&-
MODELS FOR THE LOCATION-ALLOCATION-PROBLEM
I N URBAN AND REGIONAL INFRASTRUCTURE PLANNING
Lueder Bach
Department o f Urban and Regional P I anning
U n i v e r s i t y o f Dortmund, U e s t Germany
I
and
Center for Metropol it a n P l a n n i n g and Research
The Johns Hopkins U n i v e r s i t y , USA
Paper presented a t t h e
I n t e r n a t i o n a l Symposium on L o c a t i o n a l D e c i s i o n s
B a n f f , A l b e r t a , Canada
A p r i l 24-28, 1978
.
TABLE OF CONTENTS
Page Number
1.
Introduction
1
2.
Concepts for !Spatial Behavioral Patterns
6
3.
Political and Societal Goals for Locational
Patterns of Central Facilities
10
Definitions for Central Locations and
Districts
13
5.
A Systematic of Flodels for the
Location- A1 1olcati on-P rob1 em
15
6.
Appendix: Plathemati cal Programming
Formulations f o r Models SE-1 t o SE-8
18
7.
Footnotes
25
8.
qources and Reference F4aterial s
27
Figure 1
31
Figure 2
32
4
..
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1.
Introduction
Many of the infrastructure f a c i l i t i e s t h a t are considered t o be
necessary prerequisites for the use of an urban o r rural region by
other a c t i v i t i e s can be considered 'central f a c i l i t i e s ' . This concept
i s derived from the notion t h a t a spatially distributed demand for
services provided b,y these f a c i l i t i e s can be offered only a t fewer
locations compared t o the many locations o f demand. Therefore, these
f a c i l i t i e s are supposed t o be located i n such a way t h a t they are
central t o the locations of demand.
Infrastructure f a c i l i t i e s t o which
t h i s essential a t t r i b u t e of central location applies range from kindergartens schools , or hospitals t o emergency call boxes rnai 1 boxes or
¶
telephone booths t o doctors' offices, pharmacies , or retai 1 centers t o
I
waste disposal s i t e s , f i r e stations, or fall-out shelters. These are
only a few examples o f the many f a c i l i t i e s t h a t are regarded as i n f r a structure f a c i l i t i e s demanding a central location.
They a l l cut
across the well accepted categories of private sector and public sector
facilities.
Though some of the infrastructure faci 1 i t i e s may be a t t r i -
buted t o one or the other of these two categories without dispute
t h e i r private or public ownership becomes less important once the
public, politicians; and planners conceive them as one activity system
among other activity systems such as residential or industrial activity
systems.
Furthermore, when f a c i l i t y location i s guided by public
means and measures w i t h i n the overall development of interrelated
activity systems o f an urban or regional s e t t i n g , ownership i s n o t a
decisive c r i t e r i o n .
/
-24
.
During the p a s t decade the theoretically 'oriented discussion on
central f a c i l i t i e s locations has been dominated by the distinction
between public central f a c i l i t i e s and private central f a c i l i t i e s .
(1)
Related t o the ownership of central f a c i l i t i e s t h i s differentiation i s
helpful i n identifying inherent characteristics of processes of decision
making i n the private versus the p u b l i c realm.
I t i s , however, of minor
importance w i t h respect t o the subject of locational decisions once
each central f a c i l i t y i s understood t o be p a r t of an activity system,
a subsystem i t s e l f of an overall activity system.
W i t h t h i s perspective
i n mind the subject of locational decisions implies two different b u t
interdependent issues:
f i r s t l y , the direct decision on a location
proper a n d , secondl:y, the indirect decision on ongoing processes of
spatial-functional interactions. The f i r s t issue of this complex
decision has been analyzed quite clearly w i t h regard t o i t s implications
i n the context of economic and social externalities.
The second issue
has usually n o t been addressed explicitly b u t has rather been covered
e i t h e r by somewhat nebulous concepts of accessibil i t y , soci a1 u t i 1i t y ,
Qoods-oriented versus consumer-oriented or ordinary versus extraordinary
public services or by a simultaneous consideration of transportation
costs and f a c i l i t y investment costs depending on the number of f a c i l i t i e s
and the s i t e s needed for locating them.
W i t h respect t o many public as
well as private infrastructure f a c i l i t i e s the consideration of tradeoffs between investment and transportation costs i s misleading for e i t h e r
one o f two reasons:
I n the case of many central f a c i l i t i e s the costs of
investment and the costs of transportation are s p l i t and p a i d f o r by two
different groups: the 'suppliers', i . e . those who invest, and the
'users I , i .e. those who uti 1ize the central services provided v i a the
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central f a c i l i t i e s .
Users pay e i t h e r directly for t h e i r expenses when
they travel t o the central f a c i l i t i e s o r they pay indirectly for
transportation costs by paying taxes and fees.
For many other central
f a c i l i t i e s , especially emergency service f a c i l i t i e s , the combined consideration of investment and transportation costs i s logically impossible
or altogether irrelevant.
Instead of integrating different and t o some
extent conflicting locational c r i t e r i a i n t o one overall objective i t may
be more appropriate t o pursue an approach t h a t t r i e s t o analyze the
trade-offs t h a t have t o be acceptable when preferring one objective
against another objective.
(2)
Instead of a r g u i n g along the private versus public dichotomy i t
i s suggested t o neglect these categories i n the context of defining
locations for central f a c i l i t i e s i n favor of reconsidering the n o t i o n
of 'central location'.
For the purpose of this analysis 'central
location' i s regarded as the doninant locational factor.
I t i s intended
t o develop some basic- operational definitions p r o v i d i n g the basis for
a series of models for the location-allocation-problem.
