### GIS and decision support (2): the crossing of two fuzzy criteria

Let’s reconsider our example regarding the towns ranking
for the Finistère region according to the following two criteria: population
and area.

How to rank entities according to a single criterion

To establish the fuzzy number allowing the
classification of the towns according a single criterion, we rely on our assessment
of two qualities: the complete satisfaction of the criterion (1) or the complete
dissatisfaction (0).

By accomplishing this task, we have generated a complete
series of intermediate values, between 0 and 1. Working with numbers is not
easy but, above all, it is not natural. Let’s say that a town meets the 0.356
surface criterions and that the town surface is rather unsatisfactory, it does
not affect the classification process but it sure does the operator.

Usually, we use a 5 satisfaction values scale:
excellent, good, average, mediocre, bad. Our brain can, sometimes, to manage a
scale of 7 classes, adding two further values, very good and very mediocre. But
beyond that, it becomes metaphysical.

Therefore we will rely on a classification with 5
classes for the rest of our example.

As you could have already noticed, classifying the towns
according to a single criterion whatever the population or area, it is not too
complicated with a fuzzy number. But you will, also, observe that it involves a
subjective aspect that is not unique. Just as much as we have only one classification
in Boolean logic when we define a range, for example 2500 to 5000 inhabitants, we
could have different low and high limits according to each operator to define
when an average town is completely excluded. In our example we used 1500 and
7500 but this is fully subjective. For someone else it could be 1500 and 6000,
etc.

Here we hit an essential point when working with
fuzzy logic: there is not ONE possible result, there are as many as possible actors.
And this is not a question of knowing which the REAL result is. They are all
true because they correspond to the vision of each concerned actor concerned.

How to rank entities according to two criteria

Once you have performed both rankings according to
each of the chosen criteria, it is time to classify the towns according to the
two associated criteria: those who fully satisfy both criteria will be at the
top of the ranking, those who do not satisfy both criteria, at the bottom. And those
that partially satisfy one of the criteria will be in the middle.
In the middle, but, where and in which order?

Let’s forget the numbers and use our five classes.
A town that wholly satisfies one of the two criteria and that does not satisfy
at all the other criterion; we’ll consider that it satisfies both as excellent,

Well, it depends! Depending on the actor it will adopt
any value. Therefore the goal is not to know which is the right way to cross
both criteria, but to determine how our actor realizes this crossing in his

To find the formula underlying mathematical the reasoning
of our actor, it is enough to propose him three crossings

• How does he classify a town that totally satisfies the population criterion and not at all the area one?
• How does he classify a town that moderately satisfies both
criteria?
• How does he classify a town that totally satisfies the population criterion and moderately the surface
one?

In our example for the towns of the Finistère area
, the ranking results from the crossing of both parameters taken into account. The
definition of the crossing function is performed by the response to the three
cases:

 Population Area Expected Result 3750 ( very good) 1500 ( bad ) average 2000 ( average ) 2250 ( average ) average 3750 ( very good) 2250 ( average ) Rather good

The mathematical translation of these responses is
: R = average ( Population, Area )

The tool we have developed allows performing this
operation as well as the crossing:

The
result is the following:

We find 20 towns located in the range of
satisfaction for both criteria 0.8-1.0

This result has to be compared with the result when
using Boolean logic (see previous article):

where we found 10 towns satisfying both criteria .

With the same data, if the operator enters different

 Population Area Result wish 3750 ( very good) 1500 ( bad ) bad 2000 ( average ) 2250 ( average ) average 3750 ( very good) 2250 ( average ) average

The mathematical translation of these responses is:
R = minimum ( Population, Area )
This function corresponds to the one used by classic tools.

With our tool we enter the following parameters:

And
we obtain the result of applying this function to the data   :

Considering these answers, now we get only 15 towns
in the 0.8 – 1.0 satisfaction range, while we had found 20 with the previous

What conclusions to draw?

The main conclusions are the following:

• The transformation of each criterion into a fuzzy function brings the notion of nuance into each criterion. We do not proceed by selection (which implies rejection and abandonment
of some elements) but by ranking the objects according to each criterion.
• Whatever the number of criteria used at any time, the set of objects is present in the crossings. An object that is misclassified by the first cross can to be ” caught up ” by later crossings.
• In any ranking there is a part of subjectivity due to the operator. This subjectivity can be measured and modelled by simple mathematical functions. According to the operator (decision maker) the result will not be exactly the same.
• In any case, this method provides richer results and closer to the reality of the decision process. It allows, also, determining the differences of appreciation between actors (different crossing methods) and highlighting
the source of divergence rather than the discordant final result.

The use of
fuzzy sets blurred on GIS existing data, allows a range of answers that offer
all the intermediate variations between completely satisfactory and completely unsatisfactory.
Therefore, the user benefits with  a
response closer to his reasoning – nuanced – and his decision criteria  is
no longer bounded, exclusively, to the wholly satisfactory responses. A special
advantage can be observed in the case where no answer fully corresponds to the
selection criteria: a traditional query gives 0 result while the answers come
closest to these criteria.
By setting mechanisms of the theory of possibility in the query tools and data
aggregation allows to introduce notions of imprecision and uncertainty that
intervene both at knowledge and decision process. The information system is
compatible with poorly known or poorly defined information: information can
become more certain or more   uncertain over time while
keeping the same measured value.

In the next article we will introduce the two tools
we have developed for ArcGis to transform an attribute as fuzzy number and to
perform the aggregation of two fuzzy criteria. You will have the chance to