Location–allocation models applied to urban public services. Spatial analysis of Primary Health Care Centers in the city of Luján, Argentina


387

Hungarian Geographical Bulletin 62 (4) (2013) 387–408. 

Location–allocation models applied to urban public services. 
Spatial analysis of Primary Health Care Centers

in the city of Luján, Argentina

Gustavo BUZAI1

Abstract

The actual digital technologies and particularly the association between the Geographical 
Information Systems (GIS) and the assistance to the Spatial Decision Support System (SDSS) 
have generated important possibilities for the treatment of spatial information. As regards 
the use of location-allocation models, this presentation assesses the possibilities of using 
such models in the fi eld of the geography of services. In this paper theoretical aspects of the 
analyzed problems are presented, as well as methodological standardized questions for their 
solution through the use of GIS+SDSS. An applied case study related to the spatial analysis 
of Primary Health Care Centers (PHCC) in the city of Lujan, Argentina is also presented.

Keywords: Spatial analysis, location-allocation models, primary health care centres, GIS, SDSS.

Introduction

The application of geographical analysis procedures, oriented towards service 
planning, is actually presented as a very dynamic fi eld of investigation starting 
from the use of Geographical Information Systems (GIS) as well as the Spatial 
Decision Support Systems (SDSS).

The models of higher application were defi ned from a conceptual 
and practical view some four decades ago (Revelle, Ch. and Swain, R. 1970; 
Austin, C. 1974; McAllister, D. 1976) and during the decade of 1990 this in-
formation slowly spread throughout the digitalization fi eld, through soft ware 
intended to support decision-making.

Digital standardization of procedures has evolved along with socio-
economic aspects in population´s basic services diversifi cation as in the ap-

1 Study Group on Geography and Spatial Analysis with GIS (GESIG), Geographical Studies 
Program (PROEG), National University of Luján. Ruta Nacional No 5 y Av. Constitución, 
6700 Luján, Argentina. E-mail: gesig-proeg@unlu.edu.ar; web: www.gesig-proeg.com.ar 



388

pearance of a post-fordist model in which small and medium enterprises 
(SMEs) supplying services to the industry and other enterprises have an im-
portant role.

In view of the above, the spatial location of services appeared to be of 
a great importance in many aspects, particularly in the fi eld of public services, 
and in att empts to improve the levels of spatial equity for the population to 
be served.

Along these lines, the present work may be considered as a later stage 
for spatial data exploration and has as a principal objective to set forth a stand-
ardization of the theoretical methodological aspects of spatial localization in 
order to prioritize, from a geographical perspective, the process of decision 
making at the moment of installing, relocating or increasing a given number 
of installations of public urban services.

The application implemented here will be centered on the spatial 
localization of the PHCCs  in the city of Luján, Argentina (34°34´13´´S and 
59°06´18´´ W), with a population of 78,500 in 2012 (Municipalidad de Luján, 
2012). Effi  cient and equitable access to public services must be guaranteed.

Theoretical background

Location-allocation models

Geographical studies have a broad tradition in the generation of theories and 
general models for the analysis of human activities. Particularly, as regards the 
tertiary activities, it is possible to consider the theory of central places proposed 
by Walter Christaller in 1933 as a model of optimum spatial localization of 
urban centres at a regional level. In its formulation, the concepts of threshold 
and reach are presented as a deductive basis from which we can explain certain 
empiric regularities that were presented in the systematization carried out by 
Beavon, K. (1980).

From a model-based view, the localizations (potential supply and de-
mand points), the distances (ideal or real) and the costs of displacements 
(spatial friction) are presented as the principal factors that produce diff erent 
territorial confi gurations in the system. A series of studies focuses on the terti-
ary activity and as regards evolution, goes forward in a change of scale from 
the analysis of urban centres (regional) towards the inside centres of the city 
(local). This materializes in the geography of marketing, a concept presented 
by Berry, B. (1971) having been widely analyzed in its current capacities by 
a series of authors (Moreno-Jiménez, A. 1995, 2004; Bosque-Sendra, J. 2004; 
Bosque-Sendra, J. and Moreno-Jiménez, A. 2004; Salado-García, M. 2004; 
Moreno-Jiménez, A. and Buzai, G. eds. 2008).



