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Analysis of Land Use Change: Theoretical and Modeling Approaches
Helen Briassoulis, Ph.D.

4.5.4. Utility Maximization Models

4.5.5. Multi-Objective/Multi-Criteria Decision Making Models

4.5.4. Utility maximization models

Utility maximization models applied to the analysis of land use and its change are discussed as a separate group not because they are based on a common mathematical solution technique but because they all share a common theoretical basis drawing from economic theory. Models in this group have been formulated mathematically usually as linear or nonlinear optimization models. In fact, the models included in this group can be classified under one of the previous model groups presented before. Moreover, utility maximization models constitute the basis for large-scale integrated models of land use change to be presented in a later section. In the following the basic concepts and approaches as well some well-known land use-related models are presented.

Welfare economics (and microeconomic theory, in general), distinguishes between producers and consumers of economic goods and services. Each group aims at maximizing some goal; producers are assumed to strive to maximize their profits from selling the goods and services they produce (the supply side of the economy) while consumers aim to maximize their utility from consuming various goods and services (the demand side of the economy). The characteristic feature of welfare economic theories of production (supply) and consumption (demand) and of the models derived from them is their emphasis on individual behavior. Utility theory of neoclassical economics is founded upon the principle of consumer sovereignty. The analysis of both producer and consumer behavior starts from the individual and then aggregates over all individuals present in the economic system to derive the aggregate behavior – the economy-wide (or, market) demand and supply conditions. The determining factor of the behavior of any individual is the price of goods and services, the signal to which both producers and consumers respond and adapt their behavior (production and consumption of goods) respectively.

In the analysis of a given economic system, two broad cases are distinguished: (a) partial equilibrium and (b) general equilibrium. In the first case, the analysis seeks to determine the market conditions which are (or, should be) met when either producers or consumers maximize their goals (profits, utility). In the general equilibrium case, the conditions which ensure equilibrium between market demand for goods (consumption) and market supply of goods (production) are sought. In both cases, the equilibrium can be either static or dynamic and competitive or non-competitive. The most frequent assumption made is that of competitive equilibrium.

The role of land and land use in the analysis of consumer and producer behavior has undergone changes over time. The French physiocrats considered the production capacity of land as the main source of welfare (Gould and Ferguson 1980). Classical economists introduced capital and labor as additional factors of production and considered the possibility of stagnant economic development due to limits on available natural resources and, in particular, agricultural land. In neoclassical economics – where the models considered here belong – the emphasis shifted to the productive capacity of labor and capital. Land is not ignored but it is considered as a fixed factor of production and as a special form, a component of capital; land and capital are considered substitutes in several cases (Nijkamp and Soeteman 1991). In this form, land (associated with certain land uses) enters the economic analysis of producer and consumer behavior. In broad terms, changes in the demand and supply of goods and services lead to land use changes. The modeling approaches can be distinguished into demand-oriented, supply-oriented and market equilibrium approaches. In the following, the broad structure of the modeling approaches which have developed in the framework of neoclassical economics are presented.

Demand-oriented modeling approaches are the most widely utilized of the three mentioned above. They draw on the micro-economic theory of consumer behavior and consider the supply of space (e.g. availability of housing, land, resources) as fixed. The individual consumer (usually the household) spends its available income (budget) on various goods, x₁, x₂…..x_n, bought in the market at market prices. One of the goods purchased is land-based such as housing. The household derives its utility from consuming different combinations of these goods; hence, the assumed purpose of its behavior is maximization of its utility subject to the available budget constraint. This is expressed in mathematical notation as:

max U = U(x₁, x₂…..x_n),      (4.45)

      (4.46)

where,

The solution to this maximization problem gives the conditions of household equilibrium. Following Batty (1976), to maximize (4.45), a Lagrangian L is constructed:

      (4.47)

where,

is the undetermined Lagrange multipliers . Differentiating (4.47) with respect to each x_i and setting the resulting equations equal to 0, gives the first-order conditions for a maximum:

      (4.48)

By manipulating (4.48), the first-order conditions can be written as:

           (4.49)

Equation (4.49) represents the well-known result that micro-economic equilibrium occurs when the ratio of the marginal utilities equals the ratio of the prices of goods i and j considered.

