J Geograph Syst (1999) 1:137±153

Coupling land use allocation models with raster GIS

Robert G. Cromley, Dean M. Hanink

Department of Geography U-148, University of Connecticut, Storrs, CT 06269-2148, USA(e-mail: cromley@uconnvm.uconn.edu; hanink@uconnvm.uconn.edu)

Received: 7 April 1998/Accepted: 2 October 1998

Abstract. As geographic information systems (GIS) have moved from infor-mation storage and retrieval operations towards more decision support func-tions, there is a need for more integration of spatial analytical modules thatcan assist in locational decisions. This paper presents a methodology for cou-pling land use allocation models with a raster GIS. For raster systems, theintegration of any decision module has been limited by the size of rasterdatasets that may contain hundreds of thousands of pixels. Therefore, decisionheuristics have been used rather than exact methods such as mathematicalprogramming models. For the problem of land use allocation, the specialstructure of the generalized assignment problem is used here to handle largescale datasets. The advantage of the mathematical programming approach isthe additional information associated with the dual variables and opportunitycosts that can be used in subsequent sensitivity analyses.

Key words: Geographic information systems, land use allocation, generalizedassignment problem, multiobjective analysis

JEL classi®cation: Q15, Q24, R14, R52

Introduction

The evolution of geographic information systems (GIS) technology pro-vides an opportunity and challenge for its integration with various forms ofspatial analysis (Goodchild 1987; Goodchild et al. 1992; Fotheringham andRogerson 1993). As the use of these systems have moved from informationstorage and retrieval operations towards more decision support functions,there is a greater need for more integration of spatial analytical modules thatcan assist in locational decisions. This paper presents a land allocation algo-rithm that can be coupled with a raster GIS to support land use locationaldecisions.

In any locational decision model, the initial problem is to de®ne the basic

( Springer-Verlag 1999

Correspondence to Robert G. Cromley

decision unit. The nature of the decision unit in a locational context corre-sponds to how geographic space is treated in the underlying model. In loca-tion theory, there are two fundamental locational decisions around whichmodels have been formulated that are based on di¨ering views of geographicspace. First, given a location, what economic activity should be placed there?This is the basic question answered by von Thunen's model of agriculturallocation and other land use models. This approach views the geography of theproblem as continuous and space-®lling. Second, given an economic activity,where should it be located? This decision is answered in Weber's model ofindustrial location and central facility location/allocation models. The geog-raphy is punctiform ± space is empty except where objects are located. Inland-use models, space has extent and only one economic activity can occupyeach land parcel; the solution determines the most e½cient arrangement ofcompeting land use types.

How one models space is also a fundamental theme in GIS. Two basicdata models ± raster and vector ± are related to our concept of space. Rastersystems are used to implement ®eld-based models whereas vector systems areused to implement entity-based systems (Worboys 1994). Raster data modelsview space as continuous and space-®lling, whereas vector models view spaceas being discrete ± space is empty except where entities are located. Rastermodels ®ll every location with activity whereas the vector model only locatesactivity were needed.

The Herbert/Stevens model (Herbert and Stevens 1960) and many otherforms of land allocation models and spatial interaction models utilize a zonalaggregation of land as the basic decision unit. These types of models are gen-erally integrated within the context of a vector GIS (see e.g. Batty and Xie1994). However, land-use allocation problems that fall within the von Thunentradition of location analysis are modeled here as a continuous space problemin land use. Modeling spatial relationships such as distance to the nearestroad, stream or market is facilitated in a raster system; whereas there is noequivalent function in a vector system ± point, line and area bu¨ers and point-to-point distances are the closest such functions. Likewise, surface modelingfunctions such as slope and aspect, used in many types of land use analysis,exist for pixels but not for polygons. A raster data model will be used here inwhich the basic decision unit is a grid cell. This permits thematic values thatvary continuously over space to be recorded at cell locations. Such cell-basedmodels have been used in di¨erent land allocation models (Hopkins and Los1979; Wright et al. 1983).

Land-use problems using mathematical programming models are generallysolved outside of the GIS environment although they may have an interfacewith either a GIS or a decision support system (Wright et al. 1983; Campbellet al. 1992; Chuvieco 1993; Moxey et al. 1995). Direct model integrationwithin a GIS is still somewhat limited. Goodchild et al. (1992) state that thereare several levels of model integration with a GIS: 1) stand alone systems withseparate data formats, 2) loose coupling through common data formats, 3)close coupling of models using macro languages, and 4) full integration withsupport by GIS vendors. Certain systems such as ARC/INFO and TRANS-CAD have integrated some common programming models such as the short-est path problem that have highly specialized algorithms but more speci®cmodels are connected using either loose or close coupling which is theapproach developed in this paper.