389

From this point of view, the theory of localization takes into consid-
eration problems in the installation of services and generates a double objec-
tive: on the one hand to fi nd the optimum localizations, and on the other to 
determine the allocation of demand for such centres. To resolve this double 
objective models of allocation-localization have been developed.

According to Ramírez, L. and Bosque-Sendra, J. (2001), the location-
allocation models meet the following characteristics: a) they are mathematical 
models since this language is considered appropriate to capture reality; b) they 
are spatial models at intermediate scale because the aspects to be solved are 
already delimited in a territory; and c) they are normative models because it 
is necessary to look for the best solution to a given problem.

In synthesis those models att empt to assess the actual locations of 
service centres on a demand distribution basis, and to generate alternatives 
to achieve a more effi  cient and/or equitable spatial distribution. Those models 
are designed to fi nd the optimum localizations and determine the best links 
of the demand (allocation).

In recent years the application of location-allocation models, even those 
operationalized based on a Geographical Information Systems basis, have been 
framed in a specifi c system called Spatial Decision Support System (SDSS).2 
According to Bosque-Sendra, J. et al. (2000), the SDSS’s principal objective is 
to supply a necessary environment in hardware and soft ware to facilitate us-
ers in spatial decision-making matt ers. In this sense, the study of exploration 
problems, the generation of various solutions and the evaluation of diff erent 
alternatives need to be assessed. .

Densham, P. (1991) presents two well diff erentiated levels as regards 
the  application of SDSS, one in which the user takes decisions through gener-
ating, evaluating and choosing solution alternatives, and the system interface 
achieving a multidirectional interaction between the data base and its pos-
sibilities for making numerical and graphical reports.

Finally, it must be noted that location-allocation models are very use-
ful methodologies to support decision-making for health care in developing 
countries (Rahman, S. and Smith, D. 2000).

2 Decision Support Systems (DSS) were initially developed in the economic and management 
sciences during the 1950s and 1960s and were widely diff used during the next two decades. 
Likewise, the concept of Spatial Decision Support Systems (SDSS) was developed at the 
same time, being the Geodata Analysis and Display Systems (GADS) developed by IBM 
(International Business Machines) during the 1970s. Since the second half of the 1980s, DSS 
have begun to be adopted as tools for the enlargement of the technology capacity GIS. 
Some aspects of this process were developed by Malczewski, J. (1998).



390

Orientation of the location-allocation

From a general point of view, the orientation supplied for the model of loca-
tion-allocation, will be infl uenced by the nature of the service. If the service is 
private, it will basically focus on improving spatial effi  ciency, on the other hand, 
if it is public, it will try to improve spatial equity. Both refer to the enhancement 
of global parameters for the access to the service: the sum of the total displace-
ments, accessibility values or diff erences among extreme values.

Similarly, a notorious diff erence is shown if the equipment to be in-
stalled is desired (benefi cial) or not desired (prejudicial). While the fi rst ones 
basically generate positive externalities (hospitals, schools, cultural centres 
etc.) the second ones generate negative externalities (cemeteries, jails, rubbish 
dumps, etc.).

Therefore, taking in to account the previous considerations, the SDSS 
will contemplate diff erent possibilities of methodological application according 
to the objective in charge of fi nding the localization of the service centres.

Methodology

Searching for candidate sites and their combinations

The application of location-allocation models implies having an off er, distrib-
uted in a point manner, and a demand which, for reasons of simplifi cation, 
may be assigned to a centroid of each area and a transport network linking 
them. However, the application of methods  att empt to fi nd new supply loca-
tions must fi rst consider the determination of possible candidate sites, that is to 
say a quantity of selected points with the purpose of selecting the best one(s) 
according to the applied model objective.

