The Gravity Polygons Method – An Operationalisation of the Central Places Theory in Marketing

Location of facilities is probably the most important decision to be taken by retailers, bankers, hotel chains and many other businesses. Location determines sales, market share and profits (Kimes and Fitzimmons, 1990). Location decisions are most important because they represent large fixed investments. Finally, location can play a crucial role in differentiating and offering competitive advantage to similar facilities (Craig, Ghosh, MsLafferty, 1984).

Research on spatial organisation of a market can be classified into three groups [1] : research on modelling spatial consumer choice behaviour [2] ; research dealing with managerial facility location decisions [3] ;  approaches based on central place theory.

The last, which are also historically the oldest, suggest a theoretical framework for the spatial distribution of retail outlets, based upon a set of hypotheses on the spatial behaviour of consumers and distribution companies and about their environment.

This paper presents and applies a new market space evaluation and partition method based upon concepts and structures from the central place theory.

It suggests a geometrical approximation of market areas, the « gravity polygons », that results in attractiveness sensitive partitions of the market space. Gravity polygons are geometrical approximations derived from proximal area polygons that include gravity aspects and spatial choice models.

The methodology that is being developed here, tries to set the central places theory analysis framework on a more flexible and measurable basis. Some solutions are suggested to integrate spatial choice models and to identify hierarchies in the spatial organisation of the market.

The application deals with the spatial distribution of bank retail networks in an urban agglomeration of one million inhabitants. The calculations traced the spatial behaviour of over 200000 customers serviced by the analysed network and produced positive results that incite to apply the method to other contexts and to other research directions.

The central places theory makes it possible to join the spatial behaviour of the customers to the one of the companies that operate in the same geographic market space. It is a third research direction dealing with spatial organisation of the market. Historically it is the oldest.

Drawing upon several hypothesis on the spatial behaviour of buyers and firms, this theory gives a representation of the spatial distribution of sales outlets within a geographic area and determines the shape of their market area.

The Central Place Theory was first proposed by Christaller in 1935 and was given a micro-economic formalisation by Lösch in 1954. Christaller suggests that there is a maximum distance the consumer is willing to travel in order to buy a good which defines the outer limit of a market area, and that the minimum amount of demand for a store to be viable defines the inner limit of its market area. Lösch considers that the demand for a store (central place) diminishes with distance due to transport cost increases. He defines the well known "spatial demand cone" who's volume determines the economical viability of a location.

According to the Central Places Theory, market areas can be represented, in a uniform market, by circles with a fixed radius. It can be shown that the space is best covered (minimising not covered areas) when the circles are distributed like in figure 1 (a).

Figure. 1 - Triangulo-hexagonal structure covering the territory with a maximum number of distribution points

The circular market areas are allowed to overlap (like in figure 1 b) in order to avoid uncovered areas, where customers are not supplied. Finally, consumers living in places where market areas overlap, and acting rationally will choose the nearest centre. This gives a hexagonal shape to market areas as illustrated in figure 1 (c).

This analysis shows that in a uniform market, distribution points are located at equal distances and have hexagonal market areas of equal size. This hexagonal structure is optimal with respect to the simplifying hypotheses formulated by central places theory:

-         the space is an unlimited plane;

-         consumers are : identical among each other, uniformly distributed and can move freely in any direction;

-         distribution points have equal size and attractiveness

The central places theory also introduces the concept of  “central place hierarchy” that states the existence of an ordered spatial distribution of equipments based on the size of the market areas. The spatial competition incites to locate new retail units that are ever nearer to each other. Their market areas overlap in order to produce ever smaller hexagonal market areas. It is then possible to determine several hierarchical levels for these central places. At each hierarchical level, the locations are forming a net of hexagonal market areas. Each level offers a given bundle of goods and services (Christaller, 1935), or, more flexibly, any combination of goods and services (Lösch, 1954).