138 R. G. Cromley, D. M. Hanink

Integrating land use modelling with GIS

Chuvieco (1993) has presented a linear programming (LP) model of land usethat can be integrated with a GIS. In his land-use model, decision variablesare established to enable certain existing types of land use to be transformedinto new land uses. However, his decision variables do not have a spatialrepresentation but refer instead to aggregate amounts of land to be changedfrom an existing land use. He notes that one could either introduce spatialentities as decision variables using zero-one programming or the results couldbe mapped out of the LP environment using spatial suitability based on se-lected auxiliary variables (Chuvieco 1993, p. 80).

Chuvieco opts for the latter approach because the former would increasethe number of decision variables and constraints. However, his suitabilityanalysis assumes that the land use changes are not in con¯ict and would nat-urally occur in di¨erent locations. Although Chuvieco uses a two-step processfor determining the locational consequences of land use change, a one stepprocess can be derived if one initially based the decision variable on a geo-graphic entity such as a pixel for which suitability scores have already beencalculated.

In raster systems, though, the type of land-use model speci®cation is also afunction of the size of the raster dataset that may contain thousands or hun-dreds of thousands of pixels. For this reason, Eastman et al. (1995) have de-veloped a decision heuristic for solving multiobjective, con¯icting land allo-cation problems within the context of a raster GIS. They note that onealternative would be to utilize a priority ranking method in which the highestranked land use would be allocated ®rst and then lower ranked land useswould be allocated in order. However, this approach would place lowerranked land uses in areas that may not be very suitable for them.

In the Eastman et al. decision heuristic, the suitability scores are ranked foreach objective. Next, the raster cells having the highest scores for each objec-tive are temporarily assigned to that activity. If any cell is assigned to morethan one activity, then the con¯ict is resolved by assigning the cell to thatobjective for which it is closer to the objective's ideal point. The ideal point isthe highest score possible after standardization of the suitability scores. Be-cause not enough land will be devoted to any activity after the reallocations,the cells having the next highest scores for each objective are added until theactivity level constraint is satis®ed. The process of resolving con¯icts andadding more pixels continues until the desired activity levels are reached foreach objective. This decision heuristic has been implemented in the Multi-Objective Land Allocation (MOLA) module of IDRISI (Eastman, 1995).

Such a decision heuristic is most e½cient when the cell scores are inverselycorrelated between objectives (Fig. 1a). In this case the number of con¯icts isminimal because each cell that is highly suitable with respect to one objectivewill likely be not well suited with respect to the other objective(s) and fewcon¯icts will arise. Also, if the scores are uncorrelated between objectives, thenumber of cells in the con¯ict area is likely to be low relative to the totalnumber of cells (Fig. 1b). However, if the cell scores are directly correlated,then a high number of potential con¯icts will arise (Fig. 1c). This situationincreases the likelihood that the heuristic will not result in a near optimalsolution because more con¯icts must be processed.

In fact, if the scores are perfectly correlated, it is possible that the worst

Models and raster GIS 139

overall solution could be found because the method only considers the directcost of assigning a con¯ict cell to the highest score and not the full opportu-nity cost associated with assigning the losing activity to a di¨erent cell havinga much poorer score. Heuristics are appropriate only if they produce near

Fig. 1a±c. Alternative correlations between objectives. (a) negatively correlated, (b) uncorrelated,(c) positively correlated

140 R. G. Cromley, D. M. Hanink

optimal results and if an exact method cannot be developed to work withinthe existing limitation of computing technology.

Consider the following example. Assume that two con¯icting objectivesexist ± Objective 1 is to locate 20 cells to land use A and Objective 2 is tolocate 30 cells to land use B. A total of 50 cells are available for the location ofthese two land uses. Assume that the suitability scores for 50 cells with respectto Objective 1 decline in a linear progression from a high value of 240 to a lowof 142 (Fig. 2). The suitability scores for the same 50 cells with respect toObjective 2 decline from a high of 230 to a low of 83 (Fig. 2). The suitabilityscores for the two objectives are perfectly correlated in this example. The idealpoint for each objective is a score of 255. Finally, assume that the weightassociated with each objective is the same and equal to one. The problem isto allocate land so that the overall suitability scores are maximized for allobjectives.

