Iterative Proportional Fitting (IPF) are disaggregation methods to obtain doubly constrained matrix cell values X_ij proportional to a given strictly positive interaction measure (aka proxy) C_ij given restrictions on row an column totals T_i. and T_.j.

We introduce a_i and b_j to express X_ij as a_ib_jC_ij.

It follows that:

$$a_i := \frac{T_{i}}{\sum\limits_{j}{b_j C_{ij}}}$$

and

$$b_j := \frac{T_{j}}{\sum\limits_{i}{a_i C_{ij}}}$$

An IPF is used to solve the Continuous Allocation of the earlier versions of the Land Use Scanner by using e^β**s_i**j as proxy.

• Doubly Constrained Gravity Models of Spatial Interaction
• Trip Distribution in Transport modelling is often modelled as doubly constrained gravity model
• Continuous Allocation where land units are sources and claims for land use are destinations.

• https://en.wikipedia.org/wiki/Iterative_proportional_fitting
• IPF RIVM/PBL paper (in Dutch) and here
• LAND USE SCANNER: An integrated GIS based model for long term projections of land use in urban and rural areas, Maarten Hilferink and Piet Rietveld Article on SpringerLink describing the Continuous Allocation in the Land Use Scanner
• video on the number type of exact solutions and related paper of Eric Rowland and Jason Wu