transportation and spatial modelling: lecture 15

17/4/13

Challenge the future Delft University of Technology

CIE4801 Transportation and spatial modelling

Rob van Nes, Transport & Planning

Spatial modelling: concept and descriptive models

2 CIE4801: Spatial modelling 1

• Two topics in balance? •  Transportation modelling 14 lectures •  Spatial modelling 2.5 lectures

• Three reasons •  Main expertise T&P •  Similar modelling aproaches •  Interaction within transport system is

dominant

•  Expertise within T&P •  3 PhD-studies •  Dr.ir. R. Verhaeghe

Transportation & spatial modelling

Transport system

(transport and traffic services)

Accessibility (ease to reach a

location)

Land use (spatial planning)

Activities (living. work. recreation)


Content

• Background •  Early (economic) models •  Some empirical findings

• Descriptive models •  Accessibility •  Hansen •  Lowry •  Immers & Hamerslag

•  Land use and transportation models (Tuesday) •  TIGRIS XL •  Choice modelling for household and firm allocation


2.

Background: classical models


Main mechanism for the transportation and spatial system

Transport system (transport and traffic

services)

Accessibility (ease to reach a

location)

Land use (spatial planning)

Activities (living. work. recreation)


Single commodity model •  (large number of) producers want to sell a certain agricultural

commodity a central market place •  (large number of) owners rent land to the producers •  land owners let the producers bid on the land space •  producers will bid a rent with which they still can make profits Question: What rents will the producers pay for the land, and how much land will be used?

land used by producer

central market

?

Classical models: Von Thunen (1826)


i j surplus for producer

quantity (units) produced selling price per unit production costs per unit transportation costs per unit per km distance to central market place

======

j

ij

Sqpckd

( )= − −j ijS q p c kd

ijd

= what is left for paying rent

land used by producers

Von Thunen: single commodity model


surplus for producer of commodity quantity (units) produced of commodity selling price per unit of commodity production costs per unit of commodity transportation costs per un

=

====

mjm

m

m

m

S mq mp mc mk it per km of comm.

distance to central market place=ij

md

( )= − −m m m m mj ijS q p c k d

ijdi

Von Thunen: Multi-commodity model


Classical models: Christaller (1932)


•  evenly distributed population of self-sufficient farmers •  farmers can produce beer for a certain area

producing for a larger area: - cheaper production process - more expensive distribution

Classical models: Christaller (1932)


Spatial levels according to Christaller

Administrative principle Transport principle Marketing principle

Basic area level 1 Central place level 1 Basic area level 2


Morphological perspective

•  Every level shows new details

Dispersion or maximal homogeniety

Concentration or minimal homogeniety


One of many possible classifications

Name Radius [km]

Surface [km2]

Inhabitants

Village 0,3 0,3 1.000

Town 1 3 10.000

City 3 30 100.000

Agglomeration 10 300 1.000.000

Metropolis 30 3.000 10.000.000


Classical models: Hotelling


Agglomeration factors

• Advantages of agglomerations

•  Positive externalities

• Associated development

•  Education, health care, financial services

• Transport is essential

• Congestion limits growth of agglomerations


3.

Some empirical findings


5

10

15

20

25

30

35

40

5.000 7.000 9.000 11.000 13.000 15.000 Car use per capita (km)

Urb

an d

ensi

ty (

pers

./ha

)

Houston

Montreal

Toronto

Chicago New York

Los Angeles

Phoenix

Density and car use (North America, 1991)


Built up area prior to introduction of mechanical transport Development consequent on (steam) railways Development consequent on tramways Development consequent on buses Development consequent on private car

Historical pattern


• The number of cities of a size greater than N is proportional to 1/N

Scale laws: Zipf


•  Local conditions play a major role for the location of settlements •  Safety •  Water and food •  Natural resources •  Accessibility

• Mechanisms/drivers change over time •  Other drivers might take over

•  Law of increasing returns •  Drivers may loose importance

•  Stand-still or decline •  Permanent or temporary

Settlement patterns are also determined by chance


2.

