Towards a More . . . A Seemingly Natural . . . • Describing preferences: empirical utility formula A More Realistic . . . u i = − 0 . 1051 · E ( T ) − 0 . 0931 · E ( SDE ) − 0 . 1299 · E ( SDL ) − Taking Uncertainty . . . S 1 . 3466 · P L − 0 . 3463 · E ( T ) , Logit Discrete Choice . . . Towards an Optimal . . . where E ( X ) means expected value, Exponential Disutility . . . • T is the travel time T , Title Page • SDE is the wait time when arriving early, ◭◭ ◮◮ • SDL is the delay when arriving late, ◭ ◮ • P L is the probability of arriving late, and Page 5 of 14 • S is the variance of the travel time. Go Back • Logit model: the probability P i that a driver will choose Full Screen the i -th time interval is proportional to exp( u i ): exp( u i ) Close P i = exp( u 1 ) + . . . + exp( u n ) . Quit
Decreasing Traffic . . . 5. A Seemingly Natural Idea and Its Limitations Traffic Assignment: . . . Towards a More . . . • Seemingly natural idea: A Seemingly Natural . . . – start with the 1st approximation O-D matrices M 1 ; A More Realistic . . . – based on M 1 , we find travel times, and use them to Taking Uncertainty . . . def find the new O-D matrices M 2 = F ( M 1 ); Logit Discrete Choice . . . – based on M 2 , we find travel times, and use them to Towards an Optimal . . . def find the new O-D matrices M 3 = F ( M 2 ); Exponential Disutility . . . Title Page – repeat until converges. ◭◭ ◮◮ • Toy example illustrating a problem: ◭ ◮ • now: no congestion, all start at 7:30, work at 8 am; • M 1 : full O-D matrix for 7:30 am, 0 for 7:15 am; Page 6 of 14 • based on this M 1 , we get huge delays; Go Back • M 2 : everyone leaves for work early at 7:15 am; Full Screen • at 7:30, roads are freer, so in M 3 , all start at 7:30; Close • no convergence: M 1 = M 3 = . . . � = M 2 = M 4 . . . Quit
Decreasing Traffic . . . 6. A More Realistic Approach Traffic Assignment: . . . Towards a More . . . • Above idea: drivers make decisions based only on pre- A Seemingly Natural . . . vious day traffic. A More Realistic . . . • More accurate idea: drivers make decisions based on Taking Uncertainty . . . the average traffic over a few past days. Logit Discrete Choice . . . • Resulting process: Towards an Optimal . . . – start with the 1st approximation O-D matrices M 1 ; Exponential Disutility . . . Title Page – for i = 2 , 3 , . . . : ∗ compute the average E i = M 1 + . . . + M i ◭◭ ◮◮ , i ◭ ◮ ∗ find traffic times based on E i ; Page 7 of 14 ∗ use these traffic times to compute a new O-D matrix M i +1 = F ( E i ); Go Back ∗ repeat until converges. Full Screen • Process converges: on toy examples, on El Paso net- Close work, etc. Quit
Decreasing Traffic . . . 7. Algorithm Simplified Traffic Assignment: . . . Towards a More . . . • Main idea: once we know the previous average E i , we A Seemingly Natural . . . can compute A More Realistic . . . E i +1 = ( M 1 + . . . + M i ) + M i +1 = i · E i + M i +1 = Taking Uncertainty . . . i + 1 i + 1 Logit Discrete Choice . . . � � 1 1 Towards an Optimal . . . E i · 1 − + M i +1 · i + 1 . i + 1 Exponential Disutility . . . Title Page • We know: that M i +1 = F ( E i ). ◭◭ ◮◮ • Resulting algorithm: ◭ ◮ – start with the 1st approximation O-D matrices Page 8 of 14 E 1 = M 1 ; Go Back � � 1 1 Full Screen – compute E i +1 = E i · 1 − + F ( E i ) · i + 1; i + 1 Close – repeat until converges. Quit
Decreasing Traffic . . . 8. Taking Uncertainty into Account Traffic Assignment: . . . � � β � � v Towards a More . . . • Deterministic model: t = t f · 1 + a · . A Seemingly Natural . . . c A More Realistic . . . • Traffic assignment: a driver minimizes the travel time Taking Uncertainty . . . t = t 1 + . . . + t n . Logit Discrete Choice . . . • In practice: travel times vary. Towards an Optimal . . . • Decision theory: maximize expected utility E [ u ]. Exponential Disutility . . . Title Page • How utility depends on travel time: u ( t ) = − U ( t ), where U ( t ) = exp( α · t ). ◭◭ ◮◮ • Conclusion: the driver minimizes ◭ ◮ Page 9 of 14 E [ U ( t )] = E [exp( α · t )] = E [exp( α · ( t 1 + . . . + t n )] = Go Back E [exp( α · t 1 ) · . . . · exp( α · t n )] . Full Screen • Deviations on different links are independent, so Close E [ U ( t )] = E [exp( α · t 1 )] · . . . · E [exp( α · t n )] . Quit
Decreasing Traffic . . . 9. Taking Uncertainty into Account (cont-d) Traffic Assignment: . . . Towards a More . . . • Minimizing E [ U ( t )] = E [exp( α · t 1 )] · . . . · E [exp( α · t n )] � A Seemingly Natural . . . n def � t i , where � ⇔ minimizing t i = ln( E [exp( α · t i )]) . A More Realistic . . . i =1 Taking Uncertainty . . . = t − t f t depends on t f and r def • � : � t = F ( t f , r ). Logit Discrete Choice . . . t Towards an Optimal . . . • If we divide a link into sublinks, we conclude that Exponential Disutility . . . t = t f · k ( r ). F ( t f 1 + t f 2 , r ) = F ( t f 1 , r ) + F ( t f 2 , r ), hence � Title Page • For no-congestion case r = 0, we have � t = t f , so k (0) = ◭◭ ◮◮ 1 and k ( r ) = 1 + a 0 · r + a 2 · r 2 + . . . ◭ ◮ • Empirical analysis: a 1 ≈ 1 . 4, b ≈ 0, so Page 10 of 14 � � β � � v t = t f · � 1 + a · a 1 · . Go Back c Full Screen • Solution: use the standard travel time formula with Close a · a 1 ≈ 0 . 21 instead of a ≈ 0 . 14. Quit
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