There are two basic possibilities for the consideration of candidate 
sites: a) obtain them through procedures of thematic superposition and multi-
criteria evaluation (MCE) techniques, and b) consider each centroid of demand 
as a possible site for the installation. The MCE techniques were extensively 
developed in Buzai, G. and Baxendale, C. (2011) and the use of the centroids 
of areas as candidate sites was studied methodologically by Fotheringham, 
A. et al. (1995). The second technique appears to be linked with the modifi able 
area unit problem (MAUP) at the moment in which a variation in the number 
of spatial units will allow possible modifi cation of the results obtained.

Therefore, avoiding the necessity of assessing the infi nite localizations, 
the models work with the combinations of p centres in n candidates points, 
being p<n; where p will be the best sites obtained (Church, R. and Sorensen, 
P. 1994; Lea, A. and Simmons, J. 1995). 



391

Even when the above-mentioned simplifi cations are carried out, the 
calculations are extensive, and thus heuristic mechanisms (for iterative proce-
dures of proof and error in a continuous approximation to the best solution) 
are being sought to obtain results (Densham, P. and Rushton, G. 1992). One 
such method is presented in this work.

The identifi cation of possible solutions based upon the application of 
combinatorial supplies extremely high values when changing the elements n 
and x applying:

For example, in a simple case of locating 2 entities among 10 candidate 
sites, the result is 45 possibilities. If we raise the number of entities to 4, the 
result will be. To obtain the best 12 sites within the 70 candidate sites (centro-
ids) in Luján, we need to use heuristic methods as there are 10,638,894,058,520 
election possibilities.

Facing the overwhelming amount of calculations, the heuristic strategy 
of approximation to the best solution is theoretically quite acceptable; another 
proposal also points to the consideration of a multiple interchange of two or 
three candidates simultaneously, however, for the substantial improvement in 
the time calculations, the advance in the computational capacities of hardware 
continues being fundamental.

Models for desirable equipments

p-median model

The p-median model is an initial and simpler form of the location-allocation 
modelling procedures. Its objective is to minimize the sum of the total of the 
products of the population displacements from the points of demand (cen-
troids that group the dispersed demand) to the supply points. The function 
objective is: 
where ai is the weight associated to each demand point dij  is the distance be-

tween potential demand i and supply j points, xij  is the allocation factor which 
is equal to 1, if the center of off er j is the closest to the point of demand i and 
0 to the contrary; n is the total amount of demand points and m the potential 
supply points (considering the existing ones).



392

The model is called p-median because it is considered that p is the 
number of installations to be located. The objective of this model is to fi nd the 
minimum value of the function objective F and with this the greater spatial 
effi  ciency in respect to the total number of displacements from demand centres 
towards the p supply points.

The p-median model can be enlarged by incorporating a restriction of 
distance. It has a similar objective as the previous one, but in this case consid-
ering none of the (dij ) surpass a determinate reach value (S).

even though dij  is the lowest value for both points.
This way, on one side it is intended to act on the global cost of displace-

ments (effi  ciency) and on the other, it is intended to minimize the maximum 
distances of transfer (equity).

Applying this restriction, it is possible that the solution does not ap-
pear from the quantity of the requested points, in this sense it is possible that 
the necessity of extending them emerges.

Maximum coverage model

The maximum coverage model has the objective to maximize the total of demand 
values within a coverage ratio (R) prefi xed for the supply points. Within these 
surfaces the largest amount of demand must remain assigned.

where I is the group of demand points (indexed by i) ai is the population in 
the demand node i and xi are 1 if the center of demand i is located inside the 
area of coverage (xi ≤ R) and 0 in a contrary case. 

Maximum coverage model with distance constraint

The maximum coverage model can be enlarged by incorporating a restriction 
of distance, whose object is to maximize the total demand values within a ratio 
of coverage prefi xed for the supply points, considering that all the demand 
exists within a radio S, greater than the reach of the goods or services.



393

The formulas which are present in Buzai, G. and Baxendale, C. (2011) 
are used to fi nd optimum locations, for non-desirable equipment will not be 
developed here. Conceptually, those seek for the inverse eff ect to those pre-
sented here.

