Continuous Allocation

Problem formulation

Continuous allocation describes a land unit as a mix: every land use type can get a share of it, rather than one type taking the whole unit. The shares can also be read as the chance that a type ends up at that location, which is why the same approach is also called probabilistic. The first versions of the Land Use Scanner worked this way.

The model is a doubly constrained logit model that rests on discrete choice theory (McFadden, 1978). Suitability takes the place of utility in that theory: the better a land unit suits a type compared with the other types, the larger the share of that type. Because suitability weighs the advantages and disadvantages of a location for a user, it can also be read as the price that user would bid for the land. Two sets of balancing factors then make the outcome fit the land available per land unit and the claim per land use type; see the specification.

The balancing factors depend on each other, as the corollaries show, so the model finds them by iteration (Dekkers & Koomen, 2007). In each round a type that falls short raises its bid and a type that gets too much lowers it, which makes the iteration a simple version of a land market. Spatial policy fits this picture as a tax or subsidy on suitability, and the balancing factors give an indication of the price of land per cell (Koomen & Buurman, 2002). Hilferink & Rietveld (1999) give the mathematics of the basic model and its extensions.

Applications usually extend this basic form in four ways:

• Some land use types, such as infrastructure and water, are read from a map instead of allocated. Planned changes such as a new railway line enter the same way.
• Claims hold per region, and each land use type can have its own regional division.
• A claim has a minimum and a maximum, so that the model also finds a solution when the claims together exceed the available land.
• Suitability maps are expressed in money, with the highest value per land use type tied to a realistic land price, so that urban uses can outbid other uses where a location suits both equally (Borsboom-van Beurden et al., 2005; Groen, Koomen, Ritsema & Piek, 2004; Dekkers, 2005).

Working with shares has two drawbacks. Small amounts of many types can end up in one cell, mainly when the suitability maps vary little across space. More distinct suitability maps reduce this. A minimum share per type would reduce it too, but then a solution becomes harder to find, as experience with the comparable Natuur Plan Generator shows (Van Eupen & Nieuwenhuizen, 2002). The second drawback appears in maps: showing only the dominant type per cell overstates some types and understates others. Discrete allocation avoids both because each land unit gets one type.

For the background of this formulation in the Land Use Scanner, see Koomen, Hilferink & Borsboom-van Beurden (2011).

Specification

Continuous allocation in the context of the GeoDMS is solving the following equation given the suitability $S_{ij}$ for each land use type $j$ and land unit $i$:

• $X_{ij} := a_i * b_j * e^{\beta S_{ij}}$

subject to:

• $L_{jr} \le \sum\limits_{i}{X_{ij}} \le H_{jr}$ for each $j$ and $r$ for which claims are specified;
  • with $b_j < 1$ only if $H_{jr}$ is binding
  • and $b_j > 1$ only if $L_{jr}$ is binding
• and for each land unit $i$: $\sum\limits_{j}{X_{ij}} = L_{i}$

in which:

• $X_{ij}$ is the land in cell $i$ that goes to type $j$;
• $S_{ij}$ is the suitability of cell $i$ for type $j$;
• $\beta$ sets how strongly suitability counts, which makes it the inverse of the temperature in the entropy reading below;
• $a_i$ is the balancing factor that makes the land allocated in cell $i$ equal to the land available there;
• $b_j$ is the balancing factor that keeps the land allocated to type $j$ within its claims;
• $L_{jr}$ is the lower claim of type $j$ in region $r$;
• $H_{jr}$ is the upper claim of type $j$ in region $r$; and
• $L_i$ is the amount of land available in cell $i$.

Compare this with discrete allocation.

When $L_i = 1$ for each land unit $i$, we call this a probabilistic allocation problem. When $L_{jr} = H_{jr}$, we call this an iterative proportional fitting problem.

Corollaries

note that

• from substituting $X_{ij}$ in the land unit restriction it follows that $a_i = L_i / \sum\limits_{j}{b_j e^{\beta S_{ij}}}$
• and similarly,
  • if $H_{jr}$ is binding then $b_j = H_{jr} / \sum\limits_{i}{a_i e^{\beta S_{ij}}}$
  • if $L_{jr}$ is binding then $b_j = L_{jr} / \sum\limits_{i}{a_i e^{\beta S_{ij}}}$
• $ln(X_{ij}) = ln(a_i) + ln(b_j) + βS_{ij}$
• $βS_{ij} = ln(X_{ij}) − ln(a_i) − ln(b_j)$

Solving

The corollaries give each set of balancing factors in terms of the other, so the model computes them in turn. It starts with $b_j = 1$, and each iteration has three steps:

1. Compute $a_i$ from the current $b_j$, which divides the land of every land unit among the land use types.
2. Sum the result per claim, which gives $T_{jr} = \sum\limits_{i}{X_{ij}}$, the land allocated to type $j$ in region $r$.
3. Correct the balancing factor of each claim with the bound $C_{jr}$ that is binding: $b_j' = b_j \cdot (C_{jr} / T_{jr})^{\omega}$.