'Central location' i s understood t o be the result of a locational
decision process and i s understood t o be realized d u r i n g the process
of plan implementatpion. The specific meaning of 'central location' i s
derived from the spatial interrelationships t h a t manifest themselves
i n ongoing hourly, daily, weekly t r i p patterns or some other periodic
system of t r i p s i n space. The evaluation of the attributes of these
spatial interrelationships i n terms of p l a n n i n g goals results in
different concepts for 'central location'.
/
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v
W
The deliberate restriction on the locational factor 'central
location' a1 lows one t o concentrate on the spatial behavioral patterns
t h a t are an expression o f performed spatial-functional interrelationships.
I t allows one as well, t o concentrate on the intended influencing
of a l l those involved i n the locational decision process, an influencing
t h a t i s directed towards the spatial-functional interrelationships
.
This influencing i s interpreted as a 'locational goal ' . S p a t i a l
behavioral patterns and locational goals form the starting points for
operational definitions of 'central location'.
They allow, t o o , the
set-up of a systematic substantive framework f o r a series of models
f o r the location-a1 location-problem.
Once this substantive-oriented framework for the locationa1 locadion-problem has been established the technical -oriented approach
t o model systemization becomes a useful concept.
c3 1
This teciinical-
oriented approach deals with dimensions of models such as s t a t i c
versus dynamic, deterministic versus stochastic, normative versus
explanative, or location on a plane versus location on a network.
(4)
T h i s approach t o model stratifica.tion commonly used i n the context of
models f o r the location-allocation-problem i s irrelevant t o the issue
of 'central location'.
I t i s , however, a valuable tool i n characterizing
the planning situation as defined by the available d a t a base on the
planning region.
I t i s , as well, a valuable tool i n characterizing
the type o f model i n terms of modelling theory.
Thus, the technical-
oriented framework may be used i n choosing the method of solution being
appropriat:
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a mode1 f o r the location-a1 location-problem.
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V
w
T h e two basic issues, the one, of spatial behavioral patterns and
the one of locational goals, are discussed i n Sections 2 and 3 .
Section
4 proposes definitions for the locational factor of 'central location'
being based on spatial behavioral patterns and locational goals. These
definitions i n connection w i t h definitions for d i s t r i c t s , i .e. service
areas of central f a c i l i t i e s , provide a framework for a series of basic
models for the location-allocation-problem.
These models are described
i n Section 5 and t h e i r mathematical programming formulation i s presented
.
i n t h e Appendix.
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rl
2.
Concepts for S p a t i a1 Behavi oral Patterns
Compared t o the broad range of different types of central
faci 1i t i e s there ex-ist only few empi ri cal studies on spatial. behavioral
patterns that can shed 1 i g h t on spatio-functional relationships between
central f a c i l i t i e s md users.
Central f a c i l i t i e s for which empirical
studies have been pub1 ished are general ly 1 imited t o retai 1 facil i t i e s ,
warehouses and prodluction plants, and health services faci 1it i e s
W i t h respect to r e t a i l f a c i l i t i e s the concept of potential i s the
explanatory concept for the spatial behavioral pattern . According t o
t h i s concept the spatial behavioral pattern of consumers as expressed
by the choice of one particular f a c i l i t y o u t o f a l l f a c i l i t i e s i s
determined by the attractiveness of the r e t a i l f a c i l i t y as well as by
the spatial distance between the f a c i l i t y and the location o f the
consume r
(5'1
W i t h respect t o warehouses and production plants the
distribution of goods and products or the shipment o f materials needed
-
a t the production plant i s steered by the c r i t e r i a of transportation
costs.
In this case the concept of a spatial behavioral pattern i s
based on the minimization of t r a n s p o r t a t i o n costs.
With respect t o
health services empirical studies have proven t h a t spatial distances
(61
influence the frequency of health services requested by patients.
These three examples suggest the following three principles
underlying spatial patterns :
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T h e principle of minimizing spatial f r i c t i o n without a n
influence on the frequency of utilization of a system of
central faci 1 -it i e s ; ( 7 )
The principle of minimizing spatial f r i c t i o h w i t h an
influence on the frequency of utilization of a system of
central f a c i l i t i e s ;
The principle of maximizing potential.
I t i s assumed t h a t one of these principles determines the spatial
behavioral pattern of e i t h e r a user or a. supplier. The user has t o
be considered when he has t o pay for surmounting spatial f r i c t i o n , and
the supplier has t o be considered when he has t o pay for surmounting
spatial f r i c t i o n .
S t a r t i n g from the assumption t h a t a l l users have t o
be taken i n t o consideration such t h a t the demand of each user for
central services i s covered, three different spatial behavioral patterns
of u s e k and one spatial behavioral pattern of suppliers can be derived.
These patterns are:
-
Spatial behavioral pattern o f users based on the principle of
minimizing spatial friction w i t h o u t influence on the frequency
o f utili’zation of a system of central f a c i l i t i e s :
Each user goes t o t h a t f a c i l i t y of a system of central f a c i l i t i e s
located next t o his own location.
The frequency of utilization i s n o t
influenced by the spatial f r i c t i o n , i.e. i t i s not decreasing w i t h
increasing spatial friction.
A spatial behavioral pattern such as t h i s
one implies t h a t the user i s able t o determine which f a c i l i t y i s nearest
t o him i n terms of metrical distance, time distance, costs of transp o r t a t i o n or a distance concept based on physical and psychological
effort.
Examples of systems of central f a c i l i t i e s for which t h i s
concept of
spatia.1 behavioral pattern i s appropriate are post offices,
emergency call boxes, t r a i n s t a t i o n s , or mass t r a n s i t stops.
-8-
b
*
I
Spatial behavioral pattern of users based. on the principle of m i n i m i z i n g spatial f r i c t i o n w i t h influence on the frequency of u t i l i zation of a system of central f a c i l i t i e s :
Each user goes t o t h a t f a c i l i t y of a system - o f central f a c i l i t i e s located
next t o his own location as he wants t o maximize the frequency of
utilization o f serviices provided by the system o f f a c i l i t i e s . The
frequency of u t i 1 i z a t i o n decreases w i t h increasing friction of space.