Modelling for distances calculations

When applying location-allocation models, the development of distance cal-
culation from the demand and supply points (dij ), that is, from the centroids 
of areas with grouped demand towards the existing installations, or towards 
the candidate points, is an important procedure.

From the coordinates of each location in an absolute space diff erent 
measures of distance, named metrics, can be calculated.3

The straight line distance or Euclidean distance which appears due to 
the consideration of an ideal space where there are no limitations to transit in 
any sense is obtained through the application of the following formula:

The Manhatt an distance or city block which assumes a displacement 
through a regular grid is given by:

In both cases, the results are obtained considering absolute coordinates 
over the geographical space and while in the fi rst case is used the Pythagorean 
solution for calculating the hypotenuse of the triangle, in the second case is 
used the sum of the measure units for both hicks.

With the aim of generating possibilities of more fl exible calculations 
which tend to surpass the Manhatt an metric considering the sub-estimation 
problems caused by localizations among blocks or the appearance of barriers, or 
overestimation from the appearance of streets in diff erent direction of the basic 
circulation grid (Hodgson, M. et al. 1995), the Lp metric has been proposed.

3 A metric that permit the calculation of distance between points, as has been expressed 
in Bosque-Sendra, J. (1992), must meet certain conditions: Positivity (dij   ≥ 0), Identity, if 
dij  = 0, Simmetry (dij  = dji) and Triangular Inequality (dij  ≤ dk + dkj).



394

Here appears a β parameter which indicates a modifi cation of the 
costs of displacements with the distance, and note that when β=1 we have 
the Manhatt an distance with p =1 and Euclidean with p = 2. The Lp metric was 
proposed by Love, R. and Morris, J. (1972) and turns out to be an excellent 
application alternative when it is not possible to calculate over the street net-
work, yet knowing the urban road structure. 

However, when the distance calculus is carried out on a geometrical 
basis from a structure of a raster layers, it is also possible to initially establish 
a correspondence with the analyzed metrics.

A step forward from absolute space to relative space happens when the 
distances between two locations are calculated in other units of measurement 
(time or any other types of cost), which are based on a map of friction which in-
corporates for each pixel a relative value to the eff ort which must be expended 
in order to traverse it.  In this way, from each point entity we can generate a 
cost surface which corresponds to the cost (eff ort) which accumulates in each 
pixel of the study area in order to reach the said entity.

The present application incorporates the thematic layer of the road net-
work of the study area. The distance calculations were made up on such a basis.

Application: location-allocation model applied to Primary Health Care 
Centers (PHCC) in the City of Luján, Argentina 4

The application carried out is based in the theoretical and methodological as-
pects carried out and it presents an analysis based on vector spatial structure 
made up by points, arcs (lines) and polygons (areas). The application focus-
ing on the analysis of the spatial justice of the health services, considered as a 
fundamental issue of the development procedure with GIS in the framework 
of geography of health studies (Buzai, G. 2009; Fuenzalida, M. 2010).

In this confi guration, the points correspond to the existing PHCC and 
centroids of census radii which work as demand points (upon assigning the 
population values of potential demands) and at the same time as candidate 
sites for new center localization; the lines correspond to the streets through 
which the demand and supply points will be spatially linked and the polygons 
are areas with diverse population values of potential utility.

Figure 1 demonstrates the study area. Figure 2 presents some of the 
mentioned components on the studied area: street net, localization and census 
4 The thematic layers used in the application, correspond to (1) 70 census radii of Luján city, 

2010 (2) centroids of census radii as points of demand and candidate sites for localization of 
the service off er (3) 12 PHCC as off er points, and (4) vial net as friction map. The numerical 
data taken in to account are: (5) Population of 6 to 14 years per census radii with potential 
off er value. Soft ware: Flower Map © Ultrecht University, Holland.



395

radii limits and localization of the 12 existing PHCC which, for the purpose of 
modelling, were assigned to the centroid of the corresponding census radii.

