Departure Time and Route Choices with Bottleneck Congestion: User Equilibrium under Risk and Ambiguity
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1 Departure Time and Route Choice with Bottleneck Congetion: Uer Equilibrium under Rik and Ambiguity Yang Liu, Yuanyuan Li and Lu Hu Department of Indutrial Sytem Engineering and Management National Univerity of Singapore July 26, 2017
2 Outline 1 Introduction 2 General Model Setting 3 Single-Route Problem 4 Two-Route Problem 5 Concluion
3 Morning Commute Problem Bottleneck Model [Vickrey, 1969] Departure time choice Determinitic travel time Extenion Route choice [e.g., Arnott et al 1992; Liu and Nie 2011; Qian and Zhang 2013] Uer heterogeneity [e.g., Newell 1987; Daganzo 1985; Liu et al., 2015] Elatic demand [e.g., Arnott, de Palma and Lindey 1993] Stochaticity [e.g., Lindey 1995, 1996; Arnott et al. 1999; Fogerau 2008; Siu and Lo 2009; Xiao et al. 2015] Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 1/25
4 Literature Rik in deciion-making Implication in route choice in tranportation Rik meaure: Mean-variance [Markowitz 1952] and Expected utility theory Stochatic routing [e.g., Nie and Wu 2009; Miller-Hook and Mahmaani 2003] Uer equilibrium [e.g., Yin et al. 2004; Connor et al. 2007; Chen and Zhou, 2010] Ambiguity in deciion-making Evidence: [Ellberg 1961; Hu et al. 2005] Ambiguity heterogeneity in traveler deciion-making [Sikka 2012] Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 2/25
5 Literature Rik in deciion-making Implication in route choice in tranportation Rik meaure: Mean-variance [Markowitz 1952] and Expected utility theory Stochatic routing [e.g., Nie and Wu 2009; Miller-Hook and Mahmaani 2003] Uer equilibrium [e.g., Yin et al. 2004; Connor et al. 2007; Chen and Zhou, 2010] Ambiguity in deciion-making Evidence: [Ellberg 1961; Hu et al. 2005] Ambiguity heterogeneity in traveler deciion-making [Sikka 2012] A unified framework of rik and ambiguity [Qi et al. 2016] Combining CARA model and Hurwicz model [Hurwicz 1951] Ambiguity-aware CARA Travel Time Route choice in a tatic network equilibirum Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 2/25
6 Motivation Uncertain travel time under ambiguity The perfect information of the travel time ditribution i rarely available to the commuter. Implication in departure time and route choice under rik and ambiguity Dynamic bottleneck equilibrium - Cloed-form olution - Homogeneou deired arrival time, value of time and unit chedule delay cot Policy Inight Provide information ytem Deign control trategie, e.g., congetion pricing Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 3/25
7 Outline 1 Introduction 2 General Model Setting 3 Single-Route Problem 4 Two-Route Problem 5 Concluion
8 Model Setting The commuter preference Identical value of time (α), unit chedule delay cot (β for early arrival and γ for late arrival) Attitude toward rik and ambiguity of cla j Rik parameter: λ j (, + ) Ambiguity parameter: µ j [0, 1] Uncertain travel time on route i T i (t) = Q i (t)/ i + T 0i + ε i The random travel delay belong to a bounded ditributional uncertainty et H H = {P E P ( ε i ) = 0, P( ε i [ i, i ]) = 1} i : the maximum variation of travel time Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 4/25
9 Ambiguity-aware CARA travel time (ACT) Definition of ACT ACT G j ( T i (t)) for the commuter in group j who depart from home at t on route i, Ω R for the commuter with parameter V = (λ j, µ j, H) i G j ( T i (t)) = { µ j up P H 1 λj ln E P (exp(λ j T i (t))) + (1 µ j ) inf P H 1 λj ln E P (exp(λ j T i (t))), λ j 0 µ j up P H E P ( T i (t)) + (1 µ j ) inf P H E P ( T i (t)), λ j = 0 G j ( T i (t)): the perceived travel time Ω: the tate-pace of the uncertain T i (t) P: the unknown true probability ditribution of T i (t) H: a ditributional uncertainty et Adopted from (Qi et al., 2016), ACT i a function of departure time now Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 5/25