This formulation of (individual consumer or household) utility maximization subject to a budget constraint has formed the starting point for a large number of theoretical and modeling contributions in urban and regional economics. In urban economics, most contributions refer to the location behavior of the firm or of the household (see Arnott 1986, Beckmann and Thisse 1986, Stahl 1986) and the resulting patterns when individual behavior is aggregated over the whole urban region. A large number of models concern the housing market – as residential land use is the most extensive, land-consuming use in urban areas. In fact, these are the only land use (and change) models which are discussed here from the large body of (urban) location models as land use is included explicitly in the models as an area of land occupied by housing and not as a point in space. It is reminded that this contribution refers to land use (and the related patters) and not to location (and the related patterns) (see also the introductory remarks of this chapter). At the regional level, the land use types analyzed are mainly agricultural and forest uses. Other important applications of the utility maximization modeling framework include travel choice, shopping behavior, and recreation behavior. It has been employed also in integrated urban and regional land use models which are discussed in a later section. Recall also that the utility maximization framework has been used as a basis for deriving spatial interaction models, hence, providing a less mechanistic, more social-science oriented rationale for their use and interpretation (Niedercorn and Bechdolt 1969 cited in, among others, Batten and Boyce 1986, Haynes and Fotheringham 1984).

The first model to be discussed is the much-celebrated Alonso model or the urban land market model. The theoretical aspects of Alonso’s model have been presented in Chapter 3. The reader is referred to the original work (Alonso 1964) for a complete presentation of the analysis. Here, the focus is on certain quantitative aspects of the model. Alonso’s work had been preceded by the research of other economists working on the same problems and in the same direction (Wingo 1961, Kain 1962 cited in Romanos 1976 and Batty 1976) but his formulation of an urban land market model (mainly a residential location model) was the first explicit application of the utility maximization approach to residential location. His model is essentially an applied refinement of von Thunen’s agricultural land rent theory (and model). This is true for many other urban models which are similar in structure and theoretical origin to Alonso’s model (Miyao 1986).

A monocentric city is assumed and households commute to the city center to shop and work. The distance from the household’s residence to the city center is denoted by u. A household spends its total income y to purchase land (for housing) q at a unit price r(u), transport to the city center T(u), and a bundle of all other commodities z (its cost set to 1). Transport costs and unit price of land depend on distance from the city center. Unit land prices decrease with distance from the city center because transport cost increases with distance. The household’s utility function is written as:

U = U (z, q, u)      (4.50)

which is subject to the budget constraint:

y = z + q r(u) + T(u)      (4.51)

Assuming the utility function has certain properties (see, for example, Alonso 1964, Straszheim 1986), the utility maximization problem involves writing the Lagrangian and setting first-order derivatives equal to zero (see, Straszheim 1986):

L = U (z, q, u) - [z + q r(u) + T(u) - y]      (4.52)

U_z - = 0      (4.53)

U_q - r(u) = 0      (4.54)

      (4.55)

z + q r(u) + T(u) - y = 0      (4.56)

Conditions (4.53) and (4.54) imply that, at the optimal location, the marginal rate of substitution between the composite good z and land q equals the ratio of prices (at competitive equilibrium):

      (4.57)

Condition (4.55) defines the household’s location equilibrium decision which entails a trade-off between the cost of land and the cost of commuting.

The level of utility a household experiences depends on the amounts of the three goods it consumes. To represent the land market allocation mechanism, Alonso assumes a competitive land market where households bid for available space and land owners offer land to the highest bidder. He introduces the notion of the "bid rent function" (or curve) which depicts the amounts of land households would bid for in the land market at various distances from the city center to attain a certain level of utility (Figure 6.8 in Hoover and Giarratani 1999). Hence, each household is characterized by a bundle of bid-rent functions, one for each level of utility. (For a mathematical exposition, the reader is referred to Alonso 1964, Straszheim 1986). In the competitive land market, bidders with different bid rent gradients arrange themselves by the height of their bid rent curve, with bidders with the steepest gradient occupying the most central locations. To assess the effects of income on the slope of the bid rent functions requires making additional assumptions about the utility functions (Straszheim 1986).

