# Discrete Allocation

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Discrete Allocation is the Allocation of resources to a set of categories.

In the context of the GeoDMS and its applications it is defined as finding the $$X_{ij} >= 0$$ for each land unit $$i$$ and land use type $$j$$ that solve the following Semi Assignment Problem for given suitabilities $$S_{ij}$$:

$$\max \sum\limits_{ij}{X_{ij} S_{ij}}$$ subject to

for each claim $$j$$: $$ClaimMin_j \le \sum\limits_{i}{X_{ij}} \le ClaimMax_j$$ and for each land unit $$i$$: $$\sum\limits_{j}{X_{ij}} = 1$$

Thus $$X_{ij}$$ represents whether land unit i is allocated to land use type j and only one single allocation per land unit is allowed.

It is used to find the allocation of land use to land units that maximizes total suitability when endogenous interactions are disregarded.

It can also be used to perform an aggregation (aka Downsampling) of a discrete map while keeping the total area's constant by using the amount of each land use type in or near a Downsampled land unit as suitability for that type and the total area's as claims (rounded down as minimum claim and rounded up as maximum claim). A script called BalancedClassAgggr.dms will become available in our code examples.

When applied iteratively and by incorporation of dynamic neighbourhood enrichment, one can simulate land use change caused by natural processes while minimum demands and/or maximum land use restrictions (as specified by the claims) are maintained.
When applied iteratively with a feedback from future (neigbourhood dependend) yields on the current suitability, one can model a time consistent market equilibrium.

In the GeoDms, discrete allocation can be done with the discrete_alloc function.

In the EuClueScanner, the suitabilities for discrete allocation are called Transition Potentials and there are three ModelTraits for calculating them: