First-Best Dynamic Assignment of Commuters with Endogenous Heterogeneities in a Corridor Network

ISTTT 22@Northwestern University

Minoru Osawa, Haoran Fu, Takashi AkamatsuTohoku University

July 24, 2017

Background

►Departure-time choice equilibrium model for commute problems has long been studied.

- Vickrey, 1969; Hendrickson & Kocur, 1981; etc..

►In the long term, travel demand patterns vary over time because of commuters’ choices (e.g. residential locations and jobs) and inevitably affect short-term dynamic.

An effective framework incorporating the short

term and long term is worth of studying.2

Purpose

►Propose a framework that synthesizes long-termtravel demand forecasting with short-term dynamic traffic assignment.

▪ Long-term equilibrium: residential location; job

▪ Short-term equilibrium: departure time

►Investigate properties of equilibrium. ▪ Temporal/spatial patterns

3

Contributions

►Integration of dynamic bottleneck congestion and Urban Economics

cf. Arnott, 1998; Gubins & Verhoef, 2014;

Takayama & Kuwahara, 2016

▪ In a corridor network with multiple bottleneckscf. Shen & Zhang, 2009; Tian, et al., 2012; etc.

▪ With endogenous heterogeneities

►Policy implication for investment on bottleneck capacity expansions

4

Outline

►Model▪ Settings: Network, Commuters’ Behavior▪ Equilibria: Short-term and Long-term▪ Integrated Model

►Analysis of Short-term Equilibrium▪ Special Case I: Corridor without Heterogeneities▪ Special Case II: Single Bottleneck with Heterogeneities▪ General Case: Corridor with Heterogeneities

►Analysis of Long-term Equilibrium▪ Spatial Sorting Patterns▪ Policy Implications

►Concluding Remarks5

Network

►A corridor with multiple bottlenecks

bottleneck (BN)

residential location

jobs at CBD

BN capacityI i 1

labor demandIA iA 1A2AiA

2i

1L

2L

JL

jL jL

6

land supply

►Commuters’ choices▪ In the long term

- Residential location i with endogenous land rent- Job j with endogenous wage

▪ In the short term- Departure time (arrival time t at the CBD) with

commuting cost ►Commuters’ objective

▪ Maximize own utility (minimize own cost)

Commuters’ Behavior

ir

jw

, ,,, ,max ( m minax }() ){ j i

i ji j

ti j ti j w r tv t C

short-termlong-term

, ( )i jC t

7

Interactions between Short and Long Terms

►Feedback structure

Long-term Equilibrium(location choice i and

job choice j)

Short-term Equilibrium(arrival-time choice t)

,{ }i j ,{ }i jQ

Travel

demand

Equilibrium

commuting

cost

Short-term policy: Travel Demand Management (TDM) 8

Long-term policies:land supply , labor demand , BN capacity{ }iA { }jL { }i

Short-term TDM Policy: TNP Scheme

►Definition▪ A tradable network permit (TNP) is a right that

allows its holder to pass through a pre-specified BN during a pre-specified time period. - Akamatsu, 2007; Wada & Akamatsu, 2013;

Akamatsu & Wada, 2017

►Assumptions▪ A competitive market is assumed for trading permits.▪ The number of issued permits is limited to be

less than or equal to BN capacity.Queues of all BNs are eliminated.

9

►Optimal-choice conditions in short-term equilibrium

▪

►Permit market clearing conditions

, , ,

, , ,

( ) 0 if

( ) 0(

if ) 0

,(

,) 0

i j i j i j

i j i j i j

C t q

C t q

ti j t

t

schedule delay free-flow travel time

permit price

supply (BN capacity) demand

commuting cost arrival-flow rateequilibrium cost

Short-term Equilibrium under TDM Policy

,

,

( ) 0 if ( ) 0,

( )

