Prescriptive modeling with map algebra for multi-zone allocation with size constraints

Map algebra is a methodology for organizing and processing digital cartographic data in a geographic information system (GIS). While its capabilities to describe patterns existing (or hidden) in data have been well studied, its capabilities to prescribe new patterns in response to specific requirements have not been much explored. The latter, prescriptive capabilities help planners address a class of spatial problems called “cartographic allocation” (Tomlin, 1990), which concerns allocation of subsets (or zones) of a cartographic space to certain uses according to one or more criteria. Taking a school districting problem as an example, this paper introduces a systematic approach to designing a map algebraic procedure for a cartographic allocation problem with capacity constraints. It is found that a classical trial-and-error heuristic can be refined to a more formal approximation method and serve as a good alternative to other solution methods when the problem involves a large number of spatial units as is often the case with a raster-based GIS.

► A map algebraic heuristic designed for a multi-zone cartographic allocation problem.
► Vidale’s solution and Lagrangian relaxation interpreted in map algebraic terms.
► Map algebra’s prescriptive (as opposed to descriptive) capabilities.