Given that the supply of space is considered fixed, the equilibrium between the demand for and the supply of space requires determining the margin of development in the city – the maximum radius from the city center – which in turn determines the rent profile according to von Thunen (Batty 1976); this is referred to as the closed city assumption (Straszheim 1986). The land rent profile (or, curve) shows how the market price of land varies with distance from the city center (Figure 6.8 in Hoover and Giarratani 1999). Using this curve, and invoking the bidding mechanism, a household’s equilibrium is determined by the tangency of the household’s highest possible bid-rent curve with the land rent curve. To obtain the equilibrium land use pattern, Alonso’s model relies on a number of critical assumptions such as the assumption of monocentricity, of continuous space, of similar housing preferences, assumptions about the mathematical form of the utility function and the budget constraint. Moreover, it treats firms and other users of urban land in a rather simplistic manner.

Muth (1969) presented the most complete analysis of residential location and his model is considered a landmark contribution in this area (Batty 1976, Romanos 1976). Although similar to Alonso’s approach, Muth’s model differs from it in two ways: (a) it deals with "housing services" (land, size of the house, and other dimensions of the value of housing) and (b) it considers a household’s income as one of the determinants of transportation expenditures. Muth bases his analysis on a set of assumptions concerning the housing market, transport costs and the center of economic activity (the monocentric city assumption) (Romanos 1976). A simple mathematical exposition of the basic structure of the model is given below following Romanos (1976).

The household’s utility function is:

U = U(x,q)      (4.58)

where,

The budget constraint is:

y = x + p(k) q + T (k,y)      (4.59)

where,

Household equilibrium is found by maximizing utility subject to the budget constraint (following a similar procedure to the one shown in the case of Alonso’s model):

U_x - = 0      (4.60)

U_q - p(k) = 0      (4.61)

q p(k) + T(k) = 0      (4.62)

y - x + p(k) q + T (k,y) = 0      (4.63)

where is the Lagrange multiplier . Solving equations (4.60) and (4.61) yields that the marginal rate of substitution between housing and other commodities is equal to their price ratio (compare to equation (4.57) above). Equation (4.62) states that in equilibrium location, the household’s marginal transport costs will be equal to its marginal housing savings. The slope of the bid-rent function for a household can be obtained from (4.62) as:

     (4.64)

Equation (4.64) shows that as distance from the city center increases, the household will bid less for each new location.

Muth analyzed also the supply side of the housing market where he treated land as any other factor of production but he did not take into account such characteristics as immobility, indivisibility, durability, etc. Moreover, he made a set of assumptions for the supply side of the housing market, the most important of which were:

1. firms and households are competitive in both the product and the factor markets
2. all firms producing a given commodity (including housing) are identical; they have the same production function and use both land and nonland inputs
3. producers employ quantities of land and nonland inputs which maximize profits at each distance
4. land rents and housing services are set by the markets so that the profits of the housing service producers equal zero everywhere the services are produced (Romanos 1976, 77).

In Muth’s analysis, the housing producer chooses the capital-land ratio, k, to maximize profits. A housing production function, Q, shows the relationship between the quantity of housing services provided, Q(k), when k units of capital are applied to a unit area of land. Denoting rents on land at a location by R, unit price of capital by _k, and unit price of housing services by p, the producer’s profit maximization problem is written as:

max p Q(k) - _k k - R      (4.65)

The first order condition for maximization yields:

p Q’ - _k = 0      (4.66)

This means that capital should be applied to the point where marginal revenue equals marginal cost. Competition among housing producers drives profits to zero and, hence, land rents are:

R = p (Q - Q’k)      (4.67)

Muth examined the effect of accessibility on housing rent and housing services per unit area of land (housing density) also and derived a set of capital-land relationships under conditions of competitive, long tun equilibrium (Arnott 1986). Despite its shortcomings, such as its simplicity, long-run static equilibrium nature, omission of important attributes of housing (e.g. durability), etc., Muth’s model of the housing market "was the first formal, general equilibrium model of the housing market, and almost all subsequent mainstream housing market theory has evolved from it" (Arnott 1986, 969).