The MOLA routine in IDRISI was used to perform this allocation usingthe suitability scores presented in Fig. 2. The resulting allocation had LandUse A located in the 20 cells that were associated with its highest scores andLand Use B located in the 30 cells associated with its lowest scores (Fig. 3a).This allocation results from the fact that for every con¯ict, Land Use A'sscore is closer to its ideal point than Land Use B's score is to its respectiveideal point. However, this land allocation minimizes rather than maximizesthe overall suitability scores. The optimal allocation would locate Land Use Bin the 30 cells associated with its highest scores and Land Use A in the 20 cellsassociated with its lowest scores (Fig. 3b). The cost associated with allowing Bto assume its best locations is less than the gain made by having Land Use Aassume its worst locations. The opportunity cost of this switch is positive. Inan allocation process based on suitability scores, if one land use can gain moreby its location than another can from its location, the overall allocation willbe more e½cient.

Land allocation as a generalized assignment problem

In an attempt to model land-use allocation in the absence of complete marketvalues, Hanink and Cromley (1998) have developed a generalized assignment

Fig. 2. Hypothetical suitability scores for Objectives 1 and 2

Models and raster GIS 141

problem for allocating land-uses based on suitability scores (see Aldrich 1981;Cromley 1994; Dobson 1979; Eastman et al. 1995; Hepner 1984; Hopkins1977; Pereira and Duckstein 1993; and Van Driel 1980 for a discussion ofdi¨erent approaches to suitability analysis). From a cost perspective, the landallocation problem is expressed mathematically as:

MinimizeXN

i�1

XMj�1

S 0ijXij �1�

subject to:XMj�1

Xij U 1; for all i; �2�

XN

i�1Xij VDj; for all j; �3�

Xij � 0 or 1; for all i and j; �4�

where, N is the number of land pixels; M is the number of land use activities;Xij is the decision variable assigning the ith land parcel to the jth activity; S 0ij isa transformed weighted composite suitability score de®ned below; and, Dj is

Fig. 3a,b. Alternative allocations. (a) The MOLA allocation, (b) the optimal allocation

142 R. G. Cromley, D. M. Hanink

the demand level for the jth land use. To ensure that the total demand equalsthe total amount of land, an extra land use (which is interpreted as fallowland) must be added if total demand is less than available land. Extra landarea (which is interpreted as the amount of total land use demand that cannotbe satis®ed) is added if the total amount of demand is greater than total landavailable.

S 0ij is de®ned as the cost value (Carver 1991):

S 0ij �Wj

maxi

Sij ÿ Sij

maxi

Sij ÿmini

Sij: �5�

S 0ij represents the standard distance that the suitability value of the ith landunit with respect to the jth land use, Sij , is from its ideal suitability value,which in this case is the maximum value of Sij for the jth land use. The allo-cation process assigns activities to land to minimize the di¨erence between theactual assignment and the ideal assignment. The trade-o¨ weights, Wj , can beeither given a priori or estimated using Saaty's pairwise comparison method(see Periera and Duckstein, 1993). The weights are included to stress the im-portance of one land use over another and usually sum to one (Eastman et al.1995).

The dual program to this model is:

MaximizeXMj�1

DjVj ÿXN

i�1Ui �6�

subject to: Vj ÿUi US 0ij ; for all �i; j�; �7�

Vj;Ui V 0; for all i and j;

where, Vj and Ui are the shadow prices associated with the demand for eachland use and supply of land respectively. The standard interpretation of Ui isthat it represents the locational rent of that site (Stevens 1961); fallow land(the marginal land) will have a rent of zero.

Implementation

Di¨erent methods exist for solving the generalized assignment problem. Be-cause the generalized assignment problem is a linear program, it can be solvedusing the simplex algorithm but this requires more than a reasonable amountof storage and computational time for thousands of decision variables andconstraints that are expected in the above land-use model. Alternatively be-cause it is also a network ¯ow problem, it can be solved using primal/dualtechniques such as the out-of-kilter algorithm (Fulkerson 1961). However,since it is also a transportation problem, primal algorithms that are extensionsof the stepping-stone algorithm (Charnes and Cooper 1954) have been shownto be more e½cient than either the simplex or primal/dual algorithms (Gloveret al. 1974). A number of other approaches for solving the transportationproblem have been developed by Harris. One is a dual method for solving the

Models and raster GIS 143

transportation problem that can signi®cantly reduce the number of columnsbeing processed, but it is only an approximate method as the number of rowsapproaches the number of columns (Harris 1979). Harris (1986) has also in-vestigated using the theoretical relationship between doubly constrainedmaximum entropy models and the transportation problem for creating a so-lution method that can be adapted to parallel processing.

The sixteen pixel area presented in Fig. 4 will be used to illustrate how amodi®ed stepping-stone algorithm can be adopted to the land-use problem.Each pixel is identi®ed by its row and column position; thus, the pixel in row 1and column 1 is identi®ed as pixel R1C1. In this example, each pixel is to beassigned to one of three land-use categories ± LU1, LU2, and LU3; there areto be three, four, and three pixels to be assigned to LU1, LU2, and LU3 re-spectively. LU4 is a category representing ``fallow'' land pixels that are notallocated to any land-use.

Any basic feasible solution to the generalized assignment problem can berepresented in a tableau format in which each cell denotes the assignmentof the jth land use to the ith pixel. With respect to land use allocation, thereexists one tableau row for each land pixel (supply) and a tableau columnfor each type of land use (demand). In this example, an extra column is addedfor LU4 because available land is greater than the total demand from allland uses. Although, there are N �M decision variables, only a total of�N �M ÿ 1� decision variables will be basic and the remainder will be non-basic at any one time. Figure 5 presents an initial basic feasible solution to aland use problem for the 16 land pixels and three land uses based on thenorthwest corner rule (see Hadley (1962) for a discussion of this rule). Zeroassignments are necessary in assignment and generalized assignment problemsto ensure a basic solution. While zero assignments are numerous in a strictassignment problem, the number of zero assignments is limited to �M ÿ 1�where M is the total number of land uses (including fallow land).

An alternative strategy for storing and representing a basic feasible solu-tion to the generalized assignment problem is to use a rooted arborescence(Glover et al. 1972). Each node of the arborescence tree corresponds to eithera row or column of the tableau (see Fig. 6). At the top of the arborescence is azero node (the root) and either a row or column node is attached to it; at eachlevel of the tree only row nodes or column nodes are possible and are alter-

Fig. 4. An example area represented by sixteen pixels.

144 R. G. Cromley, D. M. Hanink

nated between levels. The connections between levels, then, correspond tobasic feasible allocations between rows and columns. In Fig. 6, LU1 is con-nected to the root node and nodes for R1C1, R2C1, and R3C1 are connectedto it at the next level down.

This information can be stored using `predecessor', `successor', and`brother' index lists for each node in the tree (Glover et al. 1972). For a giventree node, its predecessor node is the node directly connected to it in the nexthigher level. Its successor node is the leftmost node directly connected to it inthe next lower level; and its brother node is the node immediately to the rightat the same level. Thus, the predecessor node for LU1 is the root node; itssuccessor node is R1C1, and its brother node is 0. The index lists for the initialbasic feasible solution are given in Fig. 7.

Fig. 5. Representing a basic feasible solution to the generalized assignment problem in a tableauformat.

Models and raster GIS 145

Given an initial basic solution, the next step in primal methods involves®nding successive basic feasible solutions that converge to optimality. Duringeach iteration, one non-basic cell replaces one basic cell such that total cost isreduced. Which non-basic element enters is determined by calculating the op-portunity costs associated with the current solution. The advantage of therooted arborescence representation is the ease with which the stepping-stonepaths for making the switch are calculated (see Glover and Klingman (1970)for a discussion of this technique). Harris (1978) also has noted that forproblems with many more columns than rows there are many demand col-umns, referred to as ``twigs''; by separating ``essential'' columns having rowsuccessors from the twigs, additional computational and storage savings canbe realized.

For the land-use model, the predecessor index list for each pixel node atoptimality contains the index of the land use to which that pixel is assigned.For a few allocations, however, this allocation may be at a zero level; in thesecases, the predecessor index for some land-use node will contain the corre-

Fig. 6. Representing a basic feasible solution to the generalized assignment problem as a rootedarborescence.

146 R. G. Cromley, D. M. Hanink

sponding pixel index and this allocation will have a value of one. For thesepixels, the predecessor index of the pixel is switched to that correspondingland-use node.

A comparative example

The above model was operationalized with the IDRISI software package.IDRISI was chosen because it is organized around a loose coupling frame-work; each IDRISI procedure operates on an image layer or pair of imagelayers and the results are written to an output layer. Each input and outputlayer is an independent data ®le that is stored outside the program in aworking directory in a common format. The pixels for each row and columnare stored column by column in a single array. The predecessor index list forpixel nodes, described in the previous section, is in the same format as anIDRISI image ®le.