10 The Cloed Form of ACT Propoition The ambiguity-aware CARA travel time G j ( T i (t)) for the commuter with parameter V = (λ j, µ j, H) admit the cloed form: G j ( T i (t)) = T i (t) + f (λ j, µ j, i ). where µ j 1+exp(2 λj i ) λ j ln( 2 ) µ j i, λ j > 0 f (λ j, µ j, i ) = 1 µ j 1+exp(2 λj i ) λ j ln( 2 ) (1 µ j ) i, λ j < 0 0, λ j = 0 Interpretation of f (λ j, µ j, i ): perceived uncertainty in travel time f (λ j, µ j, i ) < 0 when λ < 0; f (λ j, µ j, i ) > 0 when λ > 0. Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 6/25
11 Ambiguity-Aware CARA Commute Cot (ACC) Definition The ACC F ( C i (t)) for the commuter who depart from home at t on route i, with parameter V = (λ j, µ j, H) i: { β(t F j( Ci(t)) = αg j( Ti(t)) t G j( Ti(t))), t t G j( Ti(t)) + γ(t + G j( Ti(t)) t ), t t G j( Ti(t)) The commuter make deciion baed on ACT At equilibrium, no one can reduce her own ACC commute cot by changing departure time or route choice. Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 7/25
12 Outline 1 Introduction 2 General Model Setting 3 Single-Route Problem 4 Two-Route Problem 5 Concluion
13 Single-Route Single-Cla Problem Homogeneou preference: V = (λ, µ, H) Queuing rate at equilibirum df ( C(t)) dt = 0 dq(t) dt = { β, α β γ, α+γ t t tm tm t te ACC commute cot: F ( C) = δn + α T 0 + α f (λ, µ, ) Perceived uncertainty cot: α f (λ, µ, ) Cumulative departure and arrival Departure curve for rik avere cla Departure curve for rik eeking cla Departure curve for rik neutral cla tp tmp tm tmo tn to Time UE tep te t teo Cumulative departure and arrival tp Departure curve for rik avere cla Departure curve for rik eeking cla Departure curve for rik neutral cla t to tm tmp tep Time Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 8/25 SO te t tmo teo
14 Single-Route Single-Cla Problem Both the ACC commute cot and the perceived uncertainty cot in time unit are increaing with ambiguity and rik The congetion hift to earlier for rik-avere and later for rik-eeking µ µ λ F ( C) λ f (λ, µ, ) Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 9/25
15 Model Setting for Single-Route Two-Cla Problem The commuter with parameter V 1 = (λ 1, µ 1, H) The rik-avere cla (j = 1): the perceived uncertainty f (λ 1, µ 1, ) The rik-eeking cla (j = 2): the perceived uncertainty f (λ 2, µ 2, ) The preference gap between the two clae f 12 = f (λ 1, µ 1, ) f (λ 2, µ 2, ) > 0 Analytically derive UE olution The UE olution depend on f 12 Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 10/25
16 Single-Route Two-Cla Problem: Uer Equilibrium Propoition (Uer equilibrium for the ingle-route two-cla problem) For the ingle-route bottleneck model with a rik-avere cla and a rik-eeking cla (parameter V j = (λ j, µ j, H), j = 1, 2, the UE olution depend on the gap between the cot of uncertainty in time unit f 12 f (λ 1, µ 1, ) f (λ 2, µ 2, ) Cae 1: f 12 [ δ γ Cae 2: f 12 [ δ γ Cae 3: f 12 [0, δ β Cae 4: f 12 [0, δ γ N 1 + δ N 2 β N 1 δ N 2 β, δ γ N 2 δ N 1 γ ] N 1 δ N 2 β ], ) N 1 + δ β N 2 ] Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 11/25
17 Single-Route Two-Cla Problem: Uer Equilibrium Small preference gap f 12 : f 12 [0, δ β N 2 δ γ Two clae hare the ame iocot curve N 1 ] Given certain parameter etting, only one of them occur We focu on the aggregated departure curve Aumption: the rik-avere cla depart earlier than the rik-eeking cla to avoid infinite poibilitie of departure order Cumulative departure and arrival Departure curve at UE for rik-avere cla Departure curve at UE for rik-eeking cla Departure curve at SO for rik-avere cla Departure curve at SO for rik-eeking cla Cumulative departure and arrival Departure curve at UE for rik-avere cla Departure curve at UE for rik-eeking cla Departure curve at SO for rik-avere cla Departure curve at SO for rik-eeking cla Time Time Cae 3 Cae 4 Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 12/25
18 Single-Route Two-Cla Problem: Uer Equilibrium Intermediate preference gap f 12 [ δ γ Two peak occur in peak-hour congetion N 1 δ β The rik-avere cla prefer earlier time lot N 2, δ N 1 γ + δ N 2 β ] Cumulative departure and arrival Departure curve at UE for rik-avere cla Departure curve at UE for rik-eeking cla Departure curve at SO for rik-avere cla Departure curve at SO for rik-eeking cla Time Cae 2 Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 13/25
19 Single-Route Two-Cla Problem: Uer Equilibrium Large preference gap f 12 [ δ γ Two eparate peak N 1 + δ β N 2, ) The rik-avere cla prefer earlier time lot Cumulative departure and arrival Departure curve at UE for rik-avere cla Departure curve at UE for rik-eeking cla Departure curve at SO for rik-avere cla Departure curve at SO for rik-eeking cla Time Cae 1 Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 14/25
20 Numerical Example for Single-route Two-clae Problem Two clae departure time window deviate with f 12 The total expected travel time (TE) i non-increaing with f 12. The heterogeneou preference toward rik and ambiguity can pread the departure and alleviate average congetion. f 1 f 2 f 12 t 1 t m1 t e1 t 2 t m2 t e2 TE Cae Cae Cae Cae Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 15/25
21 Outline 1 Introduction 2 General Model Setting 3 Single-Route Problem 4 Two-Route Problem 5 Concluion
22 Model Setting for the Two-Route Problem Two parallel route: A highway (route 1) and a local arterial (route 2) The maximum variation: 1 > 2 The free flow travel time: T01 < T 02 Homogeneou commuter: (λ, µ, i ), i = 1, 2. Flow on route i: n i ACC commute cot on route i: F i ( C) = α(t 0i + f (λ, µ, i )) + δn i i The gap of uncertainty cot between two route: f 12 = f (λ, µ, 1 ) f (λ, µ, 2 ) Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 16/25
23 Flow Ditribution at UE Propoition 4 For the two-route problem with homogeneou commuter, the commuter ditribution (n 1 and n 2) between route 1 and route 2 at UE i: Condition Flow ditribution n 1 n 2 Cae 1 N1 0 and N N1 N 0 Cae 2 N1 0 and N > N N N Cae 3 N2 0 and N N2 0 N Cae 4 N2 0 and N > N (N N2 ) where N 1 = 1 α δ (T02 T01 f 12) and N 2 = 2 α δ (T01 T02 + f 12) (N N 1 ) N N The impact of uncertainty on flow ditribution by comparing the interior olution with determinitic model N1 more commuter on route 1 in Cae N2 more commuter on route 2 in Cae 4 Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 17/25
24 Impact of Uncertainty Propoition 5 For the two-route bottleneck model with homogeneou commuter, the route choice of commuter with parameter V = (λ, µ, H) between route 1 and route 2 depend on the attitude toward rik, i.e., the flow on route 1 (n 1) with repect to maximum variation ha the following tendency: Rik attitude 1 2 Flow pattern at UE λ > 0 Any one of Cae 1-4 λ < 0 Any one of Cae 1-2 λ = 0 Any one of Cae 1-2 For the rik-avere commuter: the larger 1 ( 2) will make the highway le (more) attractive. For the rik-eeking commuter: the larger 1 ( 2) will make the highway more (le) attractive. For the rik-neutral commuter, the uncertaintie have no impact on the route choice. Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 18/25