To obtain the market equilibrium, Muth assumed that the city extended from its center as far as it was necessary for the demand to equal the supply of housing services; this is the open city assumption (Straszheim 1986). A market land rent function is derived in an analogous fashion and a bidding land allocation process is basically followed as in Alonso’s model. It is interesting to note that at equilibrium, all households have identical levels of utility, a result achieved through the spatial variation of the unit rental price of housing. For more mathematically elaborate presentations and discussion of Muth’s model, see also Straszheim (1986), Brueckner (1986), Arnott (1986).

E. S. Mills’ (1967) model of residential location operates in the Alonso-Muth utility maximization spirit. Mills, like Muth, considers land as an intermediate factor in the production of housing, which is the final consumption good (Brueckner 1986) in contrast to Alonso who considered only the land area occupied by a house. Mills (1972) analyzed also the location of employment assuming that the whole urban area is used for the production of a single commodity with an aggregate production function; i.e. he dropped Alonso’s assumption that all employment is concentrated at the city center and that all product is produced there. This urban land use (associated with the production of the single commodity) competes with transportation for land. In equilibrium, urban land at each distance from the city center is exhausted in production and transportation. Mills derives the equilibrium rent-distance function, a negative exponential form similar to that derived by Muth (Romanos 1976). Muth and Mills’s models have been analyzed in a unified manner by Brueckner (1986). Frequently, also, the literature refers to all three models as the Alonso-Muth-Mills (see, for example, Miyao 1986) or the classical or the standard model of the urban land market. This is because their analysis: (a) shares the same theoretical basis, (b) employs the same methodological framework of budget-constrained utility maximization to derive the relationships between land use and price of land, (c) arrives at similar bid-rent functional forms (the negative exponential), and (d) employs essentially the same mechanism for allocation of land to its users – the bidding process.

The first generation of this genre of analysis presented above has been criticized, on the one hand, on philosophical/epistemological grounds – the adoption of the utility maximizing theoretical framework and the associated model of the rational economic man – and, on the other, on methodological grounds – the many, frequently unrealistic, assumptions upon which it rests. The later refer to those made to derive equilibrium land use patterns or to perform dynamic analysis of the land market which make it difficult to generalize the results of the analysis to urban areas with many centers of employment and other imperfections in the real market. Representative lines of criticism include:

1. the narrowly rational logic for action these models postulate at a conceptual level ( deductivism ) which is then transcribed unproblematically upon real world processes (Cooke 1983)
2. with their reliance on the concept of utility and its propensity to stimulate action, these models conceive of humans as if they exist to express the utility-maximizing quality and nothing else (Cooke 1983)
3. the excessive emphasis these models place on accessibility as the most important determinant of urban spatial structure and the neglect of many other determinants (see, for example, Romanos 1976, Cooke 1983)
4. the neglect of many important particularities of housing itself and of the neighborhood characteristics (see, for example, Arnott 1986, Straszheim 1986)
5. the assumption of a perfectly competitive land market – i.e. without imperfections such as various forms of externalities (Batty 1976, Cooke 1983)
6. the monocentric city assumption
7. the their static nature, and
8. the assumption that location is continuously variable.

Subsequent studies and models of the housing market and of residential land use patterns in urban areas attempted to respond to these criticisms and alleviate some of the restrictions imposed by the classical model. These studies employ alternative forms of spatial demand and supply and of price and rent functions and dynamize the static versions by introducing dynamic factors such as changes in population, income, transport costs, opportunity costs of land the transportation component (see, among others, Beckmann (1969), Solow (1972), Casetti and Papageorgiou (1971) cited in Batty 1976, 259-261; also, Arnott 1986, Brueckner 1986, Miyao 1986, Straszheim 1986). Moreover, as discussed in chapter 3, attempts have been made to relax the monocentric city assumption as well as to consider the incidence of externalities (Solow 1973, Romanos 1976, Shieh 1987, Engle et al. 1992). The development of discrete choice models attempts to provide more realistic representations of consumer choices in the urban land use context and relax the assumption of continuity of location (see, for example, Anas 1981, Anas 1982, Batten and Boyce 1986, Clark and van Lierop 1986, McFadden 1973, McFadden 1978, Quigley 1985, Smith 1975, Straszheim 1986). Discrete choice models are discussed in sections 4.3.1 on statistical models and 4.6.3A on integrated land use-transportation models.