Although Central Place Theory remains the "best developed normative theory" (Craig, Ghosh and McLafferty, 1984) and the "best known and most widely accepted" (Mason and Mayer 1981)  theory of the location and spacing of retail centres, some shortcomings need to be emphasised with regard to a geometrical construction of market areas which supposes uniform servicing of the spatial network and ordered distribution of facilities. Critical aspects (like complexity of the environment, consumer sensibility to distance, specific strategic options taken by companies, dynamics of site locations) have often been discussed in contributions incumbent to the other two research directions already discussed. The next sections show that the central place theoretic approach remains interesting for certain situation and suggest some further developments based on the "gravity polygons" method.

Some empirical studies  (Berry 1967, Skinner G.W.1964) seem to show a more flexible structure of polygonal market areas. This less regular geometric shape (as compared to hexagons) can be obtained by relaxing certain conditions imposed by the central place theory:

• the space is not an endless plane, it provides only a limited number of possible locations.
• the consumers  are not uniformly distributed
• the distribution points are not identical, they mark some differences in mass and attractiveness.

The hexagonal structure used in the central places theory seems inappropriate in real world market partitions among retail units.  Therefore we suggest a more flexible triangulo-polygonal structure that gives convenient geometrical approximations of the market areas and helps central places theory application to spatial distribution in marketing.

There is a variety of methods to estimate trade areas of an outlet. All rely on some measure of accessibility, based on the consumer's travel time, distance, of cost or a combination of these. To evaluate accessibility some use simple geometric approximations based on Euclidean (straight line) measurement of distance others rely on more detailed (and costly) often survey based techniques to record travel patterns and derive there from the travel cost (time, distance, etc.).

Ghosh and McLafferty (1987) in their review of trade area estimation methods consider three approaches: the space-ratio method that uses the relative selling space to determine the market area, the proximal area method that uses Dirichlet(1850) or Thiessen and Alter (1911) polygons to compute the nearest distance area around an outlet, and a method based on Reilly's Law of Retail Gravitation.

The space-sales method uses initial circular approximations of trade area, subjective or experience based radii equal to expected travel distance to separate the total market area. It then estimates the distribution of buying power within the joint market area using secondary or internal data and finally divides the joint area among outlets proportionally to size of each outlet in the total retail space.

The proximal area method defines nearest point polygons around each outlet based on the nearest centre hypothesis of classical central places theory (Christaller, 1935, Lösch 1954). It identifies for each outlet the nearest neighbouring outlets (normally using some nearest point triangulation) and intersects bisectors to the imaginary adjacent lines linking the outlet to the neighbours. In this way the method demarcates the area that is closer to an outlet than to any other..

The method based on Reilly's Law of Retail Gravitation is demarcating trade areas for outlets fixing the boundary between two neighbouring outlets there where attractiveness of the two outlets is equal. The formula used was derived by analogy to the Newtonian law of planetary attraction. It states that shoppers from an area between two cities are attracted by each city in direct proportion to the city's population and in inverse proportion of the square of the distance between the shopper's location and the city.

In spite their success these methods are known to have as major limitation the lack of theory explaining spatial choice behaviour.

Spatial choice models are based on Luce’s (1959) fundamental axiom, telling that individuals choose among several alternatives stochastically, depending on the utility offered by each choice alternative. They insure the expected theoretical support and have therefore gained large acceptance.

The gravity polygons (Figure 2a), introduced by Calciu and Salerno (1996), are geometrical approximations based upon proximity areas that are able to integrate gravity aspects and spatial choice models. They use Delaunay’s (1932, 1934) triangulation method in order to identify nearest neighbour retail units (Fang and Piegl, 1992) and to fix an attraction limit between two neighbouring units. The Voronoi [4] (1907, 1908) polygons that fix this limit  at the middle of the distance between  two neighbouring units (Figure 2b), can be seen as a special case of gravity polygons.

Figure 2. Attraction limit to obtain Voronoi polygons and gravity polygons (a)When attraction limit tends towards 0.5 it intersects the straight line between A and B through the middle and Voronoi polygons are obtained, if attraction limit is variable along AB gravity polygons are obtained.