A spatial behavioral pattern such as t h i s one implies that w i t h respect
t o his frequency of utilization a user i s influenced by the spatial
f r i c t i o n he has t o surmount.
I t i s implied, t o o , that the individual
user i s able t o evaluate the spatial f r i c t i o n between his location and
the locations t o al'l central f a c i l i t i e s enabling him t o determine the
nearest central f a c i l i t y as the one he wants t o go to.
The amount of
utilization is solely influenced by the spatial f r i c t i o n and n o t by the
a t t r a c t i v i t y of the central f a c i l i t i e s .
Examples of systems of central
f a c i l i t i e s for which this concept of a spatial behavioral pattern i s
appropriate are health- service f a c i l i t i e s such as hospitals and health
care centers or leisure f a c i l i t i e s , play-grounds, and sports f a c i l i t i e s .
-
Spatial behavioral ps.ttern o f users based on the principle of
maximizing potential:
Each user goes t o t h a t f a c i l i t y of a system o f central f a c i l i t i e s t h a t
exerts the highest potential a t this location.
The potential of a
f a c i l i t y a t a user location results from i t s a t t r a c t i v i t y and the f r i c t i o n
of space considered t o be a factor of resistance. A spatial behavioral
pattern such as t h i s one implies t h a t the user i s able t o evaluate and
rate, f i r s t l y , the a t t r a c t i v i t y of each of the central f a c i l i t i e s a n d ,
secondly, t o determine the amount of spatial f r i c t i o n between his
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-9
.
location and the locations of a l l the central facil.ities.
I t i s assumed
t h a t the amount of ciemand for central services f o r each user i s neither
increased nor decreased by the attractiveness or the spatial frictions.
Examples of central f a c i l i t i e s systems f o r which t h i s concept of a
spatial behavioral pattern i s appropriate are retai 1 centers.
Further-
more, t h i s concept may be appropriate, t m , for f a c i l i t i e s such as l i b r a r i e s ,
neighborhood center:; for senior c i t i z e n s , f a c i l i t i e s for teen-agers or
leisure f a c i l i t i e s .
c
Spatial behavioral pattern of a supplier or a group o f suppliers
based on the principle of minimizing spatial friction without
influence on the frequency of utilization o f a system of central
facil i ties :
Each user i s served by a supplier or a group of suppliers v i a t h a t
f a c i l i t y of a system of central f a c i l i t i e s which i s located next t o the
user location.
The amount of service i s not influenced by the amount of
spatial f r i c t i o n , i.e. i t i s n o t decreasing w i t h increasing spatial
friction.
A spatial behavioral pattern such as this one implies that
the supplier or the group o f suppliers is able t o determine the nearest
central f a c i l i t y for each user location.
Examples of systems of central
f a c i l i t i e s for which t h i s concept of a spatial behavioral pattern i s
appropriate are warehouses , fire-stations or ambulance dispatch centers
-10-
c
.
.
)
3.
Political and Societal Goals for Location'al Patterns of Central
Facilities
In the context: of locational decisions for systems o f central
f a c i l i t i e s the foreniost goal
-
i n s p a t i a l terms
- i s t o assure the
availability of central services by an appropriate locational pattern
of these f a c i l i t i e s , T h u s , the locational pattern has t o guarantee
a 'spatial a v a i l a b i l i t y ' .
Criteria for measuring spatial availability
have t o be derived from a consideration of spatial behavioral patterns
as well as from a rationale underlying the political and/or societal
goals for locational patterns o f systems o f central f a c i l i t i e s .
Starting from the p o i n t of view o f spatial behavioral patterns
the spatial availability of a system of central f a c i l i t i e s has t o be
measured by e i t h e r one of the following three c r i t e r i a :
0
Cost of surmounting spati a7 f r i ction :
The cost o f surmounting spatial f r i c t i o n i s the appropriate criterion
for such systems of central f a c i l i t i e s for which the principle of
minimizing s p a t i a l f r i c t i o n w i t h o u t an influence on the frequency of
utilization i s appropriate w i t h respect t o the spatial behavioral
pattern e i t h e r of users or of suppliers.
-
Amount of utilization of central services:
The amount o f utilization of central services provided v i a f a c i l i t i e s
i s the applicable criterion for such systems of central f a c i l i t i e s
for which the princ:iple o f m i n i m i z i n g spatial f r i c t i o n w i t h an influence
on the frequency of: utilization i s appropriate w i t h respect t o the
spatial behavioral pattern o f users.
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.
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Access opportunity for users:
The criterion of access opportunity for users t o central services applies
for such systems of central f a c i l i t i e s for which the principle o f
maximizing potential i s appropriate w i t h respect t o the spatial
behavioral pattern of users.
The spatial availability o f a system of central f a c i l i t i e s can be
measured i n e i t h e r one o f the following two ways depending on the
locational goal purl;ued:
F i r s t l y , the spatial availability can be
measured by looking a t the t o t a l i t y of spatio-functional relationships
existing between a l l users and the system of central f a c i l i t i e s .
Secondly,
i t can be measured Iby looking a t the spatio-functional relationship
existing between each i n d i v i d u a l user and the system of central f a c i l i t i e s
i n comparison t o a l l users or between each individual f a c i l i t y and a l l
users i n comparison t o a l l f a c i l i t i e s o f the system of central f a c i l i t i e s .
In the f i r s t case, the c r i t e r i a f o r spatial f r i c t i o n are:
-
the sum of costs of surmounting spatial f r i c t i o n ;
the total amount of utilization of a l l central f a c i l i t i e s ;
the t o t a l amount of access opportunity for the sum of a l l
user locations.