It is also interesting to query, from the perspective of the analysis of land use change, how land, land use and its change are represented in these models. These concepts did not escape the deductive mode of analysis which is applied to all other concepts with which these models deal. Land and land use are treated explicitly but at a very high level of abstraction (even when attempts at disaggregation are made) and with very little differentiation of their intrinsic qualities. Note also that usually land (or housing or housing services) is one item which is consumed together with a bundle of all other items which appear to have no relationship with land and its use, i. e. to be independent of one another. People are behaving as if their choice of a house (or of any "landed" piece of property in other applications of the utility maximization framework) is affected only by the accessibility of its location and the particular mode of transport. The host of other constraints and considerations which enter land use decision making at the individual and the collective level are treated in a limited way or not at all (such as institutional factors like planning, legislation, etc., socio-cultural, and political factors, the heterogeneity of the landscape and the environment). Consequently, land use change is treated in a very limited sense as if it is subject only to the influence of the economic determinants which the land market theory postulates. This may be true for market economies perhaps (to which these models refer mostly) but not for other types of economies where the influence on non-economic factors may be stronger and land use decision making is based on a different rationale and value system. However, even in market economies, the influence of environmental and landscape features, at least, on land use decisions is not negligible and recent modeling efforts in urban and regional economics have started to address this issue (see Bockstael and Irwin 1999).

Despite efforts to model both the demand and the supply sides of the land market, most applications are demand-oriented making the implicit assumption that land will adjust to any magnitude of market demand. When models attempt to simulate market clearing, they rely on the particular mode of land use decision making which is characteristic of these models – the bidding process. Although this may be a rough and intuitively appealing approximation of several real world situations in many market and non-market economies, it is far from a satisfactory representation of the land market as there do exist several types of land markets and various types of landed interests and not simply two undifferentiated categories of landlords and households (see, among many others, Form 1954, Cooke 1983, McNamara 1983, 1984, Piore and Sabel 1984, Harvey 1985, Healey and Nabarro 1990).

From a practical point of view, several of the criticisms discussed above address also the limitations of the Alonso-type urban models to provide explicit formulations for operational models as it is the case with discrete urban models (Batty 1976, Wilson 1974). However, they have provided the theoretical underpinnings and the economic rationale for the development of several (potentially and actually) operational programming models (the Herbert-Stevens and the Harris models being well-known cases). This is true for the broader utility maximization (and profit maximization) analytical approach upon which these models rest which has provided the framework for the development of operational models at larger scales – ranging from the regional to the global – and for uses of land associated with non-urban economic activities such as agriculture and forestry (see, for example, Takayama and Labys 1986, Fischer et al. 1996a). The concept of utility consumers derive from purchasing and consuming bundles of goods and services produced by economic activities which are, more or less, associated with given types of land, constitutes the basis for formulating single or multiple objective optimization models which have been discussed previously. In these models, utility has been expressed operationally by means of either mono-dimensional measures such as income (e.g. farmers income or the value of agricultural or forest products) or more composite measures which reflect the multi-attribute nature of utility (including, for example, in addition to economic, other aspects of quality of life, of the environment, etc.). In addition, the bidding process introduced by the utility maximization models is being used in several land use modeling contexts as a land allocation mechanism underlying land use change at least in market (or, approximations of) economies.

Applications of the utility maximization approach on larger scales are made basically in the context of larger, integrated modeling exercises. The FASOM non-linear programming model has already been mentioned previously and other integrated models are discussed in a latter section. The characteristic of the contemporary applications of the utility maximization approaches is that they do not make several of the restrictive assumptions characteristic of the early theoretical models. In particular, they do not assume monocentric and uniform flat plains within which economic activities take place; they account for several of the environmental features of the geographic environment under study; they consider the interactions (e.g. trade) and interdependencies between spatial entities and among economic agents (Anas and Kim 1996, Anas et al. 1998, Fujita et al. 1999). Frequently, they provide for considerable disaggregation (e.g. by consumer group, product type, land use type) which render them more realistic representations of reality.