The fi rst calculus carried out successfully determines the best place-
ment for a total of 12 PHCC considering that each one of the 70 census radii, 
through its centroid, is presented as a candidate site for the localization. The 
result presents the spatial confi guration of 12 selected points from a total of 70 
that is a unique solution of the immense number of possible combinations.

To obtain this, we have used a model of maximum coverage of de-
mand, one of the practical orientations of great possibilities in the environment 
of Geographical Information Systems (Spaulding, B. and Cromley, R. 2007). 
In this model, candidate sites which may capture all the population demand 
were selected considering that no inhabitant of the study area is located more 
than 1,500 meters from the nearest PHCC, since through a previous explora-
tory analysis, it was possible to determine that with this distance restriction 
one can obtain, the same quantity of existing PHCC, which would permit its 
perfect comparison.

Figure 3 presents the map with the 12 optimum centres obtained; and 
Figure 4 is the spider map (also called desire map), made up from the centres 
and the demand allocation from the census radii which are included within 
each infl uence area, is presented in the following fi gures.

The alphanumerical data correspond to this spatial confi guration can 
be seen clearly in Table 1 from which the following information is presented:

Column 1 – Label: Number of census radii/centroid.
Column 2 – CAPS 1: Existing PHCC. 
Column 3 – CAPS 2: Optimal PHCC.
Column 4 – Demand (around a distance of 1,500 meters): Population 

UBN of potential demand to each optimum center.

Fig. 1. Location of sutdy area of Luján. Source: Edited by the author



396

Column 5 – Center (around a distance of 1,500 meters): Centroid to 
which the data of the census radii are assigned.

Column 6 – Distance (around a distance of 1,500 meters): Distance 
between centroids.

Column 7 – UBN: Population of Unsatisfi ed Basics Needs in each cen-
sus radii (Original variable from the Argentine national census: NBI Necesidades 
Básicas Insatisfechas, Synthesis of poverty).

The 12 existing PHCC are in the centroids of the census radii 18, 21, 24, 
26, 27, 35, 41, 43, 51, 60, 62 and 65, while the optimum localization PHCC with 
a maximum coverage of 1,500 meters, would be located in 20, 21, 26, 36, 42, 
48, 49, 51, 52, 59, 65 and 71. Upon being coincident the census radii centroids 

Fig. 2. City of Luján. Census radii, streets and twelve Primary Health Care Centers (PHCC). 
Source: Edited by the author (performed with Flow Map)



397

of 21, 26, 51, and 65, there is a 33% of spatial correspondence between the real 
and the optimum spatial confi guration.

The data about the population of potential demand was adjusted to the 
inhabitants with UBN. The localization 36 has the highest value of potential 
demand with UBN, with 1,552 inhabitants and localization 52 has the smallest 
value with 72 inhabitants, in their respective infl uence areas. The entire values 
are useful in order to fi gure out the calculations of distance and the popula-
tion assignation to the localization of the 12 optimum PHCC. The calculations 
made on the total values give out the following results (Table 2).

Figure 5 shows the accumulative values of population with UBN as re-
gards of distance to the optimum centres as a group, and Figure 6 presents the 

Fig. 3. 12 PHCC optimum location identifi ed and infl uence areas obtained as model of 
coverage with a solution of 1,500 meters. Source: see Fig. 2. 



398

Table 1. Results of the allocation solution

Label PHCC-1 PHCC-2 Demand Center Distance UBN

1
2
3
4
5
6
7
8
9

10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45

–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
x
–
–
x
–
–
x
–
x
x
–
–
–
–
–
–
–
x
–
–
–
–
–
x
–
x
–
–

–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
x
x
–
–
–
–
x
–
–
–
–
–
–
–
–
–
x
–
–
–
–
–
x
–
–
–

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

1,320
467

0
0
0
0

568
0
0
0
0
0
0
0
0
0

1,552
0
0
0
0
0

535
0
0
0

59
59
36
36
36
36
49
59
59
59
49
49
59
48
48
48
51
20
20
20
21
21
21
26
26
26
26
26
26
42
26
36
36
36
36
36
36
36
42
42
36
42
49
49
49