In the second case, the c r i t e r i a for spatial availability are:
.-
the equalized costs of surmounting spatial f r i c t i o n w i t h respect
t o each individual user;
the equalized amount of utilization w i t h respect t o each
individual faici 1i t y of the system of central faci 1i t i e s ;
the equalized amount of access opportunity w i t h respect t o
a l l user locations.
These six c r i t e r i a imply a differentiation w i t h regard t o a
collective locational rationale versus an equalizing locational rationale.
By introducing t h i s differentiation i t i s intended t o delimit three
spheres of locational choices.
Each one of these three spheres
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i s based on one particular principle of spatial behavioral pattern.
Furthermore, t h i s differentiation allows the evaluation of locational
patterns of systems of central f a c i l i t i e s w i t h respect t o locational
efficiency versus locational equity.
choices are indicated i n Figure 2.
(8)
The three spheres of locational
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.,
4
Definitions for Central Locations a n d Districts
4.
Using the principles of spatial behavioral patterns as well as
the locational goals as points of relevance -a1together nine definitions
are derived for central locations and four definitions for d i s t r i c t s ,
i.e. service areas, of f a c i l i t i e s located a t central locations (See
Figures 1 and 2)
(9)
.
The definitions for the central locations are:
Model S-1 :
Central P o i n t :
The p o i n t m i n i m i z i n g the sum of weighted distances;
Model S-2:
Median Point:
The p o i n t minimizing the sum o f weighted rectangular
d i s t a n ce s ;
Model S r 3 :
Arithmetic Mean Point:
The p o i n t minimizing the sum of weighted squared
d i s t an ce s ;
Model S-4:
Center P o i n t :
The p o i n t m i n i m i z i n g the maximum unweighted distance;
Model S-5:
Facility-Oriented P o i n t of Potential:
The p o i n t maximizing f a c i l i t y u t i l i z a t i o n ;
Model S-6:
User'-Oriented P o i n t of Potential :
The p o i n t maximizing access opportunity f o r the sum of
user locations;
Node1 S-7:
User-Oriented P o i n t of Equalized Potential :
The p o i n t equalizing access op o r t u n i t y for the i n d i v i d u a l
user location w i t h regard t o a 1 other user locations;
B
Model S-8:
Radial P o i n t :
The p o i n t from which each user location can be reached
w i t h i n a maximum allowable distance;
-1 4-
Model S-9:
Constrained Radial Point:
The p o i n t from which the maximum number of users can
be ireached w i t h i n a maximum allowable distance.
The definitions f o r the d i s t r i c t s o r t h e i r boundaries !, respectively,
(101.
are :
Model E-1:
Districts delimited by boundaries formed by points of
equal distances between neighboring central f a c i l i t i e s ;
Model E-2:
Districts delimited by boundaries formed by points o f
equal potential between neighboring central f a c i l i t i e s ;
Model E-3:
Districts as attractivity-dependent and distancedependent spheres of influence of central f a c i l i t i e s ;
Model E-4:
Districts delimited by boundaries formed by p o i n t s of
a maximum allowable distance.
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5.
A Systematic of Models f o r the Location-Allocation-Problem
Based on the nine models for the location-problem and the f o u r
models for the allocation-problem, three groups of altogether ten
models for the loca-tion-allocation-problem are derived.
Each one of
these models can be traced back t o one particular principle of spatial
behavioral pattern md can be attached t o one of the three spheres of
locational choices as shown i n Figure 2.
Mathematical programming
(11)
forqulations for these ten models are presented i n the Appendix:
Model SE-1:
Locations and d i s t r i c t s for central points;
Model SE-2:
Locations and d i s t r i c t s f o r center p o i n t s ;
Model SE-3:
Locations and d i s t r i c t s for facility-oriented
p o i n t s of potential ;
Model $E-3/Variant 1 :
Locations and d i s t r i c t s for facility-oriented p o i n t s
o f equalized potential ;
Model SE-3/Vari a n t 2 :
Locations and spheres of infl uence for faci 1i tyoriented points of equalized potential ;
-
Model SE-4:
Locations and d i s t r i c t s for user-oriented points of
potential ;
Model SE-5:
Locations and d i s t r i c t s f o r user-oriented points
o f equalized potential ;
Model SE-6:
Locations and spheres of influence for useroriented p o i n t s of equal ized potenti a1 ;
Model SE-7:
Locations and d i s t r i c t s for radial points;
Model SE-8:
Locations and d i s t r i c t s for constrained radial
points.
The models SE-1 t o SE-8 as presented here and i n the Appendix have t o
be regarded
3s
basic versions.
The only locational factor t h a t i s
-1 6-
represented by these models i s the one of 'central location'. Though
t h i s locational factor i s considered t o be a dominating one i n the context
of locational decisions for systems of central f a c i l i t i e s an extension
of the basic models may be desirable.
T h i s can be done primarily along
two lines: Firstly, the models may be extended by including those other
locational factors which are t o some degree directly related t o the
central location of f a c i l i t i e s .
(12)
of i n this connection are:
c1
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Locational factors t h a t could be t h o u g h t
minimal and maximal capacities of central f a c i l i t i e s have t o be
observed;
a maximum a l l owable distance between central locations and user
locations must n o t be surpassed; (131
already existing central f a c i l i t i e s have t o be integrated
i n t o the new, extended system o f central f a c i l i t i e s ;
I
functional and spatial-functional , hierarchical structures o f
systems o f central faci 1 i t i e s must be represented.
Secondly, the models may be extended by including time-dependent
dynamic s h i f t s of the- supply-component and/or the demand-component o f
systems o f central f a c i l i t i e s .
Up t i l l now, the implicit assumption
has been made t h a t a l l f a c i l i t i e s of a system o f central f a c i l i t i e s will
be i n existence a t the same p o i n t of time.
dropped for two reasons.
T h i s assumption s h o u l d be
On the one h a n d , the implementation of
a t o t a l system of c:entral f a c i l i t i e s depends on the availability o f
financial , man-power, and spatial-physical resources.