In a nearest points triangle or Delaunay (1932, 1934) triangle the attraction limit between two points on fixed on each edge bilaterally. We suggested a general formula for computing the attraction limit between two points A and B of a triangle (limit noted L_AB):

                                                (1)

where a_A and a_B are attractiveness indexes of the points A and B and is the elasticity coefficient towards distance.

The attraction indexes and the value of can be fixed by objective measures (spatial choice models) or subjective ones (expert judgements).

A first application, using expert judgements (Calciu, Salerno and Vanheems, 1996), has been done in a multi-network environment. It dealt with dividing the market territory in a town among 22 agencies belonging to nine bank retail networks and tested how gravity polygons were functioning in real situations, where the spatial distribution of retail outlets is irregular and their size varies.

The resulting territorial partitions have shown that small bank agencies were relatively uniformly distributed and that the bigger ones tended to concentrate near town centres or on main commercial roads. This showed that central place hierarchy is manifest in retailing of bank services.

The spatial organisation of supply cannot be thoroughly analysed without taking into account demand and without objective measures…

This justifies the development of a territorial portioning methodology base on the central places theory and on empirical estimation of the consumer spatial choice behaviour.

The methodology that is being developed here, tries to set the central places theory analysis framework on a more flexible and measurable basis. It tries to go beyond the purely theoretic and normative aspects of central places and to integrate empirical measurement models in order to prepare the development of spatial decision support systems in marketing.

At this stage our approach focuses on the development of an estimation procedure of retail units attractiveness and on the development of a market space partitioning method reflecting this attractiveness.

With this occasion the relevance of a deviation from the nearest centre hypothesis is twice verified: once through the predictive precision of a extended spatial choice model compared to simple one and through the efficiency of a attractiveness sensitive territorial partition compared to an insensitive one.

The envisaged task flow chart is represented in figure 3.

Figure 3 - Flow chart of  a territorial market partitioning method taking into account the consumers’ spatial choice behaviour.

Preceding stages aspects like the taking into account of a hierarchy of retail units (in the spirit of the central places theory) as well as extensions of this method will be discussed in the final part of this paper.

Estimation of attractiveness sensitive market area partitions, supposes previous estimation of patronage sensitivity to the outlet's location and to non-locational marketing variables. MCI models are well adapted to incorporate multiple attractiveness and proximity indicators and are also computationally easy to estimate.

Two model forms were tested using the available data a simple model, the basic spatial model having as unique explanatory variable distance and an extended spatial model including besides distance a set of available non-locational variables.

The basic spatial model assumes equal attractiveness for each outlet, implying that the outlet's market potential depends uniquely on proximity to consumers. The following relation expresses it:

                                                                                                         (2)

where:

            d_ij = distance (or other measure of spatial separation) between consumers in point i and outlet j

            = parameter reflecting consumer patronage sensitivity to distance.

The extended spatial model adds non-locational variables, essentially marketing variables, whose effect on patronage is assumed similar among all outlets.

This model is the Nakanishi and Cooper (1974) generalisation of Huff’s (1964) model:

                                                                                            (3)

where:

            a_kj = marketing variable k (where k = 1, 2 .. K) for outlet j and

            _k = parameter reflecting consumer patronage sensitivity due to variable k.

In order to estimate these MCI (Multiplicative Competitive Interaction) models a matrix that captures the spatial behaviour of clients grouped by census tracts. It crosses the number of clients by retail unit and by census tract. This matrix is a valuable means to study clients’ spatial choice behaviour because it offers complete geographic coverage and gives information about actual outlet patronage and not only intentions.

The formula (1) of the attraction limit between two retail units can be adapted in order to accept previously calibrated spatial choice model parameters (see also Calciu and Delagrande, 1996).

We suggest a probabilistic approach to the attraction limit between two outlets considering that this limit can be fixed on a straight line in point i, where the probability of patronising the outlet A (p_iA) is equal to the one patronising the outlet B (p_iB).

