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Analysis of Land Use Change: Theoretical and Modeling Approaches
Helen Briassoulis, Ph.D.
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The application of mathematical programming and optimization techniques to urban and regional analysis, spurred by the post-1950s developments in solution techniques and computer technology, has an impressive record and continues to attract significant research contributions as well as to offer significant decision support in various circumstances, notably in planning. As their name denotes, optimization models are exclusively oriented towards producing solutions which optimize certain objectives defined by (interested) users/decision makers. In other words, they are fit to provide support in decision situations where the question is to choose a solution to a decision problem which satisfies one or more objectives and takes into account various constraints. Hence, they are prescriptive models although they are used also as evaluation tools. They have found important applications in the analysis of land use – especially land use planning applications – and, recently, they appear to be useful tools in the search for land use solutions which contribute to sustainable development and use of environmental and human resources. Examples of their use by public and private bodies cover the whole range from the scale of urban regions up to the global scale.
The principal criterion for classifying these models appears to be the particular mathematical programming/optimization technique they employ and this is used in the following presentation. An exception is the group of utility maximization models which are treated as a separate group as they are expressly based on and derive directly from economic theory. Models in this latter group can classified also under the particular programming techniques they utilize. The following principal categories of optimization models are presented below (Table 4.1b):
Several of these models are used in the context of larger integrated models also as it will be discussed in the respective section below. As it was the case with the previous model groups, some of the of optimization models are direct land use (and change) models while in some others land use is treated in a more indirect fashion.
4.5.1. Linear programming models
Linear programming (LP) is one of the most widely used techniques in model building since the mid-1950s as it is more manageable, understandable and computationally easier than other optimization techniques. Its use in the analysis of land use is marked perhaps by the widely known Herbert-Stevens Linear Programming Model designed for the Penn-Jersey Transportation Study (Herbert and Stevens 1960). Other similar models were built in the same period such as the Southern Wisconsin Regional Plan Model (Schlager 1965) and Britton Harris’ Optimizing Model – a modification of the Herbert-Stevens model (Harris 1962, 1966 cited in Romanos 1976, 63). More applications ensued in the following decades up to the present (Bammi et al. 1976, Bammi and Bammi 1979, Campbell, et al. 1992; Stoorvogel et al. 1995; Jansen and Schipper 1995; Latesteijn 1995). In the following, the basic structure of a LP model is presented first and, then, details of its applications drawing on the published literature are offered.
There are two main groups of LP models, the single and the multiple objective (or, multiobjective). The first deal with problems in which there is one objective to optimize and the second address the more realistic situation of finding solutions which satisfy more than one objective. In both cases, the structure of the optimization problem includes one or more (in the case of multiple objectives) objective functions and a set of constraints. The objective function(s) for land use problems expresses in mathematical form the question: "how much land to allocate to each of a number of land use types in order to optimize objective A (or, B, C, D)." The objectives may be, for example, maximization of (household or individual) rent-paying ability, minimization of environmental impacts, maximization of population income, minimization of the cost of development (or maximization of the benefits of development), etc. The constraints which can be taken into account depend on the case but representative objectives include: lower and upper limits on land use (reflecting, for example, zoning or natural constraints such as land suitability), other constraints on development, availability of labour, and so on. Examples of particular LP models are given below.