This implies t h a t
a system of central1 f a c i l i t i e s can usually be implemented only step
by step.
On the other h a n d , a phased implementation of a system of
central f x ' 1 i t i e s might be necessitated by spatial and/or qualitative
shifts of the demand for central services.
-1 74
In o r d e r t o hiandl e time-dependent changes caused
by changing
r e s t r i c t i o n s o f resources o r by changes o f demand, f o u r d i f f e r e n t
s t r a t e g i e s o f implementation can be suggested.
These s t r a t e g i e s d i f f e r
from each o t h e r b y t h e u n c e r t a i n t y about f u t u r e l e v e l s o f demand as w e l l
as by r e l a t i v e inde:pendence o f each implementation phase w i t h i n t h e
(14)
sequence o f phases. These f o u r stra-teiiies are:
t h e s t r a t e g y of
development phases
t h e s t r a t e g y o f a development sequence, t h e s t r a t e g y
o f completion phases, and t h e s t r a t e g y o f a completion sequence.
I n the
d i f f e r e n t i a t i o n o f 'development' and 'completion' t h e attempt i s made
t o g i v e c o n s i d e r a t i o n t o t h e growth o f a system o f c e n t r a l f a c i l i t i e s
out o f preceding t i m e phases i n t h e f i r s t case, and i t s gradual completion
(15)
towards a predetermined- f i n a l s t a t e i n t h e l a t t e r case.
i
/
-18-
I
6.
Appendi x: Mathematical Programming Formul ations for Models S E - 1 t o S E - 8
The following n o t a t i o n s are used throughout .the Appendix:
dij
dji
d+
distance from the i - t h central location t o the j - t h user location
distance from the j - t h user location t o the i - t h central location
maximum allowable distance between a central location and a user location
8
distance exponent for h a n d l i n g a l i n e a r relationship ( ~ = 1o)r a
non-linear relationship ( 0 ~ o~r 4p 1 ) between distances and transportation
costs
m
number of central locations
'i
attractiveness o f the f a c i l i t y a t the i - t h central location
n
number o f user locations
r
number o f users located a t the j - t h user location
j
a i j 3 aji 0-1-vari ab'l e (a11oca t i on coe f f i ci en t )
a
ij
= 1
= 1
"ji
3
whe re :
i f the j - t h user location i s w i t h i n the service area
( d i s t r i c t ) o f the f a c i l i t y a t the i - t h central location
and
b
number o f given potential central locations
t
number o f possibilities t o form different combinations containing m central
locations o u t o f b potential central locations
d!.
31
aii
distance from the j - t h user location t o the i - t h central location which
i s identical w i t h one o f the b given potential central locations
0-1-variable (allocation coefficient) where:
a j' i = 1
aji
= 0
i f the j - t h user location i s w i t h i n the service area
( d i s t r i c t ) o f the f a c i l i t y a t the i - t h central location
which i s identical w i t h one o f the b given potential central
locations and
otherwise
1
-
-1 9-
I
I
i j
i
.
amount of u t i l i z a t i o n o f a f a c i l i t y located a t the i - t h central location
by a l l users located a t the j - t h user location
amount o f u t i l i z a t i o n of a f a c i l i t y located a t the i - t h central location
by a l l users located a t a l l n user-locations
I
probability indicating t h a t a user located a t the j - t h user location
will u t i l i z e central services offered a t the i - t h central location
i@j
access-opportunity a t the j - t h user location w i t h regard t o the f a c i l i t y
a t the i-th central location
I
j i
A
standard deviation o f the amounts o f u t i l i z a t i o n o f a l l m central f a c i l i t i e s
S
5 - i s . calculated by applying the following formula f o r the
standard deviation S o f u observations vw:
s
U
=
((u (U.-l))-l (u
c v;
w= 1
-
U
(
c
w=l
2
vw) 1)
112
0-1-variable representing potential central locations , where :
i f a central f a c i l i t y i s located a t X i , and
xi = 1
o the rwi s e
xi = 0
'i
-
MODEL SE-1: LOCATIOiNS AND D I S T R I C T S FOR CENTRAL P O I N T S
The locations and d i s t r i c t s o f m central f a c i l i t i e s have to be determined i n such
a way t h a t : ( a ) each user i s allocated t o exactly one central f a c i l i t y and ( b ) the
sum o f weighted distances i s a minimum:
Mini m i ze :
m
n
..
8
(1.1)
subject to:
f o r a l l i , i = l ,...,m
for all j , j=l,...,n
m
c a i j = l
. .(1.2)
for a l l j , j=l,...,n
....(1.3)
for all i , i=l,...,m
for a l l j , j=l,...,n
....(1- 4 )
i=l
dij 1 0
I
-20-
MODEL S E - 2 : LOCATIONS AND D I S T R I C T S FOR CENTER P O I N T S
The locations and d i s t r i c t s of m central f a c i l i t i e s have to be determined i n such
a way t h a t : ( a ) each user i s a l l o c a t e d t o a t l e a s t one central f a c i l i t y and ( b ) the
maximum distance i s minimized t h a t e x i s t s i n any o f the m d i s t r i c t s between the
central f a c i l i t y and the user l o c a t i o n s allocated t o i t :
Mi n i m i ze :
Z = max min
j
i
B
dij
ui j
.. .(2 .1)
subject t o :
01
ij
=("
1
m
c
aij 2 1
i =1
dij 2 8
f o r a l l i , i = l ,...,m
f o r a l l j , j = l ,...,n
....(2.2)
f o r a l l j, j = l, . . . , n
.. .(2.3)
f o r a l l i , i=l,...,m
for a l l j , j = l , . . . , n
....(2.4)
MODEL S E - 3 : LOCATIONS AND D I S T R I C T S FOR F A C I L I T Y - O R I E N T E D P O I N T S OF POTENTIAL
The locations and d i s t r i c t s o f m central f a c i l i t i e s have t o be determined i n such
a way t h a t : ( a ) each user i s a l l o c a t e d t o exactly one central f a c i l i t y and (b) the
u t i l i z a t i o n o f a l l central f a c i l i t i e s i s maximized:
Maxi m i ze :
m
n
z= c
c
j=1
i I j - aji
-
. ..( 3.1)
subject t o :
01..