( ) 0 if 0m j

m j

i m j ii

i m j ii

tq t p

q t pi t

t

, ( ) ( )[ ] ( )i j j i mm iC t s t d p t

10

value of time

►Optimal choice conditions in long-term equilibrium

▪►Land/labor market clearing conditions

supply demand

demand supply

equilibrium utility level travel demand

Long-term Equilibrium

land market

labor market

11

, ,

, ,

0 if

0,

0

0 if j i i j i j

j i i j i j

Vi

w r Q

w r Qj

V

,

,

0 if

0 if

0

0i i j ij

iji i j

A r

A r

Qi

Q

,

,

0 if

0 if

0

0j i j ji

j i j ji

L

Lj

w

Q w

Q

, , ( )di it

j jQ q t t

►Equivalent linear programming problem (LP)

▪

Efficiency of The Integrated Equilibrium

permit supply

land supply

labor demand

total cost

Proposition 1: The integrated equilibrium is socially optimal

in the sense that it minimizes the total transportation cost.

12

, ,

,

,

,

min ( ) ( )d

. . ( ) ( )

( )d ( )

( )d ( )

)

( ,

i j i ji

m j i im i

i j i i

i j

j t

j

j

j

t

tji

c t q t t

s t q t p

q t t A r

q t t L

t i t

i

jw

q 0

, ( ( )[) ]i j j ic t s t d

The long-term problem (master problem)

The short-term problem (subproblem)

Benders Decomposition of the Integrated Model

13

,

1,

,

min

. . ( )

( )

j i i ji

i j i i

i j j ji

j

j

Q z

s t Q A r

Q L w

d

Q 0

1 ,

,

, ,*,

min ( ) ( )d

. . ( ) ( )

( )d

( )

( )

j i ji

m j i im i

i j

j t

j

i jt

i j

z s t q t t

s t q t p

q t

t

t Q

q 0

Outline

►Model▪ Settings: Network, Commuters’ Behavior▪ Equilibria: Short-term and Long-term▪ Integrated Model

►Analysis of Short-term Equilibrium▪ Special Case I: Corridor without Heterogeneities▪ Special Case II: Single Bottleneck with Heterogeneities▪ General Case: Corridor with Heterogeneities

►Analysis of Long-term Equilibrium▪ Spatial Sorting Patterns▪ Policy Implications

►Concluding Remarks14

►Model settings▪ Single job class but multiple BNs

►Equivalent LP

Special Case I: Corridor without Heterogeneities

I i 1

IA iA 1A2A

{ }jL

2

{ }jLL

15

min ( ) ( )d

. . ( ) ( )

( )d (

( ) ,

)

i ii

m i im i

i i

t

it

c t q t t

s t q t p

q

t

iA

i

t t r

t

q 0

Analytical Solution

3

2

1

t

*( )iiq t

1T

2T

3T

1Q

2Q

3Q

[Special Case I]

Equilibrium arrival-flow rates

at the CBD

*3

*2

*1

t

( )s t

1T

2T

3T

Equilibrium commuting costs

Proposition 2-A (location-based temporal sorting pattern):Commuters residing closer to the CBD arrive at times in

a time window closer to the desired arrival time.

16

*2 ( )p t

*3 ( )p t

*1 ( )p t

T

t

( )s t

Schedule delay function

( )p t

i 1

iA 1A2A

2

1L

2L

JL

jL

►Model settings▪ Single BN but multiple job classes

►Equivalent LP

Special Case II: Single BN with Heterogeneities

17

1

1, 1,

1,

,

1

1

( )

min ( ) ( )d

. . ( ) ( )

( )d ( )

jj tj

j

j

j

tj j

c t q t t

s t q t t t

j

p

q t t L w

q 0

Analytical Solution

*1 ( )p t

*1,1

*1,2

t

1,2T

1,1T

1

*1, ( )

j jq t

t1,2T1,1T

1,1Q

1,2Q

[Special Case II]

Equilibrium arrival-flow rates

at the CBD

Equilibrium commuting costs

and permit prices (isocost curve)

Proposition 2-B (job-based temporal sorting pattern):Commuters of higher value of time arrive at

times closer to the desired arrival time.