4.5.1A The Herbert-Stevens Linear Programming Model
The Herbert-Stevens Linear Programming Model (Herbert and Stevens 1960) was built with the purpose to obtain optimal distributions of households to available residential land in the context of the larger, comprehensive model of metropolitan structure designed to locate land-using activities for the Penn-Jersey Transportation Study (Wilson 1974, Romanos 1976). It assumes a study region subdivided into zones and it operates iteratively. It receives exogenous forecasts of the amount of available residential land as well as of the number of households to be located in the study region within each iteration period. Basic assumptions upon which the model rests include the following:
The objective to be maximized is the aggregate rent-paying ability which corresponds to maximization of savings of each household. In mathematical form:
subject to:
(4.23)
where,
| U | total number of zones of the study region, K=1……..U |
| n | household groups, i = 1,…….. …n |
| m | residential bundles, i = 1,………..m |
| bih | residential budget allocated by a household in group I to purchase a residential bundle h |
| cihk | annual cost to a household of group i of the residential bundle h in area K – exclusive of site cost |
| sih | number of acres in the site used by a household of group i if it uses residential bundle h |
| Lk | L number of acres available for residential use in area K in a particular iteration |
| Ni | number of households of group i that are to be located in the zone during a particular iteration |
| Xihk | number of households of group I using residential bundle h located by the model in area K |
Its designers as well as its commentators contend that it is a simple, operationalized expression of Alonso’s urban land market theory (Wilson 1974, Romanos 1976). The term (bih - cihk) "represents the bidding power for site rent, which is clearly equivalent to Alonso’s bid-price, and so the model maximizes bid-prices subject to constraints in land availability and finding everyone a house. An analysis of the dual shows that, if bid-rent is maximized, actual rent paid is minimized" (Wilson 1974, 198). (Note: Alonso’s model is described in this section under the utility maximization models group).
The Herbert-Stevens model has many desirable features such as: (a) the fine level of aggregation with respect to households which allows the inclusion of households of different behavioral characteristics and permits a more realistic land allocation process, (b) the simulation of the market clearing mechanism through a simple linear programming action, (c) the inclusion of policy constraints on the amount of available land, (d) an operational form which makes its real world application possible. However, it has several disadvantages as well. The linear programming formulation imposes the linearity assumption on both the objective function and the constraints which may not be always the case in the real world (a general problem with all LP models). The (good quality) data requirements of the model are heavy. The iterative nature of the model ensures that the allocation of households within the given iteration period will be optimal but it does not ensure that the allocation will be optimal in the aggregate. Overall, these and other constraints such as available time and resources, and the capabilities of available computers and computing methods at the time it was designed, prevented the model from becoming operational (Wilson 1974, Romanos 1976, Kain 1986).
Britton Harris and his students attempted to modify the original Herbert-Stevens formulation to overcome certain of the problems it posed to easy operationalization. To assess the desired or actual allocations of households’ budgets to housing and nonhousing goods and services, Harris (1962, 1966 cited in Romanos 1976, 62-63) constructs a preference function following Alonso’ theory of the urban land market (see Chapter 3). He uses also the housing preference structure of the population to predict residential behavior and to evaluate the relative differences in utility among consumers under alternative arrangements of the housing stock. However, there is not theoretical justification of the housing preferences (which are assumed to be homogeneous); only the mathematical expression of the preference function. The linear programming part of the Harris’s model is similar to the Herbert-Stevens model presented above (Romanos 1976).
The Herbert-Stevens and the Harris’ optimizing model can be considered land use change models in the sense that, although they do not assess changes in the uses of land directly, they allocate households to available residential land on the basis of particular behavioral assumptions. The prescribed optimal allocations provided by the models (within each iteration period) can be compared with actual allocations at the base period to obtain forecasts of land use changes of a qualitative nature; i.e. differences in the residential uses of land occupied by household of different socio-economic profiles. Apparently, such assessments are more meaningful in the context of broader models of land use and its change. As regards the Herbert-Stevens model, this was part of the Penn-Jersey Transportation Study model which is presented in the section on integrated models.