Accessibility


Choice destina-

tion

Choice mode

Choice route/time

Travel time & costs

Accessibility

Attractiveness

Choice location

investors

Build

Choice location

users

Move

Activities

Ability to travel

Choice trip

Mechanism in more detail: Wegener’s circle

M. Wegener, 1995, 2004


Definitions of accessibility?

•  Ease to reach a location……..

•  From where •  Specific location? •  From all locations? •  From all locations within a certain distance/travel time? •  ……

• By whom? •  Individuals/households/companies at that location? •  Possible clients/workforce? •  ……….


Generic indicator: Potential value

•  Sum of all clients/jobs/etc. that can be reached within a certain distance/time:

• Different options for f(cij):

( )( ) max, :i j ij ijj

PV g M f c j c c= ⋅ ⋅ ∀ ≤∑

( )

( ) ij ij

ij ij i j ijj

c cij i j

j

f c c PV g M c

f c e PV g M eβ β− −

= ⇒ = ⋅ ⋅

= ⇒ = ⋅ ⋅

∑

∑ Maximum is best?

Minimum is best?


• Choice for M depends on perspective •  Active accessibility: where you can go to

e.g. the number of jobs you can reach from your home •  Passive accessibility: who can reach you

e.g. number of clients that can reach you

• Choice for cij: •  Time, distance, generalised costs, logsum over modes….

• Notice the similarity with trip distribution topics

So what to use for the atributes?


4.1

Descriptive models: Hansen-model


• Given the network and the location of the jobs

• What is the location of the inhabitants?

• Main assumption: people prefer a location having the highest accessibility

• However, there’s not always enough space available

What is it for?


Hansen-model

Assuming an increase of the population, e.g. ΔR inhabitants, Where will they settle?

Number of jobs, Mj

Available space, Lj

Costs between zones i and j Cij 1 2

3

C12

C23 C13 Accessibility of zone i is e.g.

1

nj

i bj ij

MPV

c=

=∑

The increase in inhabitants in zone i is

1

i ii n

j jj

PV LR RPV L

=

⋅Δ = Δ

⋅∑Note that if you distribute jobs

you should use inhabitants for M


4.2

Descriptive models: Lowry-model


Lowry model: different type of jobs

• Distinction between basic jobs and service jobs •  Basic jobs: location bound e.g. industry, government, •  Service jobs: dependent on population e.g. education, health

care, retail

• Two main assumptions •  Fixed ratio between population and jobs: u •  Fixed ratio between service jobs and population: v


• Given is the network en the distribution of the basic jobs, as well as both ratios u and v

•  Stepwise approach 1.  Determine and distribute population based on basic jobs 2.  Determine and distribute service jobs based on the population 3.  Determine and distribute population based on total of service

jobs and basic jobs 4.  Repeat steps 2 and 3 until sufficient convergence is achieved

•  For each step for each zone the distribution is based on the contribution of a zone to the potential value

Lowry model: the method


Lowry-model: flow chart

Basic employment per zone

Service employment per zone

Total employment per zone

Number of inhabitants per zone

Inhabitant allocation

model

Service employment allocation

model

€

E jb s

jE

€

E j

€

Ri

Is equal to 0 for the first iteration


iR

jE

Lowry-model: Allocation inhabitants for workers in zone j

Cij

€

Li1

1

( )

( )

ii ij

ij j ni

k kjk

n

i ijj

L f cR u E

L f c

R R

=

=

⋅= ⋅

⋅

=

∑

∑

u = ratio between population and jobs Ej = total employment in j Ri = inhabitants in i Li = attractiveness for inhabitants in zone i cij = transport costs between i and j

Potential value of zone j


Ri

E js

Lowry-model: Allocation service jobs for zone i

Cij

Wj

( )( )