A sample area containing 175� 175 pixels was selected to test the resultsof the MOLA heuristic against the results of the generalized assignmentproblem. Of the 30,625 grid cells, 13,102 cells are to allocated to Land Use A,8840 cells to Land Use B, and 8525 cells to Land Use C. An initial investiga-tion found a high degree of spatial con¯ict among the three land uses based onthe location of cells best suited for each land use; only 14,750 of the 30,467cells needed were not in spatial con¯ict (Table 1). Figure 8a shows the landuse pattern associated with the MOLA heuristic; Land Use B retained thelargest number of cells best suited for it given that it had the steepest rise

Fig. 7. Storing the rooted arborescence as predecessor, successor, and brother index lists.

Models and raster GIS 147

suitability scores from those best suited for it. Figure 8b shows the patterngenerated by the generalized assignment problem. A comparison of the twopatterns reveals a fairly similar pattern of land use. The cost value for the as-

Table 1. Number of grid cells best suited for each land use

Land use category Number of cells

Land use A only 7902Land use B only 3607Land use C only 3241Land uses A and B 2128Land uses A and C 2179Land uses B and C 2212Land uses A, B, and C 893

Fig. 8a,b. Alternative land-use solutions. (a) The MOLA allocation (b) the generalized assignmentsolution

Fallow Land

Land Use A

Land Use B

Land Use C

Fallow Land

Land Use A

Land Use B

Land Use C

148 R. G. Cromley, D. M. Hanink

signment land-use pattern is 3.4% less than the MOLA pattern and 9.3% ofthe cells changed categories between the two allocations (see Table 2). It is acommon occurrence with programming problems that two solutions will havemore similar values in the objective space than in their geographical space.

An added advantage of the generalized assignment problem is the abilityto also map the patterns of locational rent and opportunity costs. Figure 9presents the associated arrangement of land rents (although rent is expressedhere in units of suitability). The darker areas are less valuable land while thelighter shades are associated with more valuable land.

The overall level of subjectivity that underlies suitability analysis meansthat while an ``optimal'' solution can be computed, it may be susceptible tomeasurement error or transformation weighting. Sensitivity analysis is neces-sary to ascertain how robust any solution is to small changes in any cost val-ues. At optimality, the basic variables will have a zero opportunity cost (op-portunity costs are measured as the quantity Vj ÿUi ÿ S 0ij); the non-basicvariables will have negative opportunity costs. The magnitude of these op-portunity costs indicate how much the original cost value of each variable

Table 2. Land use di¨erences between the two allocations

Generalized assignment problem

Land use A Land use B Land use C Land use DLand use A 11965 470 535 132Land use B 508 8116 216 0

MOLA Land use C 630 237 7658 0Land use D 0 16 117 25

Fig. 9. Distribution of land rent

Models and raster GIS 149

would have to change before the optimal solution would change; pixels withsmall magnitudes could have been allocated to that land use with only a smalladjustment in the relevant suitability scores.

Figure 10 presents the opportunity costs associated with the three respec-

a

b

Fig. 10a±b. Opportunity cost patterns. (a) Land Use A, (b) Land Use B, (c) Land Use C,(d) Fallow Land

150 R. G. Cromley, D. M. Hanink

tive land uses and the fallow land. Since the results from any GIS-basedanalysis is only one input into any real land-use decision, it is also useful toinclude these opportunity cost maps as auxiliary information in case the de-cisionmaker(s) desire to investigate alternative solutions.

c

d

Fig. 10c±d.

Models and raster GIS 151

Conclusions

One major problem in developing a solution method that can be used as adecision module in association with a raster GIS is that the method must beable to handle thousands of decision units. For land use models, the methodsbased on mathematical programming are generally solved outside the GISand the results may or may not be coupled with a GIS. The methods that havebeen fully incorporated within a raster GIS are based on decision heuristics.Eastman et al. (1995) have developed the MOLA procedure for the IDRISIsystem and a practical test here shows that it generates reasonable solutionseven when the land-use objectives have a high level of spatial competition.

However, the specialized nature of the generalized assignment model andthe data structures it employs permits the coupling of exact programmingmethods for solving large scale problems not only e½ciently but with theadded bene®t of post-optimality analysis in terms of both land rent dis-tributions and patterns of opportunity costs. This added information can beused to assist decisionmakers in evaluating the robustness of any initial planand to identify areas were modi®cations could be made without signi®cantimpacts.