4.5.1B The Southern Wisconsin Regional Plan Model
Another LP model, the Southern Wisconsin Regional Plan Model (Schlager 1965) has an objective function which minimizes the total cost of developing urban land in a given zone of the study area under land availability constraints. More specifically, the model is as follows:
(4.24)
(4.25)
(4.26)
(4.27)
where,
| Xi | units of a given land use type in the zone |
| ci | cost of developing a unit of land of a given type in the zone |
| Ek | total land use demand requirement for land use k |
| di | service ratio coefficients which provide for supporting service land requirements necessary for primary land use developments such as streets |
| Fm | upper limit on land use of a particular type in zone m |
| G | ratio of land use type n allowed relative to land use type m with the land use types m and n in the same or in different zones |
According to its designer, this model is a design aid and makes no claims of modeling personal preferences of families or other aspects of human economic and social behavior (Schlager 1965 cited in Romanos 1976, 61). The model emphasizes mostly the determining effect of development costs (which are affected, among others, by environmental factors such as soil conditions, etc.) on the land use distribution in an area. Romanos (1976) observes that this model, while inheriting all the problems associated with LP models, does not represent any improvement over the Herbert-Stevens models; on the contrary, it lacks the detail and ingenuity of the latter model as well as the economic nature assumptions about individual behavior.
4.5.1C The Du Page County Regional Planning Commission’s Model
Another linear programming model suggested in Bammi et al. (1976) has the basic structure of the prototype LP model and, in addition, incorporates environmental considerations in both the objective functions and the set of constraints, an important omission of past models (see, also, Bammi and Bammi 1979). Bammi and Bami (1975 cited in Bammi et al. 1976) developed a multiple objective linear programming model for the Du Page County Regional Planning Commission to provide optimal allocations of land use types by simultaneously considering several objective functions and constraints. This land use model has been used in developing the County’s Comprehensive Plan for 1990.
The model has seven objective functions each corresponding to one of the following objectives set by the county’s planners:
The environmental cost of land development was calculated for each of the 147 analysis regions in the county on the basis of their particular topographic, geologic, natural resources, wildlife and floodplain characteristics (Bammi et al. 1976).
The constraints taken into account in the multiple objective optimization solution procedure included:
The efficiency of an objective function was defined as its minimum value divided by its current value for a particular solution; this reflects the requirement that all objective are equally satisfied – i.e. they have the same efficiency. The constraints on upper and lower limits of each land use type are obtained by forecasting population and employment, by choosing a particular mix of residential dwelling units for new housing, and by setting open space standards. The undeveloped acreage in each region is measured by totaling all vacant parcels of land within the region. Local commercial zones relate total (new plus existing) commercial land to total population according to a desired standard of commercial acreage per thousand people. Institutional zones relate total institutional land to total population in the region. Local open space zones and the open space constraints are detailed in Bammi et al. (1976). A mathematical formulation of the optimization model and a list of the environmental conditions which were taken into account are included in the same work. It is noted, finally, that the above model has been used by Brill et al. (1982) in the context of a broader effort to develop a method (the HSJ – Hop, Skip, Jump) for discriminating among the solutions produced by mathematical programming models which, although they perform equivalently in terms of the modeled objectives, they are associated with drastically different land use patterns.
