## GIS and Decision Support (5): Theoretical Fundamentals (Part 1)

In the previous series of articles we have discussed examples of
application for the Boolean and fuzzy logic, as well as two commands for ArcGis
allowing the application of fuzzy logic to geographic information. We will discuss
now the theory that underlies this type of treatment.

Introduction

We have a set Ω of objects to classify according to a set C
of criteria. The number of objects is finite. The partial evaluations of the objects
according to each criterion take values ​​in easily identifiable sets.
A partial objective will be seen as a fuzzy set restricting the acceptable
values ​​of the associated criterion. Therefore, we accept the implicit hypothesis
that each objective defines a total order for Ω .
We will use as an example the case of a set Ω representing the
pixels of a study area that we wish to classify according to their ability to
receive aquaculture breeding sites. The criteria set C is the dataset
layers available: bathymetry, slope, substrate, productivity, etc. Each of
these info layers adopts easily identifiable values : favourable, somehow favourable,
unfavourable, and so on.
For each layer of information we will set a goal, for example, for bathymetry
that is at least favourable, for productivity that is at least unfavourable,
and so on. The goal is none other than the subset of the acceptable values ​​of
the info layer.
Finally, we accept the hypothesis that each layer of information can be
classified in its entirety by the set goal, that is to say that we are able for
each pixel to determine the corresponding value of the layer.