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