1

1

sj ijs

ij i ns

k ikk

ns sj ij

i

W f cE v R

W f c

E E

=

=

⋅= ⋅

⋅

=

∑

∑

v = ratio service jobs and population Es

j = service employment in j Ri = inhabitants in i Wj = attractiveness of j for service jobs cij = transport costs between i en j

In order to account for agglomeration effects Wj equals Es

j of the previous iteration


Concluding comments Lowry-model

•  In every iteration the number of service jobs and thus the population increases

• As the algoritm convergences, it leads to an equilibirum •  Rij is in fact the OD-matrix for commuters (i.e. when

corrected for u)


4.3

Descriptive models: Immers & Hamerslag


ij i j ijT Q X Fρ=

ij jiT A=∑

( ) ( )ij i j ij j i ij ji i iT Q X F X Q F Aρ ρ= = =∑ ∑ ∑

ij ijT P=∑

( ) ( )ij i j ij i j ij ij j jT Q X F Q X F Pρ ρ= = =∑ ∑ ∑

( )j

ji ij

i

AX

Q Fρ=∑( )

ii

j ijj

PQX Fρ

⇒ =∑

and

Recall the doubly constrained model


ri

i i j j ijj

sj

ji i j ij

i

Pl Q m X F

Am l Q X F

ρ

ρ

⎛ ⎞⎜ ⎟= ⋅ ⋅ ⋅ ⋅⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟= ⋅ ⋅ ⋅ ⋅⎜ ⎟⎝ ⎠

∑

∑

Doubly constrained model revisited

Alternative formulation

Kind of accessibility

Introduce ‘elasticities’ or flexible constraints r and s

ij i i j j ij

ii

i j j ijj

jj

i i j iji

T l Q m X FPl

Q m X F

Am

l Q X F

ρ

ρ

ρ

= ⋅ ⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅

=⋅ ⋅ ⋅ ⋅

∑

∑


•  In the standard model they make sure row totals equal the production and the column totals equal the attractions

• Their value is reversely related to the accessibility of the zone

• The production and attraction might not be considered as a hard constraint, e.g. because: •  They’re an estimate •  They might change due to the actual accessibility

• Note that if r and s equal 1, we have the doubly constrained model, and if one of them equals 0 we have the singly constrained model

Interpretation of balancing factors


• Model for the Randstad having zones of 150.000 inhabitants

• Reference case: everyone works in the zone they live

• Model with elastic constraints (r=-2, s=0.2) for evening peak (commuters only) •  r=-2 suggests that highly accessible locations grow

(agglomeration effect) •  s=0.2 suggests a limited effect of accessibility on residential

areas

• Model is implemented in an incremental procedure •  Matrix update using elastic constraints (1 iteration) •  Travel cost update

Example Immers & Hamerslag


High quality interurban PT network

Results for 2 scenarios

High quality freeway network


•  If r and s equal 1, we have the standard doubly constrained model

•  If one of them equals 0, we have the singly constrained model

•  If they are between 0 and 1, it takes longer to converge or otherwise, if you use fewer iterations the matrix will have a poorer match (or more flexibility) with the constraints

•  If they are greater than 1, they might oscillate the iteration process (balancing factor >1: higher values, <1 lower values)

•  If they are smaller than 0, they might oscillate as well, however, the mechanism is reversed (>1: lower values, <1 higher values)

Interpretation of ‘elasticities’ r and s


• Values between 0 and 1 are justified •  Unconstrained model •  Singly constrained model •  Models having flexible constraints

•  Note that a limited number of iterations is required (e.g. the way it used in OD-estimation (see Lecture 9) where the

match with the a priori matrix is an important criterion) •  Doubly constrained model

• Values smaller than 0 or greater than 1 are tricky •  Effect depends on the ratio for li and mj

•  Therefore: not recommended

Conclusion for ‘elasticities’

transportation and spatial modelling: lecture 15

Business

spatial levels

spatial planningcie4801

cie4801 transportation

hotelling cie4801

temporary cie4801

tigris xl choice modelling

empirical findings cie4801

transportation models