To compute the probability of a point i to patronise an outlet j we use the MCI model in formula (3):

The attraction limit between A and B can then be deducted from the following expression:

                                                      (4)

that symbolises the point i with equal probability of patronising outlet A and B

Here from we obtain the formula for the attraction limit between A and B (L_AB) seen from A as a fraction of the distance between A and B:

                                                                                       (5)

It depends upon the attractiveness indexes of the two outlets.  It is easy to see that the bigger the value of the bigger the influence of distance and smaller the effect of marketing variables. For big s the L_AB tends towards 0.5, meaning the middle of the distance between A and B.

By using gravity polygons and this probabilistic definition of the attraction limit we obtain new market areas around an outlet. They regroup customer locations having a higher probability to patronise that outlet than patronising other neighbouring outlets.

The analysed data deal with the retail network of a bank in a large metropolitan area in France. The network consists of 70 bank agencies, (with 1 to 20 employees) and covers an area containing 5600 census tracts representing a population of nearly one million inhabitants. The bank agencies were dispersed over the territory with densities proportional to the ones of the population.

These data contain information on the spatial behaviour of customers that are grouped by census tract and some information helping to evaluate the attractiveness of each bank outlet.

A visual and more detailed description of the given spatial situation can be found in Appendix A1.

The spatial behaviour results from observing patronage flows, that is number of persons from each census tract patronising each agency. Marketing data concerning each agency included total accounts and total active accounts at the moment of the study.

Other attractiveness variables were screened but rejected due to weak relationship with the criterion variable of outlet patronage probability.

In the end the matrix of outlet patronage by tract included 29770 patronage directions (flows) with intensities ranging from 1 to 480 customers per flow. In this way the travel directions towards 60 retail outlets of the bank for 206535 people were identified.

Ten other units have not been included in spatial choice model calibration calculations because they were situated on the edges of the analysed territory were the geometric shape of the partitions had to be distorted and attraction data were biased because only unilaterally available.

Unfortunately we had only patronage data for each census tract and no information about actual sales, which could have revealed more accurate flows.

^a p < 0.001

^b p < 0.05

Table 2. Estimated parameters for other marketing variables screened but rejected due to week relationship to consumer spatial choice.

^c rejected variables because not significant

The best fitting MCI model was selected to build an attractiveness index based on previously estimated parameters and to compute attractiveness scores for each of 70 available outlets.

Two partitions have been computed [5] , one sensitive to attractiveness (gravity polygons) and the other insensitive (Voronoi polygons). From the 70 resulting polygones, in each partition, 54 have been retained for comparative analysis of their effectiveness [6] .

Gravity polygons captured 51,69% of their outlets’ customer patronage flows while Voronoi polygons captured only 49,15% of those flows [7] . This results show that gravity polygons are slightly more effective than Voronoi polygons in circumscribing clients. This is consistent with the comparative results from spatial choice models, there too the extended model had better predictive precision 0,553 (adjusted R²) then  the  simple model  0,513.

Allthough comparative results of choice behaviour models and partitions converge indicating that the gravity polygons method gives better results than proximal area methods, this advantage is relatively feeble. This weakness is probably due to the context of application and to the relevance of variables used to specify attractiveness of the retail outlets and partially to the way the method was applied.

Patronage behaviour in bank retail networks could be fundamentally governed by proximity criteria.  This could explain prevalence of the distance coefficient in the extended spatial choice model and the small difference in effectiveness between the simple model and the extended model.

Applying the « gravity polygons » to other contexts where the spatial behaviour of consumers is more sensitive to marketing variables should be useful in order to reveal the advantages of this method.

The data that were available in this study are exclusively internal data of the company. Their relevance in expressing outlet attractiveness is limited. Other variables [8] measuring customer perceptions of accessibility, visibility or visual attractiveness of the same outlets have been collected retroactively by a field research company on behalf of the analysed bank in order to complete the measures of attractiveness. Due to the lack of transparency on the data collection  methodology, the resulting data have not been introduced in the evaluation the “gravity” partition.

Differentiated treatment, by clustering retail outlets on their customers’ average patronage distance, can improve the performance of gravity polygons partitions and of spatial choice models. It is also a way to test whether there is specialisation and/or hierarchy in the spatial organisation of the market and to analyse these phenomena according to the central places theory. The classification of bank agencies separates rural from urban unit and urban ones into central and proximal outlets.

