Alpha-reliable combined mean traffic equilibrium model with stochastic travel times

Abstract

Based on the reliability budget and percentile travel time (PTT) concept, a new travel time index named combined mean travel time (CMTT) under stochastic traffic network was proposed. CMTT here was defined as the convex combination of the conditional expectations of PTT-below and PTT-excess travel times. The former was designed as a risk-optimistic travel time index, and the latter was a risk-pessimistic one. Hence, CMTT was able to describe various routing risk-attitudes. The central idea of CMTT was comprehensively illustrated and the difference among the existing travel time indices was analyzed. The Wardropian combined mean traffic equilibrium (CMTE) model was formulated as a variational inequality and solved via an alternating direction algorithm nesting extra-gradient projection process. Some mathematical properties of CMTT and CMTE model were rigorously proved. Finally, a numerical example was performed to characterize the CMTE network. It is founded that that risk-pessimism is of more benefit to a modest (or low) congestion and risk network, however, it changes to be risk-optimism for a high congestion and risk network.

Keywords

travel behavior / risk attitude / travel time reliability / combined mean travel time / wardropian user equilibrium

References

[1] FrankH. Shortest paths in probabilistic graphs [J]. Operations Research, 1969, 17(4): 583-599

[2] FanY, KalabaR, MooreJ. Arriving on time [J]. Journal of Optimization Theory and Application, 2005, 127(3): 497-513

[3] LevyH. Stochastic dominance: Investment decision making under uncertainty [M]. New York: Springer, 2006147-152

[4] NieY, WuX. Shortest path problem considering on-time arrival probability [J]. Transportation Research Part B, 2009, 43(6): 597-613

[5] NieY, WuX. Reliable a priori shortest path problem with limited spatial and temporal dependencies [C]. Proceedings of the 18th International Symposium on Transportation and Traffic Theory, Hong Kong. Springer, 2009169-195

[6] NieY. Multi-class percentile user equilibrium with flow-dependent stochasticity [J]. Transportation Research Part B, 2011, 45(10): 1641-1659

[7] HallR W. The fastest path through a network with random time-dependent travel time [J]. Transportation Science, 1986, 20(3): 182-186

[8] WuX, NieY. Modelling heterogeneous risk-taking behaviour in route choice: A stochastic dominance approach [J]. Transportation Research A, 2011, 45(9): 896-915

[9] LoH K, TungY K. Network with degradable links: Capacity analysis and design [J]. Transportation Research Part B, 2003, 37(4): 345-363

[10] LoH K, LuoX W, SiuB W Y. Degradable transport network: Travel time budget of travellers with heterogeneous risk aversion [J]. Transportation Research Part B, 2006, 40(9): 792-806

[11] ArtznerP, DelbaenF, EberJ M, HeathD. Coherent measures of risk [J]. Mathematical Finance, 1999, 9(3): 203-228

[12] ChenA, JiZ W. Path finding under uncertainty [J]. Journal of Advanced Transportation, 2005, 39(1): 19-37

[13] ShaoH, LamW H K, TamM L. A reliability-based stochastic traffic assignment model for network with multiple user classes under uncertain in demand [J]. Network and Spatial Economics, 2006, 6(3/4): 313-332

[14] SiuB W Y, LoH K. Doubly uncertain transport network: Degradable link capacity and perception variation in traffic conditions [J]. Transportation Research Record, 2006, 1964: 59-69

[15] ShaoH, LamW H K, TamM L, YuanX M. Modelling rain effects on risk-taking behaviours of multi-user classes in road networks with uncertainty [J]. Journal of Advanced Transportation, 2008, 42(3): 265-290

[16] LamW H K, ShaoH, SumaleeA. Modelling impacts of adverse weather conditions on a road network with uncertainties in demand and supply [J]. Transportation Research Part B, 2008, 42(10): 890-910

[17] ChenA, ZhouZ. The alpha-reliable mean-excess traffic equilibrium model with stochastic travel times [J]. Transportation Research Part B, 2010, 44(4): 493-513

[18] RockafellarR T, UryasevS. Optimization of conditional value-at-risk [J]. Journal of Risk, 2000, 2(3): 21-41

[19] FosgerauM, KarlstromA. The value of reliability [J]. Transportation Research Part B, 2010, 44(1): 38-49

[20] FosgerauM, EngelsonL. The value of travel time variance [J]. Transportation Research Part B, 2011, 45(1): 1-8

[21] OrdonezF, Stier-mosesN E. Wardrop equilibria with risk-averse users [J]. Transportation Science, 2010, 44(1): 63-86

[22] NgM W, SzetoW Y, WallerS T. Distribution-free travel time reliability assessment with probability inequalities [J]. Transportation Research Part B, 2011, 45(6): 852-866

[23] GervaisS, HeatonJ B, OdeanTOverconfidence, investment policy, and executive stock options [R], 2002

[24] BilingsleyPProbability and measure [M], 1995, New York, John Wiley and Sons, Inc.: 359-363

[25] FacchineiF, PangJ SFinite-dimensional variational inequalities and complementarity problems [M], 2003, New York, Springer: 41-46

[26] NagurneyANetwork economics: A variational inequality approach [M], 1993, Dordrecht, Kluwer Academic Publishers: 14-15

[27] HanD. A modified alternating direction method for variational inequation problems [J]. Applied Mathematics and Optimization, 2002, 45(1): 63-74

[28] KorpelevichG M. The extra-gradient method for finding saddle points and other problems [J]. Matecon, 1976, 12: 747-756

[29] HanJ-y, XiuN-h, QiH-duoNonlinear complementarity theory and algorithms [M], 2006, Shanghai, Shanghai Scientific & Technical Publishers: 132-141

[30] ZhouZ, ChenA. Comparative analysis of three user equilibrium models under stochastic demand [J]. Journal of Advanced Transportation, 2008, 42(3): 239-263

[31] LiuT-l, HuangH-j, TianL-jun. Microscopic simulation of multi-lane traffic under dynamic tolling and information feedback [J]. Journal of Central South University of Technology, 2009, 16(5): 865-870