4.5.1D Multiple Objective Linear Programming (MOLP) Models
Multiple objective linear programming models (MOLP) address the question of land use solutions which meet more than one objective. Of particular importance in this context are environmental objectives and constraints. The role of environmental factors in determining the optimal allocation of land uses in a region has always been of high importance in the context of planning in agricultural regions. In addition, the need for detailed information on spatial data as well as for the spatial representation of the optimal land configurations always figured high on the researchers wish lists. Progress on and diffusion of GIS techniques and technology since the 1980s mostly has made possible the use of information of better spatial detail and specificity. Linear programming models for agricultural regions appeared which are sensitive to the distribution of environmental conditions in the study areas and which are linked to GIS to provide for mappings of the optimal solutions produced by the models. Representative applications are found in Campbell, et al. (1992), Stoorvogel et al. (1995), Jansen and Schipper (1995), and Stoorvogel (1995). In Campbell, et al. (1992), the purpose is to match the planned or anticipated demand for agricultural products with the ability of the agricultural sector (which includes its natural resources endowment and land suitability) of the study area to meet the demands. The objective function of the (multiple objective) LP model seeks to minimize the cost of meeting these demands and includes two components: (a) the cost of local production and (b) the cost of imports to complement local production to meet local demand. The assumption is that the economic costs of production determine whether local demand will be met by local production or by imports subject, among others, to the natural resources constraints facing the study region. A summary mathematical formulation of the LP problem, is shown below following Campbell, et al. (1992):
minimize CX + MY (4.28)
subject to:
AX 
B,
(4.29)
KX + TY
D,
(4.30)
X, Y 
0
(4.31)
where,
| C | a 1Xn vector of variable costs associated with local production on 1 acre of land |
| X | a nX1 vector of the number of acres used by each of the local production activities |
| n | the number of different production technologies for various crops – the technologies depend on farm size, geographic region, soil and environmental conditions |
| M | a 1Xm vector of unit costs on the m import possibilities |
| Y | a mX1 vector of the number of units of goods imported to complement local production to meet local demand |
| A | the rXn matrix of input coefficients required to produce 1 acre of local output for the n production activities |
| K | the mXn matrix of outputs per acre produced by the n local production activities and by using the inputs in A |
| T | an mXm identity matrix which allows imports of goods to be added to local production |
| B | a rX1 vector of resources available for local production and |
| D | an mX1 vector specifying the national demand for agricultural outputs |
C, M, A, K, T, B and D are all fixed. The object of the LP model is to choose a set from X and Y so that the national demands, D, are satisfied, the local resources, B, are not exceeded and the total cost is minimized. The inputs to the LP problem are obtained from a GIS data base created from map and statistical information using a GIS software. The results of the LP problem, the optimal crop allocations to the regions of the study area, were mapped using the GIS following a rule-based procedure developed for this purpose and using expert knowledge (Campbell et al. 1992).
Stoorvogel et al. (1995) followed a similar LP modeling procedure with the exception of the specification of the objective function and the set of constraints. The LP model employed is part of a broader methodology developed for the quantitative analysis of land use scenarios and which is operationalized by means of a specially developed software called MODUS. MODUS transforms databases from one of the models or tools to the specific requirements of the others. The study region is subdivided into a number of farms and the objective function of the LP model maximizes farm income in the study area. The constraints of the model describe the availability of resources (e.g. land and labor) and restrictions on sustainability parameters. The latter include the soil nutrient balance and a biocide index.
The study employed a detailed and elaborate methodology for distinguishing LUSTs – Land Use Types of a Specified Technology – which provides a detailed picture of the combinations of land using units, land use types and quantitative descriptions of the technology and corresponding inputs and outputs (Jansen and Schipper (1995). The idea of LUSTs borrows from FAO’s methodologies for land evaluation (see, for example, FAO 1976, 1978, 1995). A spatial database was set up using data from farm surveys, field surveys, literature surveys, field experiments, expert knowledge, and maps of the area. The results of the model runs – optimal crop allocations among the farms of the study area – were mapped with the use of the MODUS software. Stoorvogel (1995) provides details on the development of a GIS-models interface which makes possible the translation of the results of external model calculations into a GIS and, hence, the visualization of their spatial distribution.
Another application of linear programming is in producing future land use scenarios and exploring their implications. An example is offered by Latesteijn (1995) who presents a multiple objective LP model which differs from the previous in that has been applied at the level of a group of nations; namely, the European Union (EU). The presence of more than one objectives necessitates the application of special optimization procedures to optimize them, either simultaneously or iteratively. In the case discussed, the latter approach was adopted, called "Interactive Multiple Goal Programming" (IMGP), which allows the optimization of a set of goals interactively. In this way, it is possible to study trade-offs among goals. The core of the procedure consists of an LP model called GOAL (General Optimal Allocation of Land Use). The broad objective of the model’s application is to arrive at optimal combinations of agricultural land use types necessary to satisfy an exogenously determined future demand for agricultural and forestry products within the EU. The particular goals which were translated into respective objective functions were the following:
Differing political philosophies and attitudes of the decision makers can be introduced in the model interactively by setting limits to the achievement of particular goals while the others are optimized. In this way, the model can generate scenarios showing the effects of different policy priorities on land use allocation among the land use types considered in the modeling exercise. The model’s constraints include the satisfaction of exogenously defined future demand as well as particular socio-economic, land use and environmental constraints applying to particular regional and local situations in the EU.
The model uses a zonal system consisting of 22,000 Land Evaluation Units for the whole area of the EU (at the time of the model’s application). The suitability of each unit for various types of farming is assessed using the Automated Land Evaluation System (ALES) and it is accomplished with the use of GIS. Production potentials for suitable locations are also calculated by means of a simulation model which are then translated into cropping systems using the notion of Best Technical Means. Expert knowledge is used to arrive at cropping systems which are acceptable from both an economic and an agronomic point of view. This information is combined with alternative policy priorities, e.g. attaining highest possible yields, reducing the environmental impacts of agriculture, preserving agricultural land in the EU, to define feasible field systems.
Latesteijn (1995) makes clear that the model is not used to produce a forecast. The scenarios generated are meant to explore technical possibilities to attain a set of objectives. Policy instruments, such as price changes and assumptions about the behavior of the actors as well as institutional obstacles are not taken into account. The results of the model indicate the technical limitations associated with policy change. The options explored through the model can be used to determine to what extent current policy can cope with the major developments generated in the scenarios. These include land use changes and their implications such as increasing productivity and decrease in land-based agriculture. In other words the results can serve as guidelines for the development of future policies.
4.5.2. Dynamic programming models
Another class of optimization techniques which have found application in problems of land use analysis are offered by dynamic programming models (DP). "Dynamic programming is a mathematical programming technique often useful for making a sequence of interrelated decisions. It provides a systematic procedure for determining the combination of decisions that maximizes overall effectiveness… In contrast to linear programming, there does not exist a standard mathematical formulation of "the" dynamic programming problem. Rather, dynamic programming is a type of a general type of approach to problem solving, and the particular equations used must be developed to fit each individual situation" (Hillier and Lieberman 1980, 266). A simplified adaptation of this approach to the case of deciding on the optimal allocation of land uses in a study area is provided below following Hillier and Lieberman (1980).
A study area is subdivided into cells which represent the "stages" of a dynamic programming problem. For each cell, a number of candidate land use types is considered; these represent the "states" of the DP problem. Each of the land use types (states) has certain characteristics such as development costs, environmental impacts, etc. which determine the value of the "policy" (of the objective function) in the DP terminology. One of the states is chosen for each stage of the problem. The solution to this problem seeks to identify the optimum allocation of states to stages (i.e. land use types to cells of the study region) which optimizes an objective such as maximization of development benefits or minimization of development costs, etc. (subject to a number of applicable constraints).
The solution procedure starts with one cell of the study area and finds the optimal "policy" for this cell; i.e. the land use type which maximizes the value of the objective function. It then gradually adds cells, finding the current optimal solution from the previous one, until all cells of the study area are considered. The decision of which land use type to choose for each cell is associated with the choice of land use types in the remaining cells. In other words, given a current land use type in a cell, the optimal "policy" for the remaining cells are independent of "policy decisions" made for the previous cells. The solution procedure starts by finding the optimal "policy" for each land use type of the last cell; a solution which is usually trivial. Then, a recursive relationship is established that identifies the optimal "policy" for each land use type in cell n, given the optimal "policy" for each land use type for cell (n+1) is available. Therefore, finding the optimal "policy" when starting with land use type s at cell n requires finding the optimizing (maximizing or minimizing) value of the objective function for this cell. Using the recursive relationship, the solution procedure moves backwards cell by cell – each time finding the optimal "policy" for each land use type of that cell – until it finds the optimal "policy" when starting at the initial cell.