1,156.43
858.08

1,000.76
1,178.04
1,342.81
1,409.72
1,449.77

776.50
808.25

1,030.10
1,275.07

442.75
1,355.58
1,241.13
1,161.43

719.14
1,288.40
1,123.14

725.80
0.00
0.00

1,340.49
1,095.67

662.02
1,031.66

0.00
702.79
656.45
869.44

1,094.01
1,184.07
1,037.71
1,411.08
1,071.01

724.51
0.00

724.31
603.73

1,280.13
853.03
991.11

0.00
1387.81
706.16

1,490.86

1
16
28
54
15
32
36
10
4

12
7

44
25
98

133
26
54

366
427
527
253
139
75

174
91

106
10
60
43
57
84
36

209
419
332
164
42
47
59

138
174
281
68

109
164



399

expansion of the calculus considering the 12 existing PHCC and it calculates 
the localization of 5 PHCC in potential optimal locations.5

In eff ect, the fi ve selected sites correspond to peripherals census radii: 
three sites in the South, one site in the North, and one site in the Eastern sector. 
The allocation of potential demand (Figure 7) generates minor displacements.

The alphanumerical data corresponding to this spatial confi guration 
can be seen in Table 3 composed by the following information:

Column 1 – Label: Number of census radii (centroid).

5 The social map of the city of Luján shows a typical confi guration of the Latin American city 
(Buzai, G. 2003; Buzai, G. and Marcos, M. 2012), where the favored areas have central 
location and the non favored areas are located in the periphery. In 2010 Luján presents 
20 peripherals census radii, therefore it was considered the possibility of including 1/4 of 
the spatial unities with vacancy of PHCC.

Table 1. (Continued)

Label PHCC-1 PHCC-2 Demand Center Distance UBN

46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72

–
–
–
–
–
x
–
–
–
–
–
–
–
–
x
–
x
–
–
x
–
–
–
–
–
–
–

–
–
x
x
–
x
x
–
–
–
–
–
–
x
–
–
–
–
–
x
–
–
–
–
–
x
–

0
0

664
506

0
951
72
0
0
0
0
0
0

319
0
0
0
0
0

773
0
0
0
0
0

84
0

51
52
48
49
48
51
52
48
59
59
59
65
59
59
59
59
51
59
65
65
65
59
59
51
51
71
65

914.01
936.32

0.00
0.00

1,286.38
0.00
0.00

458.37
1,286.35

653.28
613.87
902.74
685.21

0.00
466.56
958.29

1,296.78
1,136.41

898.86
0.00

925.70
1,043.33

485.96
962.81

1,398.99
0.00

1,282.49

224
23
63
78

213
264
49

131
32
14
2

124
7
5

33
66
64
62

121
124
167
28
2

136
209
84

237

Source: Calculation by the author



400

Table 2. Calculations on the basis of total values of Table 1

Calculations Meters
Distances to centroids
Distances to population
Averages of distances to centroids
Averages of distances to population
Minimum distance
Maximum distance
Standard deviation to distances to centroids
Standard deviation to distances to population

59,853
5,961,586

831
83,949

0
1,490

453
104,409

Source: Calculation by the author

Fig. 4. 12 optimum locations and allocations of potential demand. Source: see Fig. 2.



401

Fig. 5. Graphic of accumulated frequencies covering the potential demand. 
Source: see Fig. 2.

Column 2 – CAPS 1: Existing PHCC.
Column 3 –CAPS 3: Optimal PHCC (result obtained considering 

PHCC-1).
Column 4 – Demand (within 1,500 meters of distance): Population 

UBN of potential demand to each optimum center.
Column 5 – Center (within 1,500 meters of distance): Centroids to 

which the data of the census radii are assigned.
Column 6 – Distance (within 1,500 meters): Distance between centroids.
Column 7 – UBN: Population with Unsatisfi ed Basics Needs in each 

census radii.
The 12 PHCC of Luján are selected in Table 3 (PHCC-1). To those were 

added the localizations 39, 48, 49, 52, and 71 (PHCC-2). The confi guration 
expands to 20 PHCC. The values of assigned potential demand decrease re-
markably. In general, although localization 35 maintains a high value of 1,166 
inhabitants, being that localization 43 has the lowest value with 68 inhabitants 
with UBN in its infl uential area. All the values are useful for calculating the 
distance and the assignation of population for the location of the 20 optimum 



402

PHCC. The calculations fi gured out upon the total values give the following 
results (Table 4).

Figure 8 graphics the accumulative values of population with UBN as 
regards the distance to the optimum centres.

Comparing the data given by the statistics presented in Table 2 and 
Table 4, we can see that there exists a decrease in the following values: Distances 
to the population of 59,611,586 to 5,091,082; Distances covered departure and return 
of 11,923,172 to 10,182,164 and Standard deviation from distances to popula-
tion from 104,409 to 101,211. This indicates that with the second solution the 
parameters of spatial effi  ciency (distances) and spatial equity (standard devia-
tion) have improved.

Fig. 6. 12 PHCC+5 proposal. Calculations made with Flow Map soft ware. 
Source: see Fig. 2.



403

Table 3. Allocation solution results

Label PHCC-1 PHCC-3 Demand Center Distance UBN
1
2
3
4
5
6
7
8
9

10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45

–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
x
–
–
x
–
–
x
–
x
x
–
–
–
–
–
–
–
x
–
–
–
–
–
x
–
x
–
–

–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
x
–
–
x
–
–
x
–
x
x
–
–
–
–
–
–
–
x
–
–
–
x
–
x
–
x
–
–

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

793
0
0

919
0
0

265
0

226
221

0
0
0
0
0
0
0

1,166
0
0
0

383
0

502
0

68
0
0

27
60
39
39
39
39
60
60
60
60
60
49
60
62
62
48
62
18
18
21
21
21
27
24
24
26
27
27
27
39
26
26
35
35
35
35
35
41
39
39
41
41
43
49
49

1,368.54
1,247.07
1,449.72
1,395.17

733.11
1,099.96
1,363.27
1,165.49

899.70
636.11

1,256.18
442.75
959.74
734.39
813.39
719.14
603.54

0.00
994.75
757.82

0.00
1,340.49

626.24
0.00

527.85
0.00
0.00

322.62
579.61

1,434.59
1,184.07
1,374.13
1,178.16

521.75
0.00

724.51
1,291.02

852.10
0.00

618.73
0.00

1,013.56
0.00

706.16
1,490.86

1
16
28
54
15
32
36
10
4

12
7

44
25
98

133
26
54

366
427
527
253
139
75

174
91

106
10
60
43
57
84
36

209
419
332
164
42
47
59

138
174
281
68

109
164



404

Table 3. (Continued)

Label PHCC-1 PHCC-3 Demand Center Distance UBN
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72

–
–
–
–
–
x
–
–
–
–
–
–
–
–
x
–
x
–
–
x
–
–
–
–
–
–
–

–
–
x
x
–
x
x
–
–
–
–
–
–
–
x
–
x
–
–
x
–
–
–
–
–
x
–

0
0

433
395

0
697
72
0
0
0
0
0
0
0

329
0

485
0
0

773
0
0
0
0
0

84
0

51
52
48
49
48
51
52
48
27
60
60
65
60
60
60
60
62
60
65
65
65
60
60
62
51
71
65

914.01
936.32

0.00
0.00

1,286.38
0.00
0.00

458.37
889.29
991.48
927.30
902.74
990.64
466.56

0.00
502.41

0.00
684.59
898.86

0.00
925.70

1,348.76
523.24
750.62

1,398.99
0.00

1,282.49

224
23
63
78

213
264
49

131
32
14
2

124
7
5

33
66
64
62

121
124
167
28
2

136
209
84

237

Source: Calculation by the author

Table 4. Calculations on the basis of total values of Table 3

Calculations Meters

Distances to centroids
Distances to population
Averages of distances to centroids
Averages of distances to population
Minimum distance
Maximum distance
Standard deviation to distances to centroids
Standard deviation to distances to population

51,505
5,091,082

715
71,686

0
1,490

485
101,211

Source: Calculation by the author



405

Fig. 8. Graphic of the accumulated frequencies coverage of potential demand. 
Source: see Fig. 2.