The spatial choice models calibrated separately for each group of agencies show significant differences in the importance given to different choice criteria and confirm the network’s specialisation. The adjustment of the spatial choice model for the proximal units is largely better (adjusted R² 0,601 for the extended model and 0,578 for the simple model) than the one for all the retail units (as presented before). The adjustment is not so good  for the rural units.

As far as the central units are concerned, it is probably the concentration in the town centres and along the main roads not integrated in the geometric approximations’ definitions that causes deviation from proximity.

This deviation explains the significant improvement of the predictive precision of the extended model (adjusted R² 0,544) compared to the simple model (adjusted R² 0,424). The very low values that were obtained for rural units (adjusted R² for the extended model and 0,208 for the simple model), is mainly due to the fact that customer location was approximated using the census tract barycentre, which can induce large deviations from real location, as rural census tracts (communes) have a much larger area than census tracts in an urban environment.

It is therefore advisable to concentrate on urban effects of the hierarchy of central places and to observe that the distinction between the two levels can improve attractiveness estimations for retail units and the effectiveness of territorial partitions.

The originality of the proposed method consists in the relocation, the computation and the estimation of the attraction limit between two neighbouring retail units.

This limit appears as a deviation from the hypothesis of the nearest centre, but remains within the neighbourhood. Estimation of attractiveness parameters using spatial choice models didn’t include  such constraints. Future research should test other estimation procedures or other spatial choice models that limit choice possibilities like nested logit or competing-destinations models (Fotheringham 1983, 1986).

The development of a hierarchy of  central places, apart its organising role, can participate in easing constraint of an horizontal partition by rarefied units achieving the higher levels and increase of territories that are affected to them.

This is a problem that needs to be better studied. A “demand cone” (respectively a pyramid of demand) could be computed for each gravity polygon, in order to deduce a part of the demand drained by higher levels in the hierarchy. It seems to us that non parametric estimation by the SSDA (Squared Surface Analysis Density Analysis, Rust and Brown, 1986) method or further more by kernel functions (Donthu et Rust,  1989) used to estimate "demande cones" for stratified retail outlets (belonging to different hierarchy levels) through continuous functions, offer interesting alternatives for evaluating for each demand point (census tract, quarter etc.) the part of demand belonging to higher levels in the hierarchy. Kernel functions can play for vertical analysis of market areas the same role that spline functions have in the horizontal analysis (in the plan).

The methods developed in this paper for computing market areas relay on thow major simplifications. The borders of the market area are discrete (geometric) and clear cut while in the real world they are continuous and stochastic. The territory on the other hand is perceived as a continous space while the real one is discrete and patronage movements follow the topology of a road network and demand is located at the nodes of this network.

These limits are shared by all mentioned geometric approximation methods of the market area.

The suggested methods have mainly a cost/benefit advantages. Once the attractiveness index are estimated, by eventualy separating several hierarchic categories, the partition and tracing of the market area is obtained almost instantaneously. In order to build the kernel of geomarketing decision support system, we developed a program written in C++ language that implement the territorial portioning method that has been presented (together with a set of connex treatments like Delaunay triangulation, Voronoi polygons and several market area congruence measurement methods) included into a dynamic link library (DLL) to the Excel spreadsheet program and to geographic informations system (GIS) MapInfo.

An illustration is given in appendix A1. In order to better understand the territorial organisation of the market and manage retail networks, marketing managers can in this way apply the principles of theory of central places to partitions of the market place that are sensitive to attractiveness.

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The geographical analysis of the patronage flows using the Geographic Information System (GIS) MapInfo on an PC computer and some self programmed procedures in MapBasic offered an interesting view on travel patterns toward the outlets forming the retail network of the analysed bank (Figure 4).

From each census tract customers were patronising an average of seven outlets, the number of outlets patronised from each tract varying from 1 to 39. The 70 outlets situated in this metropolitan area were attracting people from 5600 census tracts.