It is important to keep in mind that the application of DP is justified when the decision problem involves making a sequence of interrelated decisions. This is the case with determining the optimal allocation of land use types to the sub-basins of a watershed for the purposes of minimizing flood hazard (evidently on the downstream areas of the basin) while maximizing economic rent to land, a problem addressed by Hopkins et al. (1978). They applied a dynamic programming formulation which "yields the optimal allocation of uses to maximize economic rent to land net of flood damage, while specifically considering the impact of upstream development on downstream flood levels and the impact of downstream development on the amount of damage given flood levels" (Hopkins et al. 1978, 95). According to the authors, the DP can be described mathematically as:
(4.32)
subject to:
Xn+1 = tn (Xn, Dn) for n=1 …….N (4.33)
where,
| fN(XN) | the function yielding the highest aggregate bid price for each final outflow level (or sub-basin) |
| Dn | the set of possible land uses for sub-basin n |
| Xn | the set of possible peak inflows to n |
| rn | the return function for each sub-basin which is
expressed as: rn = vjn an - cjnk dnk (4.34) |
| where, | |
| vjn | bid price per acre of use j in sub-basin n |
| an | number of acres in sub-basin n |
| cjnk | present worth of flood damage per acre of use j in sub-basin n at depth k |
| dnk | acres flooded to average depth k in sub-basin n |
It seems that several land use allocation problems may have a similar structure to the one addressed by the above study, justifying, hence, the application of dynamic programming for finding optimal land use patterns which satisfy economic and environmental objectives.
4.5.3. Goal programming, hierarchical programming, linear and
quadratic assignment problem, nonlinear programming models
Three other programming techniques have been used to build land use optimization models which are discussed in this section but they are not as widespread as those presented before; namely, goal programming, hierarchical programming, linear assignment problem, and nonlinear programming.
Goal programming (GP) is a mathematical programming technique which addresses the issue of striving to satisfy more than one goals simultaneously. According to Hillier and Lieberman (1980) "The basic idea is to establish a numerical goal for each of the objectives, formulate an objective function for each objective, and then seek a solution that minimizes the (weighted) sum of deviations of these objective functions from their respective goals" (p. 172). A simplified presentation of the basic mathematical formulation of a GP problem is presented below following Hillier and Lieberman (1980).
Assume that k objectives are considered, expressed in terms of a number of decision variables (X1, X2,..Xn). For each objective, let cjk be the coefficients in its objective function and gk the goal for this objective function. The solution being sought is the one that comes as close as possible to attaining all of the following goals:
(goal
1) (4.35)
(goal
2) (4.36)
(goal
k) (4.37)
Because it is not possible to attain all goals simultaneously, it is necessary to make explicit the meaning of the "as close as possible". In the simplest case, under the assumption that deviations from goals are equally important for all goals, the composite function for the goal programming model takes the following form:
Minimize the sum of deviations from goals:
(4.38)
Depending on the details of the particular GP formulation, various solution techniques have been developed.
Goal programming models have been applied to private sector decision problems but their application to public sector decision situations (such as those involving issues of land use allocation) has been criticized as, for example, it is not easy and straightforward (nor politically expedient) to specify the values of the goals required for the GP formulation. Nevertheless, they have found applications in forest management, agricultural and recreational resource planning, and industrial and residential location problems (Lonergan and Prudham 1994). Lonergan and Prudham (1994) cite Dane et al.’s (1977) application of a goal programming model to "assist with planning decisions for the Mount Hood National Forest in Oregon. The model was able to provide information on the sensitivity of land allocations to combinations of planning goals, the goal constraints that had the greatest effect on model solutions, the sensitivity of allocations to goal priorities, and the trade-offs between goals" (Lonergan and Prudham 1994, 429). Lonergan and Prudham (1994) cite also an application of a similar model they have built for resource management purposes in Eastern Ontario, Canada. The model included 6 planning goals and a set of constraints which referred to: (a) technical and resource constraints, (b) economic efficiency, (c) regional income and employment generation, (d) energy efficiency, and (e) environmental quality. For our purpose, the important aspects of this application are the spatial resolution of the application – the model considered 27 townships in the region and included land use variables as well as land availability constraints.