More importantly, progress in GIS and spatial data management systems has fostered the building of spatially explicit models which can capture the intricate interrelationships in space between the activities modeled as well as important economic-environmental interactions. In this context, it is possible to estimate spatial production functions which provide for greater representation of the variability in the conditions of production and of interactions in space compared with the coarse, aspatial production functions which are commonly used. (see, for example, Keyzer 1998, Keyzer and Ermoliev 1998). This is an extremely important improvement as, in principle at least, it facilitates the inclusion of policy variables and the study of the spatial variability of their impacts in terms of the land use changes they produce. However, all this gain in detail, enhanced representation, and potential greater validity of the model results has a considerable cost; namely, the difficulties of finding appropriate spatial data to use the models in all their detail and the computational difficulties stemming from the nonlinearities of the modeled spatial relationships. Finally, it is worth mentioning that the utility maximization approach has been proposed as an evaluation method and it has found applications in the analysis of the desirability of alternative land use plans (see, among others, Bell et al. 1977, Nijkamp 1980, Nijkamp and Rietveld 1986).

4.5.5. Multi-Objective/Multi-Criteria Decision Making models (MODM/MCDM)

The last group of optimization models to be discussed represent a rather recent trend in the analysis of land use and its change which involves the combination of optimization techniques with elaborate, multidimensional techniques of land use assessment/evaluation in a spatially explicit modeling environment. The roots of this modeling direction go back to broader developments in the methods and techniques of multi-criteria and multi-objective decision making methods and their subsequent adoption and application to various fields of the social sciences. Their introduction in the analysis of land use issues dates approximately since the mid-1970s but they gained momentum in the 1980s when improvements in information technology and refinement of the techniques combined to produce more user-friendly and versatile tools for decision making. There are several applications which are impossible to cover in this review, first, because they come from diverse sources and, second, because the developments in this area are very rapid currently (although the main idea on which they are built is basically the same). The interested reader is referred to the major sources of literature cited below for further study. In the following, two applications are presented briefly to illustrate the basic idea of this genre of models.

A note on terminology is necessary as the terms "multi-criteria" and "multi-objective" are frequently used interchangeably although they are not semantically identical. Nijkamp (1980) suggests a distinction: "Discrete models are characterized by a finite number of feasible alternative choices or strategies (for example, in the case of plan evaluation or project evaluation problems); discrete models are often called multi-criteria models. Continuous models are based on an infinite number of possible values for the decision arguments and hence for the objective functions; they are usually called multi-objective optimization models" (Nijkamp 1980, 30-31). Nijkamp (1980) indicates also representative models from each category. Moreover, the literature on multi-criteria and multi-objective techniques exhibits also a methodological and philosophical differentiation as regards the development of the associated techniques; the former are usually associated with the "outranking method" school established by B. Roy (1973) at LAMSADE at the University of Paris-Dauphine and the latter with the Multi-Attribute Utility Theory (MAUT) school of R. Keeney and H. Raiffa (1976) (see, also among others, Korhonen et al. 1992, Roy and Bouyssou 1993, Steuer 1986). Naturally, several other approaches exist which are not easy to classify in either one of the dominant schools such as Saaty’s Analytic Hierarchy Process method (Saaty 1994).

The first model presented below is a multi-objective model which has been used to support agricultural land re-allocation decisions in the Netherlands (Janssen 1991). It is a representative application of multi-objective optimization techniques to similar land allocation decision problems. The problem this application addresses relates to the choice of the most suitable land development strategy for each of the 118 agricultural regions into which the country has been subdivided. Three land development alternatives were taken into account: change of land use, agricultural use, combined land use. Each region was described in terms of several environmental and socio-economic characteristics stored in a GIS in use at the National Physical Planning Agency of the Netherlands. The factors which determine a region’s suitability for a change in land use were distinguished into "demands" and "opportunities". These factors were translated into 18 evaluation criteria which were used to characterize each region. The table of scores of each region on each criterion was set up (the effects table according to the authors). For each criterion, weights were determined based on interviews with experts. For each region, a utility index was calculated using the weighted summation technique. This procedure is shown schematically in Figure 4.2a below:

Figure 4.2a. The Effects Table

The utility score U_j for region j is calculated as follows:

     (4.68)

where,

In order to identify the optimal allocation of the three candidate land use types in each of the 118 agricultural regions of the study area, a linear programming approach was followed. Two LP models were used: a single-objective and a multiple-objective. The single objective LP model took the form of maximization of the total utility of the study area; i.e. the sum of total utilities of the 118 regions. For each region its total utility, TU (not to be confused with the utility score above), was calculated as the sum of utilities of the region for each of the three candidate land use types as follows:

     (4.69)

where,

The single-objective LP model was written as

     (4.70)

The second model used was a multi-objective LP model which, instead of maximizing an aggregate utility function for all 118 regions and the three land use types, it consists of maximizing simultaneously the sum of utilities of all regions by land use type. For practical purposes, the 118 regions were aggregated into 14 regions in the multi-objective model which was written as follows:

      (4.71)

The solution technique used produced the whole efficient set ; i.e. the set of non-dominated (or, Pareto optimal ) solutions. A solution to a MODM problem is characterized by the set of values of each of the n objectives of the problem. A non-dominated or pareto optimal solution is one "in which any further improvement in any one of the n objectives can be achieved only at the price of ‘worsening’ the value of at least one of the remaining objective functions" (Zeleny 1982, 53). In this case, note that the linear formulation of the objective functions means that the interactions and interdependencies of the allocations produced among the regions are not taken into account, an assumption which may be untenable in the real world. However, the inclusion of interaction effects leads inevitably to non-linear models which present considerable computational difficulties and have heavy data requirements.

The characteristic difference between single and multiple objective models is that the first produce only one, the optimal solution, while the second produce a whole set of solutions among which decision makers can choose depending on their particular preferences and priorities. In the above land use allocation problem, the solutions are characterized by the proportions of each of the three land use types to be allocated to each of the regions. The decision makers can choose those solutions in which the proportions of land allocated to each of the three land use types agree with other (implicit) criteria they may consider. Several variations of the multi-objective LP problem have been devised as regards the generation of the efficient set (for another land use decision problem example see, Briassoulis and Papazoglou 1993). In addition, because usually the efficient set contains a large number of non-dominated solutions, it is not practical for decision makers to have to inspect the whole set in order to indicate their preferred solution. This was one of the reason for the development and application of interactive optimization techniques which have found applications in land use allocation problems (see, for example, Rietveld 1977, Nijkamp 1980, Voogd 1983). A more advanced, contemporary application of a multi-objective optimization model with an interactive component is presented below.

The second model presented here is part of a Decision Support System (DSS) developed for Sustainable Agricultural Development Planning at the International Institute for Applied Systems Analysis (IIASA) at Laxenburg, Austria, in cooperation with FAO (Fischer et al. 1996b). It combines a multi-objective model with the AEZ methodology developed by FAO for land evaluation purposes (FAO 1976, FAO 1978; see also Chapter 1). Application of the AEZ methodology in a study region produces a spatially explicit inventory of its land resources – i.e. land resources by Agro-Ecological Zone – which can be used to produce a land productivity raster (or, grid cell) data base (see, Fischer et al. 1996b for the case presented here as well as for other related applications). This data base is used to provide data used in the formulation of the optimization model.

The multi-objective model considered 10 objectives (i.e. it had 10 corresponding objective functions) whose decision variables included: (a) the land use proportions in each cell of the study area devoted to a cropping, a grassland, or a fuelwood activity, (b) the number of animal units of a given livestock system kept in a given zone, and (c) the feed ratio of a given feed from a given crop allocated to a particular livestock system in a given time period in a given zone. The constraints specified included "the preferred demand baskets, crop specific production targets, risk aversion, economic constraints, land use by individual crop, crop mix, input use, quality of human diet, environmental conditions, seasonal feed demand-supply balances, feed quality, and distribution of livestock systems" (Fischer et al.1996b, 9).

The particular methodological approach followed to apply the above multi-objective model (MODM) is called Aspiration-Reservation Based Decision Support (ARBDS) method. Its main purpose is to provide a mechanism by which the large number of non-dominated solutions contained in the efficient set produced by the MODM is reduced to a set with certain properties which satisfy the preferences of the decision maker. These preferences are defined interactively by the decision maker who expresses aspiration and reservation levels for each criterion. A summary of the ARBDS as a two-stage approach is given below following Fischer et al. (1996b).