J1
m
c
=
{
0
1
f o r a l l i , i=l,...,m
for a l l j , j=l,...,n
aji = 1
f o r a l l j , j = l ,...,n
i =1
dji i 0
for a l l i , i = l ,...,m
for a l l j , j=l,...,n
where :
....(3.2)
....(3.3)
*..-(3.4)
....(3.5)
-21-
v
.
3
MODEL SE-3 / VARIANT 1: LOCATIONS AND DISTRICTS FOR FACILITY-ORIENTED POINTS
OF EQUALIZED POTENTIAL
-
The locations a n d d i s t r i c t s o f m central f a c i l i t i e s are t o be determined i n such
a way t h a t : ( a ) each user i s allocated t o exactly one central f a c i l i t y , ( b ) the
amount o f utilization i s as equal as possible when comparing the central f a c i l i t i e s
w i t h each other, a n d ( c ) the central f a c i l i t i e s are located on potential central
locations t h a t are given:
O u t o f the s e t o f b potential central locations altogether t combinations w i t h
m central locations can be formed:
t = .
b!
-
-
....(4.0)
m)! m!
For each one o f the t combinations the following objective function can be
determined so t h a t the allocation coefficients a'.= are known:
(b
J'
Maxi m i ze :
n
c
m
z= c
i = l j=1
iIj
....(4.1)
aji
subject t o :
m
C
aji = 1
i=l
dji - d i i
f o r a l l i , i=l,...,m
for a l l j, j = l , . . . , n
-...(4.2)
for a l l j, j = l , . . . , n
....(4.3)
for a l l i , i=l,...,m
for a l l j, j=l,...,n
....(4.4)
.- .
In a next round of calculations f o r each one o f the t combinations the
standard deviation S i s calculated:
. .(4.5)
where :
a.. =
J1
and
dji
-
.
a!.
for a l l i , i = l , . . ,m
for a l l j, j = l ,...,n
dii
for a l l i , i = l , . . . , m
for a l l j, j = l , . . . , n
J1
us d t o calc d a t e iIj.
In equations ( 4 . J ) and (4.5) equation (3.5) i
T h a t combination h a v i n g the m i n i m u m value f o r
for
.
2
i s the optimal solution looked
.
MODEL SE-3 / VARIANT 2: LOCATIONS AND SPHERES O F INFLUENCE FOR F A C I L I T Y - O R I E N T E D
POINTS O F E Q U A L I Z E D POTENTIAL
The l o c a t i o n s and spheres o f i n f l u e n c e o f m c e n t r a l f a c i l i t i e s have t o be determined
i n such a way t h a t : ( a ) t h e t o t a l demand o f each u s e r i s a l l o c a t e d p r o p o r t i o n a l l y
t o a l l c e n t r a l f a c i l i t i e s a c c o r d i n g t o a p r o b a b i l i t y which c o n s i d e r s t h e a t t r a c t i v e ness o f and t h e d i s t a n c e t o each o f t h e c e n t r a l f a c i l i t i e s , ( b ) t h e amount o f
u t i l i z a t i o n a t t h e c e n t r a l f a c i l i t i e s i s as equal as p o s s i b l e when compared w i t h
each o t h e r , and ( c ) t h e c e n t r a l f a c i l i t i e s a r e l o c a t e d on p o t e n t i a l c e n t r a l l o c a t i o n s
t h a t a r e given:
.
A c c o r d i n g t o e q u a t i o n (4.0) t combinations w i t h m c e n t r a l l o c a t i o n s can be
formed o u t o f a s e t o f b p o t e n t i a l c e n t r a l l o c a t i o n s . For each one o f t h e t
combinations t h e f o l l o w i n g equations (5.1) , (5.2), and (5.3) a r e c a l c u l a t e d :
t h e sum o f u t i l i z a t i o n a t t h e i - t h c e n t r a l f a c i l i t y :
-
n
c
iI =
rj
pj
.....(
5.1)
j=l
- with
. a . being a p r o b a b i l i t y factor:
1 J
m
pj =
ci
.. ( 5 - 2 )
( Z - ci / ( l t d .B. ) )
J1
i= I
/ ( l t d j iB)
-
t h e s t a n d a r d d e v i a t i o n S o f t h e sums o f u t i l i z a t i o n :
S
= ((m (m-1))-'
A
m
(m
c i ~ 2
i=l
-
m
(
i ~ ) * ))1/2
..e.(
5.3)
i= 1
A
The combination h a v i n g t h e minimum v a l u e f o r S i s t h e o p t i m a l s o l u t i o n l o o k e d
for.
MODEL SE-4: LOCATIONS AND D I S T R I C T S FOR USER-ORIENTED POINTS O F POTENTIAL
The l o c a t i o n s and d i s t r i c t s o f m c e n t r a l f a c i l i t i e s have t o be determined i n such
a way t h a t : ( a ) each u s e r i s al1oc.ated t o e x a c t l y one c e n t r a l f a c i l i t y and ( b ) t h e
sum o f a c c e s s - o p p o r t u n i t i e s o f a l l u s e r l o c a t i o n s i s a maximum:
Maximi ze :
.. .(6S)
subject to:
a.. =
J1
{:
f o r a l l i, i=l,...,m
f o r a l l j, j=l,...,n
m
c
aji
= 1
i=l
d..