Hierarchical optimization is a multidimensional (or, multiobjective) programming approach which is appropriate to problems in which the objective functions can be ranked in an ordinal way from, say, "important", to "next most important", etc. The solution procedure is based on sequential optimization of the objective functions according to the established rank order. The set of constraints at each stage of the optimization is co-determined by the optimal results obtained in previous stages (Nijkamp 1980). A formal presentation of the hierarchical programming model is given below following Nijkamp (1980).
Assume a set of objectives which are translated into a set of objective
functions: 
= {1, 2….n} expressed in terms of a set of decision variables X =
{X1, X2,….Xn}. In order to construct a
hierarchical programming model, the objective functions have to be rank
ordered:
1 É
2 É
……. É n (4.39)
where the symbol É denotes "preferred to". The goals are assumed to be conflicting, such that:
n
(Xn0) É n
(Xn’0) for every n
(4.40)
where,
Xn0 is the kX1 vector of optimal values of the
decision variables related to the maximum of the objective function
n.
If a trade-off between the objectives is possible, the purpose is to find compromise values of the decision variables X, X*, such that at least the following conditions are satisfied:
where n has to be maximized and where the
tolerance parameter 
n should be smaller than or equal to 1. The reverse holds if
n has
to be minimized.
The parameter n is associated with the maximum
tolerance deviation from the absolute optimum n (Xn0). Therefore,
n indicates the maximum proportion
of the original objective function n which be traded-off against other
objective functions. The n
coefficients are called trade-off coefficients. Depending on the
specification of the goal priorities and the trade-off coefficients, various
solution procedures are available.
Nijkamp (1980) offers an example of the application of hierarchical
programming to an industrial land use problem in a newly established industrial
area near Rotterdam. Seven candidate activities (related to seven different
land use types) were considered. For each activity the following were
specified: minimum and maximum land use requirements, employment coefficient
(employment per hectare of land occupied) and air pollution coefficient (total
emissions per year per hectare). The problem was to find the optimal mix of
uses in the area which satisfied two conflicting objectives: maximization of
regional employment (1) and
minimization of total air pollution (2). In this
case, two rank orders of the goals were possible; either
1É2 or
2
É
1. Hence, two different
hierarchical models could be solved. The solution of each model depends on the
values of the trade-off coefficients assumed and it is not unique, in general,
unless additional information is provided or the coefficients are specified a
priori. It is noted that in this example application of hierarchical
optimization the model was not spatially explicit.
Linear and quadratic assignment models is another group of programming models which are based on the prototype assignment problem – a special case of the transportation problem in operations research. The prototype problem (adapted to the case of land use) answers the question of how to match available land-using activities to available sites so as to optimize an objective such as minimization of development costs (total or net), maximization of benefits (total or net), etc. In addition to the intuitive appeal of this prototype problem, its suitability to modeling optimal land use allocations lies in that it is suited to analysing efficient allocations of indivisible resources – such as land use, plants, etc. (Koopmans and Beckman, 1957), a condition which is not met by most programming techniques which assume divisible resources (i.e. fractions of the quantities modeled are meaningful in practice). A number of related theoretical model formulations have followed the original contribution by Koopmans and Beckman (1957) which would purportedly assist planners in making land use decisions. However, no real world applications seem to have appeared; only illustrative examples are offered by the authors in the context of the theoretical model formulations. In the following, an elementary form of the linear assignment problem is presented and the most important aspects of the related theoretical model proposals are discussed.