25 Numerical Example The impact of uncertainty on route choice at UE Flow pattern n 1 rik avere n 2 rik avere n 1 rik eeking n 2 rik eeking Flow pattern n 1 rik avere 0 n 2 rik avere 500 n 1 rik eeking n rik eeking The flow ditribution with 1 The flow ditribution with 2 The monotonicity of flow on the highway Rik-avere cla: decreae with 1 and increae with 2 Rik-eeking cla: increae with 1 and decreae with 2 Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 19/25
26 Numerical Example The monotonicity of ytem performance at UE Rik avere and peimitic ambiguity Rik eeking and optimitic ambiguity Rik neutral Rik-avere and peimitic ambiguity Rik-eeking and optimitic ambiguity Rik-neutral TE TC Total expected travel time given 1 Policy inight: Sytem cot at UE given 1 Rik-avere commuter: reducing uncertainty on the highway will improve the two ytem performance meaure imultaneouly. Rik-eeking commuter: the trade-off between TE and TC hould be conidered. Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 20/25
27 Numerical Example The enitivity of price of anarchy (PoA) given rik λ and ambiguity µ TC UE /TC SO µ =0.5 µ=0.6 µ=0.7 µ=0.8 µ=0.9 µ=1 TC UE /TC SO µ=0 µ=0.1 µ=0.2 µ=0.3 µ=0.4 µ= The PoA for rik-avere cla λ The PoA for rik-eeking cla The upper bound and lower bound in thi example can be oberved For rik-avere commuter, the more peimitic preference toward ambiguity reult in higher efficiency lo. For rik-eeking commuter, the more optimitic preference toward ambiguity reult in higher efficiency lo. λ Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 21/25
28 Outline 1 Introduction 2 General Model Setting 3 Single-Route Problem 4 Two-Route Problem 5 Concluion
29 Concluion Single-route ingle-cla problem Both the perceived uncertainty in time unit f (λ, µ, ) and the ACC commute cot are monotonically increaing with ambiguity and rik. Compared with the rik-neutral cla, - The rik-avere (rik-eeking) cla will overetimate (underetimate) the commute cot. - The departure time window i hifted earlier (later) for the rik-avere (rik-eeking) cla. Single-route two-clae problem Four congetion pattern at UE baed on the preference gap between two clae A mixture of heterogeneou preference toward rik and ambiguity may pread the departure time choice and relieve the average peak hour congetion Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 22/25
30 Concluion Two-route problem Decreaing maximum time variation on the highway, e.g., by providing traffic information, - will increae it flow when commuter are rik-avere, but will decreae it flow when commuter are rik-eeking. - will reduce total expected travel time and perceived ytem cot for rik-avere commuter Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 23/25
31 Thank! -The end-
32 Reference
33 Single-route Single-cla problem At UE The critical time t = t T 0 f (λ, µ, ) δ β t m = t T 0 f (λ, µ, ) δ α t e = t T 0 f (λ, µ, ) + δ γ N N N, Total expected travel time at UE: TE = 1 2 δ α N2 + N T 0
34 Single-route Single-cla problem At SO The critical time The minimum ytem cot at SO: t = t T 0 f (λ, µ, ) δ β t m = t T 0 f (λ, µ, ) t e = t T 0 f (λ, µ, ) + δ γ TC = δn2 2 N N + α T 0N + α f (λ, µ, )N In the determinitic bottleneck model, the ytem cot (excluding the free flow travel time αt 0 N) i reduced by half at SO In thi model, the ytem cot i reduced by le than a half for rik-avere commuter and more than a half for rik-eeking commuter at SO.