J1
2
0
*
f o r a l l j, j=I,
...,n
for a l l i, i=l,..*,m
for a l l j, j=l,...,n
'
. ..(6- 2 )
....(6.3)
....(6.4)
-23-
.
where :
I = ci / (1 + d B. . )
j i
MODEL SE-5:
t
....(6.5)
J'1
LOCATIONS AND DISTRICTS FOR USER-ORIENTED POINTS O F EQUALIZED POTENTIAL
The locations and d i s t r i c t s o f m central f a c i l i t i e s have t o be determined i n such
a way t h a t : ( a ) each user i s allocated t o exactly one central f a c i l i t y and ( b ) the
access-opportunity of a l l user locations i s as equal as possible when compared
w i t h each other:
I n applying equation ( 6 . 5 ) for the access-opportunity I.a n d the concept o f
j 1
the standard deviation the objective function i s :
Minimize:
subject t o :
a ji
m I
c ' aji
i=l
dji 2 8
= 1
f o r a l l i , i=l,...,m
for a l l j , j = l , . . . s n
. .(7.2)
for a l l j , j = l , . . . , n
....(7.3)
f o r a l l i , i=l,...,m
for a l l j , j = l , . . . , n
. ..(7.4)
MODEL SE-6: LOCATIONS AND SPilERES O F INFLUENCE FOR USER-ORIENTED POINTS OF
EQUALIZED POTENTIAL.
The locations and spheres o f influence o f m central f a c i l i t i e s have t o be
determined in such a way t h a t : ( a ) each user i s considered t o be allocated t o each
central f a c i l i t y and ( b ) the access-opportunity a t a l l user locations i s as equal
as possible when compared w i t h each other:
i and the concept o f
In applying equation (6.5) for the access-opportunity j I.
the standard deviation the objective function i s :
M i n i m i ze :
n
m
n m
(8.1)
z = ( ( n ( n - 1 ) l - l ( n j=1
c ( c -I.)* - ( c c j ~ i2 )1) 1/2
i = l = J1
j=1
subject t o :
f o r a l l i , i=l,...,m
dji I 0
(8.2)
for a l l j , j = l , . . . , n
. ..
..
/
I
-24-
MODEL SE-7: LOCATIONS AND D I S T R I C T S FOR RADIAL POINTS
The locations and d i s t r i c t s o f an unknown number o f central f a c i l i t i e s have t o be
determined i n such a way t h a t : ( a ) each user i s a l l o c a t e d to a t l e a s t qne central
f a c i l i t y w h i c h i s n o t f a r t h e r away than a maximum allowable distance d and ( b ) the
number o f central f a c i l i t i e s i s minimized:
M i n i m i ze :
m
xi
z= c
i =1
....(9.1)
subject to:
.
m
c
i =I
xi
for a l l j, j=l,...,n
....(9.2)
1
0
for a l l i , i = l , . . . , m
0...(9.3)
1
for all
for all
for a l l
for a l l
a
-
ij >1
[
0
xi
=
aij
;I
i , i=l,...,m
j, j=l,...,n
i , i=l,...,m
j , j=l,...,n
for all i , i=l,...,m
for a l l j , j=l,...,n
d Bi j a
< d+
ij
-
dij
L 0
-...(9.4)
.*..(9.5)
....(9.6)
MODEL SE-8: LOCATIONS AND D I S T R I C T S FOR CONSTRAINED R A D I A L POINTS
The locations a n d d i s t r i c t s o f m central f a c i l i t i e s have to be determined i n such
a way t h a t : ( a ) a user i s not a l l o c a t e d t o more than one central f a c i l i t y and ( b )
the sum o f users i s maximized t h a t can be reached w i t h i n a maximum allowable
distance dt.
Maxi m i ze :
m
n
z= c
~
i = l j=1
j
r
ij
a
subject t o :
ai j
;("
1
m
r:
d.. 2 0
1J
I 1
aij
i=l
dCj a i j
"
5
...
....(10.2)
...,n
....(10 .3)
for a l l i , i=l,...3m
....(10 - 4 )
f o r a l l i , i = l , ,m
for a l l j , j = l , . . . , n
dt
for all j , j=l,
for a l l j , j=l,...,n
for a l l i , i = l , . . . , m
for a l l j , j=l,...,n
....(10.5)
-25-
CL
*
7.
FOOTNOTES
(1)
See e . g , : T E I T Z (1968), NARKS ET AL, (1970), REVELLE ET AL. (1970).
(2)
For an a p p l i c a t i o n o f t h i s approach in the context o f f i r e s t a t i o n
l o c a t i o n c o n s u l t S C H I L L I N G (1976, pp. 26-40 and pp. 81-109) and
REVELLE ET AL,, (1977).
(3)
See e.g. : WAGNER (1968)
(41
For a d e t a i l e d d i s c u s s i o n o f dimensions of models see HARRIS (1961,
SCHEURWATER (1976) , SHEPPARD (1974)
.
1967) o r LOWR’I (1965).
See BENTELE & MUELLER-TRUDRUNG (19701 o r BUNGE (1970, pp. 80-91)
for a review.
See DROSNESS & L U B I N (19661, MORRILL (1967), MORRILL & EARICKSON
(19681, MORRII- ET AL. (1970), EARICKSON (1970, pp. 8-39 and pp.
55-58] , SHANNON ET AL. (1969) , WEISS & GREENLICK (1970) SCHULTZ
(1970J_, WEISS ET AL. (1971), ABERNATHY & SCHREMS (1971), DE V I S E
[1973], SHUMAI’4 ET AL. (1973, pp. 123f) ROTHMEL & HAMMER (1974) 9
PYLE & LAUER (1975).