Each outlet had customers originating from 33 to 1640 census tracts, with an average of 468 census tracts per outlet. The number of customers visiting each outlet varied from 41 to 16213, with an average of 3246 customers per outlet.

Figure 4. Decision support system based on gravity polygons

The software we developed uses these data, with outlet attractiveness indexes and elasticity to distance coefficients, that have been measured objectively (estimated spatial choice models) or subjectively (expert judgements) in order to produce gravity polygons partitions (Figure 4b). Additional facilities like Delaunay triangulation, Voronoi polygons and several market area congruence measurement methods have been integrated to the same software (Figure 4a).

Statistical analysis of the distribution of travelled distances gave a better insight to patronage behaviour.

The majority of the travelled distances is concentrating towards the minimum in a range which varied between 0 and 28 kilometres. The average distance travelled by the 254744 customers first analysed was 1,5 km. Nearly three quarters of the customers (70%) travelled less than 2 km and only 10% of the customers travelled more than 4 km.

Automatic Hierarchical Clustering (AHK) of the 67 distribution outlets on the average patronage distance is shown in Figure 5. It separates rural units (with the largest average patronage distance) from urban units. Among the urban units, two categories are distinguished: central units (with longer average distances) and proximity units (with shorter average distances).

The three outlet categories can be easily identified on the map in figure 5. The circular symbols used for each unit are sized proportionally to the average patronage distance.

Figure 5. Obtaining a hierarchy of retail outlets

The resulting classification allows for slicing the competitive space into three levels that can be separately analysed and to which distinct strategic solutions can be applied.

The table below summarizes the influence of distance and of several marketing variables (using regression coefficients) on the customers’ spatial choice behaviour for all the outlets off the analysed network and for each of the previously identified categories. The importance of the marketing variables varies from one category of retail bank agency to the other *.

Tableau 3. - Importance des critères de choix spatial par catégories d'agences bancaires exprimés par les coefficients de régression des modèles MCI

A = Pedestrian accessibility; B = Car accessibility; C = Parking accessibility; D = Visual Impact vs. Compet. ; E = Distance; F = Employees; G = Location vs. Compet.; H = Employee turnover; I = Commercial Environment; J = Net Bank Product; K = Habitation proximity; L = Product total; M = Cash point visibility; N = Logo visibility; O = Facade visibility; P = Visibility, spotability; * = variable rejected as non significant.

Proximity agencies become more attractive with increasing employee numbers, employee turnover, car accessibility and visibility, spotability. The product total, logo visibility and commercial environment tend to have a slightly negative impact on attracting proximal customers.
For central agencies pedestrian accessibility and product total are not appreciated but parking accessibility and location compared to competitors are important. The look compared to competitors appears to have a slightly negative impact. Employee turnover and cash point visibility have a positive impact.
For rural agencies, parking accessibility is even more important than distance and employee turnover counts while the number of employees has a reverse impact.

L’article présente et applique une méthode originale d'évaluation et de partition du territoire commercial fondée sur des concepts et structures proposées par la théorie des places centrales. Il s’agit d’une approximation géométrique de l’aire de marché, les "polygones gravitaires", qui permet une partition de l’espace commercial sensible à l’attractivité. La méthodologie développée place le cadre d’analyse offert par la théorie des places centrales sur des bases plus flexibles et mesurables. Des solutions sont proposées pour intégrer des modèles de choix spatial et pour identifier des hiérarchies dans l’organisation spatiale du marché. L'application porte sur la distribution spatiale d’un réseau d’agences bancaires dans une agglomération d’un million d’habitants. Les calculs effectués en s’appuyant sur le comportement de choix spatial de plus de 200000 clients de ce réseau donnent des résultats encourageants qui conduisent à recommander l’application de la méthode dans d’autres contextes.

Mot clés : géomarketing, partition territoriale, modèle gravitaire, choix spatial, théorie des places centrales

Unlike other service firms, which by definition, lack patent protection for their products, banks have historically had a certain monopoly for the services they offered. This monopoly vanishes under the pressure of deregulation and technology and causes fundamental change in the way retail banks territorially compete.