With $\omega = 1$ this is the basic iterative proportional fitting of Hilferink & Rietveld (1999), also known as the Sinkhorn algorithm. Each round moves the allocation only part of the way, because a correction for one type also changes the $a_i$ and with them the land of the other types. Some claims are therefore reached only after many iterations.

A value of $\omega$ between 1 and 2 is called overrelaxation: each round corrects further than the deviation asks for. This does not change the solution, because at the solution $C_{jr} / T_{jr} = 1$ and the power has no effect. It can save many iterations. Close to the solution the iteration converges for $0 < \omega < 2$ (Lehmann et al., 2022). That is a local result, so whether a value works from the start of a calculation has to be tested: a value of 2 or more can make the balancing diverge.

A GeoDMS loop runs a fixed number of iterations, so that number decides how closely the claims are met, and the deviation from the claims is the thing to check. The Land Use Scanner demo gives an example of these settings.

Shadow price interpretation

• $β^{−1} ln(a_i)$ can be interpreted as the shadow price of land unit $i$, thus very suitable land units have a high price.
• $β^{−1} ln(b_j)$ can be interpreted as the shadow price of Claim $j$: taxation or subsidy of the claimed land may be required to get the allocation not to exceed the maximum claim, nor remain below the minimum claim respectively.

Entropy maximization

The solution $X_{ij}$ maximizes the following entropic quantity:

$E := \sum\limits_{ij}{X_{ij} (1 + \beta S_{ij})} - X_{ij} \ln(X_{ij})$ subject to the same restrictions.

This is shown by the fact that ${∂E \over ∂X_{ij}} = βS_{ij} − ln(X_{ij})$ and the KKT condition ${∂E \over ∂X_{ij}} + ln(a_i) + ln(b_j) = 0$

implies that $βS_{ij} = ln(X_{ij}) − ln(a_i) − ln(b_j)$

Usage

Continuous allocation (also probabilistic allocation) is used to find the allocation of land use to land units that fit best to the suitability maps when endogenous interactions are disregarded, but some form of beta-dependent entropy is allowed.

Given suitabilities $S_{ij}$ for land unit $i$ and land use type $j$, the share of land of unit $i$ allocated to type $j$ is usually defined by the logit transformation ${\exp(\beta S_{ij})}\over{\sum\limits{k}\exp(\beta S_{ik})}$.

This solution can be considered as the expected amount of land use for each type when the actual suitabilities are assumed to have a Weibull distributed error term, and the land users with the highest suitability always get to use the land unit.

In the GeoDms, continuous allocation can be implemented using the loop or for_each operator. An earlier version of the Land Use Scanner applied only continuous allocation. Later versions also included discrete allocation.

Bibliography

• Borsboom-van Beurden, J. A. M., Boersma, W. T., Bouwman, A. A., Crommentuijn, L. E. M., Dekkers, J. E. C., & Koomen, E. (2005). Ruimtelijke Beelden. Visualisatie van een veranderd Nederland in 2030. RIVM rapport 550016003.
• Dekkers, J. E. C. (2005). Grondprijzen, geschiktheidkaarten en parameterinstelling in de Ruimtescanner. Technisch achtergrondrapport bij Ruimtelijke Beelden. MNP report, 550016005.
• Dekkers, J., & Koomen, E. (2007). Land-use simulation for water management: Application of the Land Use Scanner in two large-scale scenario studies. Modelling land-use change: Progress and applications, 355-374.
• Groen, J., Koomen, E., Ritsema van Eck, J., & Piek, M. (2004). Scenario's in kaart; Model-en ontwerpbenaderingen voor toekomstig ruimtegebruik.
• Hilferink, M., & Rietveld, P. (1999). Land Use Scanner: An integrated GIS based model for long term projections of land use in urban and rural areas. Journal of Geographical Systems, 1, 155-177.
• Koomen, E., Hilferink, M., & Borsboom-van Beurden, J. (2011). Introducing Land Use Scanner. In E. Koomen & J. Borsboom-van Beurden (Eds.), Land-use modelling in planning practice (pp. 3-21). Dordrecht: Springer.
• Koomen, E., & Borsboom-van Beurden, J. (2011). Land-use modelling in planning practice (p. 214). Springer Nature.
• Koomen, E., & Buurman, J. (2002, April). Economic theory and land prices in land use modeling. In 5th AGILE Conference on Geographic Information Science, Palma (Balearic Islands Spain) April 25th-27th (Vol. 7).
• Lehmann, T., von Renesse, M.-K., Sambale, A., & Uschmajew, A. (2022). A note on overrelaxation in the Sinkhorn algorithm. Optimization Letters, 16, 2209-2220.
• McFadden, D. (1978). Modelling the Choice of Residential Location, Spatial Interact ion Theory and Planning Models (A. Karlqvist et al., eds.).