# Continuous Allocation

### From ObjectVision

## Contents |

# Specification

Continuous Allocation in the context of the GeoDMS is solving the following equation given the Suitabilities \(S_{ij}\) for each Land Use Type \(j\) and Land Unit \(i\):

\(X_{ij} := a_{i} b_{j} e^{\beta S_{ij}}\)

subject to

- for each claim j: \(ClaimMin_j \le \sum\limits_{i}{X_{ij}} \le ClaimMax_j\)
- with \(b_j < 1\) only if ClaimMax is binding
- and \(b_j > 1\) only if ClaimMin is binding

- and for each land unit i: \(\sum\limits_{j}{X_{ij}} = L_{i}\)

Compare this with Discrete Allocation.

When \(L_{i} = 1\) for each Land Unit i, we call this a Probabilistic Allocation problem. When \(ClaimMin_j = ClaimMax_j\), 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 \(ClaimMax_j\) is binding then \(b_j ~=~ ClaimMax_{j} / \sum\limits_{i}{a_i e^{\beta S_{ij}}} \)
- if \(ClaimMin_j\) is binding then \(b_j ~=~ ClaimMin_{j} / \sum\limits_{i}{a_i e^{\beta S_{ij}}} \)

- \( \ln X_{ij} = \ln a_{i} + \ln b_{j} + \beta S_{ij} \)
- \( \beta S_{ij} = \ln X_{ij} - \ln a_{i} - \ln b_{j} \)

# shadow price interpretation

\(-\beta^{-1}\ln a_{i}\) can be interpreted as the shadow price of land unit i, thus very suitable land units have a high price.

\(-\beta^{-1}\ln b_{j}\) 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 \({\partial E}/{\partial X_{ij}} = \beta S_{ij} - \ln X_{ij}\) and the KKT condition \({\partial E}/{\partial X_{ij}} + \ln a_{i} +\ln b_{j} = 0\)

implies that

\(\beta S_{ij} = \ln X_{ij} - \ln a_{i} - ln b_{j}\)

# Usage

Continuous Allocation (aka probabilistic allocation) is used to find the allocation of land use to land units that fits 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 by using the loop or for_each operator. Earlier version of the Land Use Scanner, applied only continuous allocation. Later versions also included Discrete Allocation.