J.E.G. Bateson (1995) considers that services firms have four basic choices when competing territorially, these choices result from four possible combinations of static and mobile clients and servuction systems.

During a long period of time, when regulations and state control limited products, pricing and location, banks had a static servuction system and were competing for reach with customers considered mobile within a relatively big urban area.

In France the deregulation process started by liberalising opening and location of agencies, 1966-67 (M. Zollinger, 1992). The approach to competition changed. The servuction system became mobile in order to attract (drain) "static" customers. In those years, banks started competing for geography. The number of agencies increased from 8000 in 1967 to more than 25000 in 1984 when it seemed to stabilise. France became one of most "bancarised" countries of the world, with less than 1300 inhabitants per bank counter, with more than 99% of the French customers of a credit institution and 96% of them having a cheque account.

This geographic expansion came to saturation in the nineties due to competition (especially the one from non bank actors like insurance companies and big retailers) favoured by further deregulation of prices and products and due to technology.

Credit cards, telephone banking and direct marketing techniques are subtly moving banks towards a situation were competing for market share is less dependent on geography.

Under these circumstances banks are feeling the burden of too large and not enough specialised (differentiated) retail networks. They are closing agencies and restructuring the network. To do this, management needs methods and tools to appreciate territorial performance and estimate market areas for their retail networks.

In this paper we apply an attractiveness sensitive method to divide a market territory among the agencies of a bank within a big metropolitan area, using outlet patronage and performance data. We analyse the sensitivity of the method compared to attractiveness neutral methods and suggest some measures of congruence between managerially defined market territories and computed market territories (using our method or other methods). We also discuss the managerial use of these methods.

Le modèle du centre le plus proche repose sur l’hypothèse que  l'acheteur fréquente l'unité de distribution la plus proche pour se procurer le bien ou service qu'il recherche.

Des critiques émises à l'égard d'une conceptualisation considérée comme trop simpliste du comportement de l'acheteur, ont amené certains auteurs à remettre en cause la notion de distance absolue et à introduire la notion de distance relative, qui  intègre la perception de l'acheteur (Devletoglou 1965, O'Sullivan et Ralston 1976). Le modèle s’adapte mal au comportement d’achat de produits anomaux (Filser, 1985) et aux déplacements multi-achats (achats groupés). Il postule une stricte équivalence des points de vente et une stratégie de minimisation de l’effort pour l'acheteur alors même que ce choix pourrait être la conséquence d'une stratégie de maximisation de la satisfaction globale retirée de l'acte d'achat (Hodgson, 1981).

Les modèles gravitaires et plus précisément les modèles probabilistes d'analyse spatiale (également appelés modèles de préférence révélée) permettent de déterminer la probabilité qu'un acheteur fréquente un point de vente à partir de l'utilité qu'il  lui associe: l'utilité varie positivement avec une mesure d'attractivité et inversement avec une certaine puissance de la distance. Les consommateurs compensent la désutilité causée par la distance supplémentaire à parcourir avec l'utilité procurée par le plus d'attractivité. Le précurseur de ces modèles est le modèle gravitaire de Reilly (1931) qui fixe la limite entre deux centres de distribution en fonction de la distance qui les sépare et de leurs dimensions respectives. Les formulations probabilistes les plus connues en marketing sont le modèle de Huff (1964, 1966) et le modèle MCI (Multiplicative Competitive Interaction) de Nakanishi et Cooper (1974). Ce dernier permet, en outre, de mesurer l'attractivité d'un point de vente comme une composante de plusieurs variables: l'image du point de vente (Stanley et Sewall 1976), la présentation des articles (Jain et Mahajan 1979), la proximité avec d'autres centres (Hansen et Weinberg 1979). Cliquet (1990) a  proposé un modèle interactif de concurrence spatiale subjectif qui permet de mesurer l'attractivité d'un point de vente à l'aide de données subjectives à la place des données objectives habituellement utilisées. Certains auteurs proposent d'utiliser le modèle logit multinomial (MNL) plutôt que le modèle MCI parce qu'il est mieux adapté à des problèmes de choix de type discret.