Fig. 7. Allocation of potential demand to solution of 17 PHCC. Source: see Fig. 2.



406

Conclusions

In this work the aptitude of the location-allocation models was exemplifi ed 
based on the calculus of coverage, mainly applied in the search of effi  ciency 
and spatial equity of the Primary Health Care Centers (PHCC) in the city of 
Luján. Also, the obtaining of solutions that start going into the process of ver-
ticalisation (Eastman, J. 2007) through the Geographical Information Systems 
(GIS) and of the Spatial Decision Support System (SDSS) supply important 
possibilities for the right support on decision making in spatial issues.

The questions answered in this work are the following: which is the 
rate of correspondence between the real localization of the supply points and 
the ideal localization based on the spatial distribution of the demand popula-
tion? In which way the spatial effi  ciency and the spatial equity is modifi ed in 
accordance to the reallocation of these points and fi nally where should new 
installations be sett led in order to satisfy the distributed demand? The appli-
cation carried out for the city of Luján has clearly answered these aspects by 
presenting key elements for the urban planning of the studied area.

By generating concrete results which allow us to answer these ques-
tions, one may note that the theoretical–methodological guidelines currently 
developed in the interior of the technologies GIS+SDSS allow a valid approxi-
mation to the solution of complex localization.

We consider that automated spatial analysis contributes to support the 
theoretical–methodological geographical basis, in decision-making processes 
in urban planning issues, thus allowing one to confront actions oriented to 
reducing aspects of socio-spatial inequalities of the population.

Acknowledgements: Claudia Baxendale (University of Buenos Aires, Argentina), Manuel 
Fuenzalida (Pontifi cal Catholic University of Valparaiso, Chile), Antonio Moreno-Jiménez 
(Autonomous University of Madrid, Spain) and David Robinson (Syracuse University, 
USA).

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    /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke.  Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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    /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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    /TUR <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>
    /UKR <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>
    /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing.  Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
    /HUN <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>
  >>
  /Namespace [
    (Adobe)
    (Common)
    (1.0)
  ]
  /OtherNamespaces [
    <<
      /AsReaderSpreads false
      /CropImagesToFrames true
      /ErrorControl /WarnAndContinue
      /FlattenerIgnoreSpreadOverrides false
      /IncludeGuidesGrids false
      /IncludeNonPrinting false
      /IncludeSlug false
      /Namespace [
        (Adobe)
        (InDesign)
        (4.0)
      ]
      /OmitPlacedBitmaps false
      /OmitPlacedEPS false
      /OmitPlacedPDF false
      /SimulateOverprint /Legacy
    >>
    <<
      /AddBleedMarks false
      /AddColorBars false
      /AddCropMarks false
      /AddPageInfo false
      /AddRegMarks false
      /ConvertColors /ConvertToCMYK
      /DestinationProfileName ()
      /DestinationProfileSelector /DocumentCMYK
      /Downsample16BitImages true
      /FlattenerPreset <<
        /PresetSelector /MediumResolution
      >>
      /FormElements false
      /GenerateStructure false
      /IncludeBookmarks false
      /IncludeHyperlinks false
      /IncludeInteractive false
      /IncludeLayers false
      /IncludeProfiles false
      /MultimediaHandling /UseObjectSettings
      /Namespace [
        (Adobe)
        (CreativeSuite)
        (2.0)
      ]
      /PDFXOutputIntentProfileSelector /DocumentCMYK
      /PreserveEditing true
      /UntaggedCMYKHandling /LeaveUntagged
      /UntaggedRGBHandling /UseDocumentProfile
      /UseDocumentBleed false
    >>
  ]
>> setdistillerparams
<<
  /HWResolution [2400 2400]
  /PageSize [612.000 792.000]
>> setpagedevice