References

Aldrich F (1981) Spatial compositing and the land use allocation problem. Proceedings, AppliedGeography Conference, pp 69±79

Batty M, Xie Y (1994) Modelling inside GIS: Part 1. Model structures, exploratory spatial dataanalysis and aggregation. International Journal of Geographical Information Systems 8:291±307

Campbell J, Radke J, Gless J, Wirtshafter R (1992) An application of linear programming andgeographic information systems. Environment and Planning A 24:535±549

Carver S (1991) Integrating multi-criteria evaluation with geographical information systems. In-ternational Journal of Geographical Information Systems 5:321±339

Charnes A, Cooper W (1954) The stepping-stone method of explaining linear programming cal-culations in transportation problems. Management Science 1:49±69

Chuvieco E (1993) Integration of linear programming and GIS for land-use modelling. Inter-national Journal of Geographical Information Systems 7:71±83

Cromley R (1994) A comparison of suitability mapping and multiobjective programming techni-ques for performing spatial searches. The Pennsylvania Geographer 32:37±46

Dobson J (1979) A regional screening procedure for land use suitability analysis. GeographicalReview 69:224±234

Eastman JR (1995) IDRISI for Windows, Version 1.0. Clark University Graduate School of Ge-ography, Worcester, Massachusetts

Eastman JR, Jin W, Kyem P, Toledano J (1995) Raster procedures for multiobjective land-useplanning. Photogrammetric Engineering and Remote Sensing 61:539±547

Elmes G, Harris T (1986) Hierarchical data structures and planning optimization: An applicationto the Sussex land-use inventory. Proceedings, Second International Symposium on SpatialData Handling, pp 289±305

Fotheringham AS, Rogerson P (1993) GIS and spatial analytical problems. International Journalof Geographical Information Systems 7:3±19

Fulkerson D (1961) An out-of-kilter method for minimal-cost ¯ow problems. SIAM Journal ofApplied Mathematics 9:18±27

Glover F, Klingman D (1970) Locating stepping-stone paths in distribution problems via thepredecessor index method. Transportation Science 4:220±225

Glover F, Karney D, Klingman D (1972) The augmented predecessor index method for locatingstepping-stone paths and assigning dual prices in distribution problems. Transportation Sci-ence 6:171±179

152 R. G. Cromley, D. M. Hanink

Glover F, Karney D, Klingman D, Napier A (1974) Implementation and computational com-parisons of primal, dual and primal-dual computer codes for minimum cost network ¯owproblems. Management Science 20:793±813

Goodchild M (1987) A spatial analytical perspective on geographical information systems. Inter-national Journal of Geographical Information Systems 1:327±334

Goodchild M et al. (1992) Integrating GIS and spatial data analysis: Problems and possibilities.International Journal of Geographical Information Systems 6:407±423

Hadley G (1962) Linear programming. Addison-Wesley, Reading, MAHanink D, Cromley R (1998) Land-use allocation in the absence of complete market values.

Journal of Regional Science 38:465±480Harris B (1978) A new algorithm for tree modi®cation in the primal transportation problem.

Transportation Science 12:271±276Harris B (1979) A rapid dual method for the Hitchcock problem. Environment and Planning A

11:1049±1054Harris B (1986) A continuous dual method for the Hitchcock problem. Transportation Research

Board 20B:71±75Hepner G (1984) Use of value functions as a possible suitability scaling procedure in automated

composite mapping. The Professional Geographer 36:468±472Herbert J, Stevens B (1960) A model for the distribution of residential activity in urban areas.

Journal of Regional Science 2:21±36Hopkins L (1977) Methods for generating land suitability maps: A comparative evaluation.

Journal of the American Institute of Planners 43:386±400Hopkins L, Los M (1979) Location-allocation algorithms for land use plan design with ®xed and

substitutable interactions. Journal of Regional Science 19:345±362Moxey A, White B, O'Callaghan J (1995) The economic component of NELUP. Journal of En-

vironmental Planning and Management 38:21±33Pereira J, Duckstein L (1993) A multiple criteria decision-making approach to GIS-based land

suitability evaluation. International Journal of Geographical Information Systems 7:407±424Stevens B (1961) Linear programming and location rent. Journal of Regional Science 3:15±26Van Driel N (1980) Computer-composite mapping for geologists. Environmental Geology 3:151±

157Worboys M (1994) Object-oriented approaches to geo-referenced information. International

Journal of Geographical Information Systems 8:385±399Wright J, ReVelle C, Cohon J (1983) A multiobjective integer programming model for the land

acquisition problem. Regional Science and Urban Economics 13:31±53

Models and raster GIS 153