18

General Case: Corridor with Heterogeneities

t t

,* ( )i jji

q t

*2

( )mmp t

2

1*2,1*1,1*2,2

*1,2

1,2T

1,1T

2,2T

2,1T

1L

2L

1A

2A

1,2T

1,1T

2,2T

2,1T

Equilibrium arrival-flow rates

at the CBD

Equilibrium commuting costs

and permit prices (isocost curve)

Proposition 3 (temporal sorting pattern):Flows of commuters admit a combination of

sorting patterns in Special Case I and II.

19

Outline

►Model▪ Settings: Network, Commuters’ Behavior▪ Equilibria: Short-term and Long-term▪ Integrated Model

►Analysis of Short-term Equilibrium▪ Special Case I: Corridor with Homogeneities▪ Special Case II: Single Bottleneck with Heterogeneities▪ General Case: Corridor with Heterogeneities

►Analysis of Long-term Equilibrium▪ Spatial Sorting Patterns▪ Policy Implications

►Concluding Remarks20

►Solving the short-term problem gives the equilibrium commuting cost for given travel demand .

►Equivalent convex optimization problem for the long-term equilibrium ignored in a naive model

The Long-term Problem

*( )ρ QQ

21

*2 ,

,

,

min

. . ( )

( )

( )di

j i i j ii i

i j i i

i j j

j

j

j

i

z Q

s t Q A r

Q L w

d

i

j

0 QQ 0ρ ω ω

Short-term Equilibrium *ρQ

Integrated Model vs. Naive Model

►Integrated model ►Naive model

Long-term

*ρ *QShort-term Nρ

C

exogenously given and fixed

{ , , | }q p ρ Q

{ , , | }Q r w ρLong-term

NQShort-term

{ , , | }q p ρ Q

{ , , | }Q r w ρ

*, Nρ ρ*, NQ Q

: equilibrium commuting cost : travel demand: commuting costC

22

TNP scheme TNP scheme

Spatial Sorting Patterns

►Integrated model vs. naive model

Integrated model Naive model

Proposition 4: The location-job distribution of

commuters is more dispersed in the integrated model.

23

Job index j Job index j

*, NQ Q

Loca

tion

inde

x i

non-zero elements of

*7 7

N

Long-term Policy: BN Capacity Expansions

►BN capacity expansions▪ A long-term policy of road construction

►Self-financing principle

- Wada & Akamatsu, 2013; Akamatsu, 2017

▪ Optimal amount of investment in total:

24

Lemma: The optimal amount of investment on BN

capacity expansions coincides with the revenue of permits.

*( ) ( | )di

ii tR tp t

Q Q

Policy Implication for BN Capacity Expansions

►Integrated model ►Naive model

25

Long-term

*ρ *QShort-term Nρ

C

{ , , | }q p ρ Q

{ , , | }Q r w ρLong-term

NQShort-term

{ , , | }q p ρ Q

{ , , | }Q r w ρ

Proposition 5: The naive model overestimates optimal

amount of total investment on BN capacity expansions,

i.e. *( ) ( ).NR RQ Q

BN capacity expansions

TNP scheme TNP scheme

BN capacity expansions

Concluding Remarks

►An integrated model effectively synthesizing the long term and short term is proposed.

►Analytical solutions for DSO problems in a corridor network are derived.

►Qualitative properties of solution in equilibrium are investigated.

►A model considering short and long terms

separately misleads long-term policies, such as investment on BN capacity expansions.

26

►Definition

►False BNs do not affect essential properties, but only cause irrelevant freeness in the solution.

►Criterion to detect false BNs

BN i is a false BN if and

only if

for some

*3

*2

*1

( )s t

overlap

(if BN 3 is false)

Appendix: False BNs without Heterogeneities

t

n nn m n i

m i

A A

.m i

A false bottleneck is a bottleneck whose permit price

is always zero in equilibrium.

*3 ( )p t

27

Appendix: Comparisons of Land Rents and Wages

►Land rent ►Wage

28

►Examples

Optimal Amount of Investment on Each BN

29

Naive model

Integrated model

Naive model

Integrated model

Naive model

Integrated model

Naive model

Integrated model