Given a number of n sites into which the study area is subdivided and an equal number of candidate land use types as well as the "costs" associated with each land use type at each site (i.e. given the nXn matrix cost coefficients), the question is which assignment of land use types to the sites of the study area minimizes the development costs (or, maximizes the benefits depending on the available data and demands of the decision environment). The mathematical formulation of this problem is as follows:
(4.43)
(4.44)where,
| aij | the cost coefficients |
| Xij | a variable taking the value of 1 if activity i is assigned to site j and the value of 0 if it is not |
The problem can be solved either as a linear programming problem by using the SIMPLEX method or as a transportation problem or by means of the more efficient Hungarian algorithm (Hillier and Liebermann 1980, Spivey and Thrall 1970). It has been shown that this particular formulation always has integer solutions (Hillier and Liebermann 1980). The Lagrange multipliers associated with constraints (4.43) and (4.44) are denoted qj and qi respectively. Their optimal values indicate how quickly the optimal value of the objective function changes as the constraint associated with the multiplier is relaxed. In mathematical programming, constraints usually represent the availability of resources. In the context of land development, these resources are usually land and capital and Lagrange multipliers are interpreted as prices or rents. More specifically, following Moore (1991): "If the owners of land and capital make these inputs available to the plant operator offering the highest bid, then the optimal values of qj and qi identify equilibrium plant (land use type in the present contribution) and site rents, respectively… At any location other than the optimal site (or, an equivalent site), the combined land and capital rents determined by the market exceed the seminet revenues available, and the plant operator would experience a loss at a suboptimal site. Thus, the configuration that maximizes the system’s seminet revenues also implies that there is no incentive for any locator i at optimal site j to consider exchanging locations with anyone else and that all locators are in spatial equilibrium" (Moore 1991, 10). The direct relationship of the assignment model formulation to the urban land market theory (see Chapter 3) is described in Moore (1991): "Plant rents accrue to the owners of the mobile factor (capital) while site rents accrue to the owners of the immobile factor (land). Lind (1973) showed that the standard (Wingo, Alonso, Muth, E. Mills) economic model of urban land use is really a special case of the assignment model in which there are large numbers of bidders and competition of such intensity that all plant rents must be bid for sites. In the case of few bidders and discrete sites it is sufficient for owners of capital (or developers) to outbid a next-highest bidder for a given site and retain any remaining profitability" (Moore 1991, 10).
The linear assignment model does not take into account the influence on a given site of the uses in neighboring sites, an untenable assumption in real world situations. This assumption is relaxed (within the structure of the assignment problem) in quadratic assignment problems which include an "interaction" term and have received interesting interpretations in terms of economic theory as well as in terms of the simultaneous nature of land use and transportation decisions (see, Koopmans and Beckmann 1957, Gordon and Moore 1989). However, although more realistic, these models are not easily applicable given their high data requirements and the computational difficulties associated with their mathematical treatment (stemming from the modeled nonlinearities). Moore and Gordon (1990) offer another extension of the linear assignment model. They develop a model of a decentralized urban development process where land development is represented as a sequence of activity shifts resulting from locators’ efforts to maximize net revenues by mitigating congestion costs and other externalities. Their sequential programming model involves solving a series of linear assignment problems that track urban land use through time. Briassoulis (1995) uses the basic idea of the linear assignment problem to propose a compromise solution procedure for locating hazardous facilities in an area by taking into account development benefits and risk associated with these facilities.
Non-linear programming models are encountered less frequently in
the literature and even less frequently in actual applications given the
computational difficulties associated with their solution. Fischer et
al. (1996a) cite FASOM (Forest and Agriculture Sector Optimization Model)
which is a dynamic, multi-market, multi-period, nonlinear programming model for
the forest and agriculture sectors of the United States built by Adams et
al. (1994 cited in Fischer et al. 1996a, 6). The model considers 11
supply regions and a single demand region – the nation – and depicts
the allocation of land to competing activities in the forest and agriculture
sectors. Its purpose was to evaluate the welfare effects on producers and
consumers of alternative carbon sequestration policies. However, it pays
limited attention to land use and land cover change and to the processes of
resource degradation (Fischer et al. 1996a).