The first (preparatory) stage consists of specifying and generating a core model which contains the objective functions and those constraints which express logical and physical relationships between the decision variables of the model. These variables should include also variables which are potential evaluation criteria (e.g. goals, performance indices, etc.). Initially, the decision maker selects a set of criteria from these variables and specifies whether it is minimization, maximization or a goal criterion (i.e. if it minimizes as deviation from a given value). Then the DSS performs a series of optimizations to produce the Utopia (best value) and the Nadir (worst value) points for each criterion. It also computes a compromise solution which corresponds to a problem for which the aspiration and the reservation levels are set to the Utopia and the Nadir points respectively.

At the second stage, during an interactive procedure, the decision maker specifies goals and preferences, including the values of the criteria (s)he wants to achieve or to avoid. The vectors composed of these values are called aspiration and reservation levels. These are used to define achievement functions which are used to select a Pareto optimal solution from the whole efficient set produced by the optimization model. This is achieved by generating additional constraints and variables which are added by the DSS to the core model. Thus, an optimization model is formed which produces non-dominated solutions which are as close as possible to the user-specified aspiration levels. The interactive procedure is repeated with the decision maker revising the aspiration and reservation levels each time after inspecting the solutions and selecting those s/he prefers. The procedure stops when a satisfactory solution is found or when the decision maker wants to discontinue it.

The above modeling procedure is one of the many efforts to develop interactive modeling packages to assist the decision makers in making land use allocation decisions, in the present case. Other versions have been designed also which handle group decision making as, normally, there is not a single decision maker in any decision context (see, for example, Levandowski 1988, Majchrazak 1988, Krus et al. 1990). The successful implementation and use of these packages depends on many factors among which the availability of appropriate data, the training of the users of the model, the cooperation and real interests of the decision makers, and the availability of high-technology computer facilities feature high.

Closing the section on optimization models for the analysis of land use change, a few general remarks are in order. First, all these models are normative or prescriptive; i.e. they indicate desirable (according to the specified objectives, preference structures, decision variables, and constraints) future land use patterns (either in a static or, less frequently, in a dynamic fashion). By comparing these future with current patterns it is possible to specify the amount and direction of change in various land use types which should take place to obtain the desirable state. For this reason they are frequently used in planning and policy making contexts as aids to making decisions about future allocations of land to alternative uses (which are frequently in conflict).

The level of detail of the prescriptions offered depends on the level of aggregation of land use types, socio-economic data and the spatial explicitness of these models. In theory, it is possible to achieve a very high level of detail in all respects but at sharply rising computational difficulties, not to mention the capacity of the human mind to conceive of all the detail offered by very disaggregate models. The level of detail offered depends critically also on the theoretical/behavioral basis supporting these models. Broadly, these are based either on neoclassical economic theory of utility maximization (normative theory) or on simplistic behavioral and other assumptions about the factors which have been observed or are assumed to impinge on land use and cause its change (the instrumental approach to theory referred to in Chapter 3). In the static versions of the models, time is implicit and they offer no guidance about the trajectory of the system which leads to the prescribed patterns. Dynamic optimization models are extremely difficult to build given the data and computational difficulties mentioned before.

Optimization models are necessarily selective as regards the environmental, social, cultural, and political factors which they include in their specification. However, they do not account for the interactions among these factors and the processes through which they combine in certain places, and within particular contexts to produce changes in land use. In other words, the prescriptions they offer do not account for the dynamics which operate to bring the land use system from the present to a desirable state. The future states prescribed are assumed to be affected by the factors taken into account in the specification of the models (mostly prices, costs, distances in the more conventional and utility maximization models) but the particular processes through which these states are generate are treated as a "black box". The more recent versions of optimization models include more detail and account for many more factors in addition to the economic ones but they still rely on either a positive or a normative theoretical framework which governs their structure. Overall, optimization models offer rough guides of desirable land use futures (usually different from the current ones) and they should be used prudently by decision makers when deciding what actions to take to achieve them. Finally, it should be noted that optimization models are frequently used as components of integrated models which are discussed in the next section.