Appendix 7I, Land Use Scanner Background

Land Use Scanner employs two approaches to simulate the probability that a certain location is chosen for a specific land use. These two approaches, continuous and discrete, are described below in different paragraphs. An extensive discussion of both allocation approaches and a comparison of their performance is provided in a report by Loonen and Koomen (2009).

A crucial variable for both approaches is the suitability $s_{cj}$ for land use of type $j$ in grid cell $c$. This suitability can be interpreted to represent the net benefits (benefits minus costs) of land-use type j in cell c. The higher the benefits (suitability) for land-use type j, the higher the probability that the cell will be used for this type. The economic rationale that motivates this choice behaviour resembles the actual functioning of the land market. The model is furthermore constrained by two conditions: the overall demand for the land-use functions which is given in the initial claims and the total amount of land which is available for each function.

Continuous allocation

The original, continuous model employs a logit-type approach, derived from discrete choice theory. Nobel prize winner McFadden has made important contributions to this approach of modelling choices between mutually exclusive alternatives. In this theory, the probability that an individual selects a certain alternative is dependent on the utility of that specific alternative, in relation to the total utility of all alternatives. This probability is, given its definition, expressed as a value between 0 and 1, but it will never reach these extremes. When translated into land use, this approach explains the probability of a certain type of land use at a certain location, based on the utility of that location for that specific type of use, in relation to the total utility of all possible uses. The utility of a location is in our case expressed as the suitability for a certain use. In combination with the constraints related to the demand for land and the amount of available land per cell the following doubly constrained logit model can be formulated:

$$ X_{cj} := a_j \cdot b_c \cdot \exp(\beta \cdot s_{cj}) $$

In which:

• $X_{cj}$ is the expected amount of land in cell c that will be used for land-use type j.
• $a_j$ is the demand balancing factor that ensures that the total amount of allocated land for land-use type j equals the sector-specific claim.
• $b_c$ is the supply balancing factor that makes sure the total amount of allocated land in cell c does not exceed the amount of land that is available for that particular cell.
• $\beta$ is a parameter that allows for the tuning of the model. A higher value for $\beta$ makes the suitability more important in the allocation and will lead to a more mixed-use land pattern, strongly following the suitability pattern. A low value will produce a more homogeneous land-use pattern.
• $s_{cj}$ is the suitability of cell c for land-use type j, based on its physical properties, operative policies and neighbourhood relations.

The outcomes of the model are thus based on various external model results, a probability approach and many operational choices of the model user. The results should therefore not be interpreted as an exact prediction for a particular location but rather as a probable spatial pattern of land-use change.

Discrete allocation

The discrete allocation approach allocates equal units of land (cells) to those land-use types that have the highest suitability, taking into account the regional land-use demand. This discrete allocation problem is defined as a form of linear programming, the solution of which is considered optimal when the sum of all suitability values corresponding to the allocated land use is maximal.

This allocation is subject to the following constraints:

• The amount of land allocated to a cell cannot be negative;
• In total only 1 hectare can be allocated to a cell; and
• The total amount of land allocated to a specific land-use type in a region should be between the minimum and maximum claim for that region.

or, mathematically:

$$ \max \sum_{c,j} ( s_{cj} \cdot X_{cj} ) $$

subject to:

• $X_{cj} \ge 0$ for each c and j;
• $\sum_j X_{cj} = 1ha$ for each c;
• $L_{jr} \le \sum_{c \in r} X_{cj} \le H_{jr}$ for each j and r for which claims are specified;

in which:

• $X_{cj}$ is the amount of land allocated to cell c to be used for land-use type j;
• $s_{cj}$ is the suitability of cell c for land-use type j;
• $L_{jr}$ is the minimum claim for land-use type j in region r; and
• $H_{jr}$ is the maximum claim for land-use type j in region r.

The regions for which the claims are specified may partially overlap, but for each land-use type j, a grid cell c can only be related to one pair of minimum and maximum claims. Since all of these constraints relate $X_{cj}$ to one minimum claim, one maximum claim (which cannot both be binding) and one grid cell with a capacity of 1 hectare, it follows that if all minimum and maximum claims are integers and feasible solutions exist, the set of optimal solutions is not empty and cornered by basic solutions in which each $X_{cj}$ is either 0 or 1 hectare.

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