35 Single-route Two-clae Problem: UE Cae 1: f 12 [ δ γ N 1 + δ β The ACC commute cot N 2, ) F ( Cj) = δ N j + α T 0 + α f (λ j, µ j, ), The queue length i given by { β α β Q(t) = (t t j ), t j t t mj γ (t α+γ tm j ) + Q(t mj ), t mj t t ej, with t j t mj t ej = t T 0 f (λ j, µ j, ) δ β = t j + 1 θ = t mj + 1 ρ δ N j β δ N j, γ N j The total expected travel time for all commuter i TE = 1 2 δ α ΣjN2 j + N T 0
36 Single-route Two-clae Problem: UE Cae 2: f 12 [ δ γ N 1 δ β The ACC commute cot { F ( C1) = 1 (β N δ N F ( C2) = 1 (γ N δ N N 2, δ N 1 γ + δ N 2 β ] βf12) + α T0 + α f (λ1, µ1, ) γf12) + α T0 + α f (λ2, µ2, ) the queue length i given by β (t α β t 1), γ Q(t) = (t α+γ tm 1) + Q(t m1 ), β α β (t t 2) + Q(t e1 ), γ (t α+γ tm 2) + Q(t m2 ), where t 1 = t T [ N 1 t m1 = t ( N 1 2θ t e1 = t m1 + 1 ( N 1 2ρ t 2 = t e1 t m2 = t ( N 1 2θ t e2 = t m2 + 1 ( N 2 2ρ + δ N β + δ N f12) β δ N + f12) β + δ β + δ γ t 1 t t m1 t m1 t t e1 t 2 t t m2 t m2 t t e2 + f (λj, µj, )] j N + f12) N f12)
37 Single-route Two-clae Problem: UE Cae 3: f 12 [0, δ β N 2 δ γ N 1 ] and N 2 N 1 β γ The ACC commute cot { F ( C1) = δ N + α T0 + α f (λ1, µ1, ) βf12 F ( C2) = δ N + α T0 + α f (λ2, µ2, ) the queue length i given by β (t α β t 1), β Q(t) = α β (t te 1) + Q(t e1 ), γ (t α+γ tm 2) + Q(t m2 ), where t 1 = t T 0 f (λ 2, µ 2, ) δ β t e1 = t 1 + α β N 1 α t 2 = t e1 t 1 t t e1 t e1 t t m2 t m2 t t e2 t m1 = t T 0 f (λ 1, µ 1, ) + β α f12 δ α t m2 = t T 0 f (λ 2, µ 2, ) δ N α t e2 = t T 0 f (λ 2, µ 2, ) + δ N, γ N N
38 Single-route Two-clae Problem: UE Cae 4: f 12 [0, δ γ N 1 δ β N 2 ] and N 2 N 1 β γ The ACC commute cot { F ( C1) = δ N + α T0 + α f (λ1, µ1, ) F ( C2) = δ N + α T0 + α f (λ2, µ2, ) γf12 the queue length i given by β (t α β t 1), γ Q(t) = α+γ (t tm 1) + Q(t m1 ), γ (t α+γ te 1) + Q(t e1 ), where N t 1 t t m1 t m1 t t e1 t e1 t t e2 t 1 = t T 0 f (λ 1, µ 1, ) δ β t m1 = t T 0 f (λ 1, µ 1, ) δ N α t m2 = t T 0 f (λ 2, µ 2, ) + γ α f12 δ α t e1 = t T 0 f (λ 1, µ 1, ) + δ N α+γ γ α t 2 = t e1 t e2 = t T 0 f (λ 1, µ 1, ) + δ N, γ N N 2
39 Finding and Contribution Single-route model with homogeneou preference The impact of rik preference Rik-avere Rik-neutral Rik-eeking The movement of departure time window - The impact of ambiguity Peimitic Neutral Optimitic Magnitude of the movement - Perceived uncertainty cot - Single-route model with heterogeneou preference Stagger the departure time choice Relieve traffic congetion Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 24/25
40 Finding and Contribution Two-route model The impact of uncertainty on highway The flow on highway The expected travel time (TE) Total ACC commute cot (TC) Rik-avere Rik-eeking Policy inight Reduce uncertainty on highway to relieve traffic congetion if the commuter are rik-avere Make the trade-off between TE and TC if the commuter are rik-eeking Yang Liu, Yuanyuan Li and Lu Hu National Univerity of Singapore 25/25
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