Tfiis p r i n c i p l e may be regarded a s a specific a s p e c t o f Z I P F ” S
(1949, pp. 5-81 more general d e f i n i t i c n o f the ’principle of
least effort’
For a f u r t h e r d i s c u s s i o n o f t h i s issue the r e a d e r is r e f e r r e d t o :
SYMONS (1971 , 1973) MORRILL & SYMONS (1974, 1977), BANERJI &
FISHER Gl974, pp. 178-180), KOLESAR & WALKER (1972, pp. 2-8)s
MORRILL (19741, MCALLISTER (19761.
See BACH (1978, pp. 52-98, pp. 246-253) for a mathematical
programming formulation as well a s methods of s o l u t i o n f o r each
of the following nine models f o r the location-problem.
See BACH (1978, pp. 99-118, pp. 254-2691 f o r a mathematical
programming formulation a s well a s numerical and graphical methods
o f s o l u t i o n f o r each of t h e following f o u r models f o r the a l l o c a t i o n probl em.
A more d e t a i l e d d i s c u s s i o n o f the ten models for the l o c a t i o n allocation-problem i s presented i n BACH (1978, pp. 119-226)
For cornputer-lprograms f o r each of these models - except
model SE-2 c o n s u l t BACH & ZOLLO (1975) , BACH & KRUEGER (1976a,
1976bJ , BACH & KNABE (1977) , BACH & SIEMON (1977).
.
The m ~ ~ h r m a t i c aprogramming
l
formulations f o r some o f these
e x t e n s i o n s a r e presentc-i i n BACH (1978, pp. 234-237).
-26-
(13)
T h i s extension i s n o t applicable t o models SE-7 and SE-8.
(141 A detailed discussion o f these f o u r strategies i s presented i n
BACH C1978, pp. 230-234, pp. 237-243).
(15)
For dynamic aspects i n models for the location-a1 location-problem
consult also: SCOTT (1971) SHEPPARD (1974) SCHEURWATER (1976,
pp. 24-27, p. 3 4 f ) .
I
-27-
8.
Sources and Reference Materi a1 s
Abernathy, W.J. , Schrems, E.L. , 1971, Distance and heaZth services: issues o f
u t i l i z a t i o n and f a c i l i t y choice for demographic s t r a t a , ( S t a n f o r d Universi t3/,
Graduate School of Business, Research Paper No. 1 9 ) , S t a n f o r d , Calif., USA.
.
Bach, L , 1978, Methoden
zur Bestirmung von Standorten und Einzugsbereichen
zentraZer Einrishtungen, (Methods f o r determining locations and d i s t r i c t s
o f central faci 1 i t i e s ) , (Birkhaeuser-Verlag, Base1 /Switzerland) (Forthcoming).
.
, Knabe, 3. , 1977, RadiaZ,umk-Ls d s Standorte fuer Systeme von Infrastruktureinrichtmgen - ModeZZ, AZgorithmus und EDV-Progrm, ( Radi a1 points
as locations f o r infrastructure f a c i l i t i e s - model , algorithm, and computer
program) , Dortmund , Germany (Forthcomi ng)
Bach, L.
.
.
Bach, L., Krueger, G . , 1976a, Einrich-t;zozgsorientierte PotentiaZpunkte aZs
Standorte fuer Systeme pr<vater und o e f f e n t Zieher z e n t r d e r Einrichtungen 3 EDV-Programme, (Facility-oriented points of potential as locations for
private and public central f a c i l i t i e s - 3 computer programs) , Dortmund, Germany.
Bach, L . , Krueger, G . , 1976b, Benutzerorientierte Potentialpunkte aZs Standorte
fuer Systeme privater w2d o e f f e n t Zicher zentraZer Einrichtungen - 3 EDVProgramme, (User-oriented points of potential as locations f o r private and
pub1 i c central faci 7 i t i e s - 3 computer programs) , Dortmund, Germany.
Bach, L., Siemon, P.,
1977, Beschraenkte RadiaZpunkte aZs Standorte f u e r Systeme
xentrazer Einrichtungen - Mode Z Z, AZgoAthmus und EDV-Program fuer die
InfrastrukturpZanmg, (Constrained radial points as locations for systems of
central f a c i l i t i e s - model, algorithm, and computer program f o r infrastructure
planning) , Dortmund, Germany .
Bach, L., Z O ~ ~ OB . , 1975, Zwei EDV-Programme zur Loesung von Standort-Einzugsbereichs-Prob Zemen fucr p r i v a t e m d o e f f e n t Ziche zentraZe Einrichtungen, (TWO
computer programs f o r solving location-a1 location-problems f o r private and
public central f a c i l i t i e s ) , Dortmund, Germany.
Banerji, S . , Fisher, H.B. , 1974, "Hierarchical location analysis f o r integrated
area planning i n rural India", Papers of the Regional Science Association,
33, 177-194.
Bentele, H. , Mueller-Trudrung, J . , 1970, "Einzelhandelsbetriebe i m Staedtebau",
(Retail enterprises i n urban development) , i n flandwoerterbuch der Ramforschung
und Rawnordnwzg, Ed. Akademie fuer Raumforschung und Landesplanung, 2nd edition,
Hannover, Germany, Columns 546-557.
.
Bunge , H , 1970, GepZmte Standorte fuer Einze Zhande 2s- wzd Hmherksbetriebe. Die
StandortpZanung privater Versorgungsbetriebe in der Mark-h-dirtschaft, insbesondere
die EinpZanung in neue WohnsiedZungen, (Planned locations for r e t a i l stores and
business enterprises. Locational planning of private service enterprises i n a
market economy w i t h special emphasis on integration i n t o new residential areas) ,
Col ogne , Germany.
-28-
Drosness, D.L. , Lulbin, J . , 1966, "Planning can be based on p a t i e n t t r a v e l " ,
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FIGURE 2: Models f o r the l o c a t i o n - a l l o c a t i o n - p r o b l e m a n d t h e i r r e l a t i o n s h i p
t o p r i n c i p l e s o f s p a t i a l behavior and l o c a t i o n a l g o a l s .