GIS and decision support (1): classifying with fuzzy numbers

Decision processes are based on information from very diverse source and
type. This information is used by the decision-makers to perform   choices,
i.e. to retain a certain number of entities and to exclude others.

Let’s discuss the following example:
An action must be performed on municipalities, but this action depends on: 1. the area of ​​the communes, between 2500 and 3000 hectares
2. the number of inhabitants of the municipality, between 2500 and 5000
The purpose of the operation is to perform a classification of objects (municipalities)
according to two criteria (population and area).



Using traditional GIS queries.

The tools offered by GIS work as follows:
A selection of the municipalities which have a population between the two limits
(in this example   between 2500 and 5000 inhabitants) is performed.
A selection of the municipalities which have a surface between the two desired
bounds (in this example   between 2500 and 3000 hectares) is performed.
The final result is those municipalities that appear in the   two previous
selection results, eliminating those that only appear   in only one of the
selections.

Let’s use as example all   municipalities for the Finistère region.

To determine which municipalities meet the criterion Population
we apply the selection request:

POPULATION
»> = 2500 AND« POPULATION
»<= 5000

The 54 municipalities with a population between
2500 and 5000 are the following 

To determine the municipalities that meet the criteria Area
we apply the following selection request:

AREA
»> = 2500 AND« AREA
»<= 3000

The 26 municipalities with an area between 2500
and 3000 ha are the following:

The
10 municipalities that meet both conditions are:

If we are a technical service, the result suits us and we pass it on to
our dear elected officials.

If we are the elected officials, our problems begin:

Why the commune of Plouenan is not in the result? Because it has a
population of 2451 inhabitants and an area of ​​3077 ha.

And the commune of Rédené? Because it has 2464 ha, but 2870 inhabitants.

And the commune of … In short, the list will be more or less long, but
at each classification performed using our all or none logic (Boolean), classic
for GIS, we will have more or less limited situations that will cause problems.

Let’s understand then that for a large part of elected officials, the
fact that they are told that our tool is a “decision  help” is far from convincing them!

Another logic, another result

A municipality having 2499 inhabitants and not 2500 will be eliminated
from the result, as any municipality having 3001 ha and not 3000.
In the decision processes, the values ​​of the variables used are always tainted
with some uncertainty. In order for the GIS to be a decision help tool, it is
essential to give the user tools that match his method of reasoning.
The value of 2500 inhabitants is used by the GIS as a strict value. Besides,
  in the decision-maker’s mind, this value is only a representative value
of a municipality “Size” (eg “average municipality”).

The use of  “fuzzy numbers” is
another possibility to classify objects.

How do we define an average municipality with fuzzy logic? Instead of
using two values ​​as minimum-maximum limits, we will use four values:

  • The two limits of the number of
    inhabitants between which the municipalities totally correspond to his perception of an average municipality: by   example: 2500 and 5000;
  • The lower limit from which the
    municipality t is completely excluded as average: for example 1500;
  • The upper limit from which the
    municipality is excluded as average: for example 7500.

This makes it possible to build a
“belonging” function that takes the following form

Here we measure the set membership “Average municipalities” between 0
and 1: the population values ​​having a   belonging 0 are completely
“excluded” from the classification, the values ​​having   a
membership of 1 are “completely included” and the values ​​between 0
and 1   correspond to a “more or less” belonging (hence the term
”   fuzzy  “).

If we apply this belonging function to our
municipalities for the Finistère region (we will discuss the tools in the next
article) to the category Municipalities, we obtain the following result:

Each municipality has a resultant value between 0 and 1.

We have grouped the municipality into 5 classes:

  • those that correspond very well
    to the criterion: resultant values ​​between 0.8 and 1.0: 63 municipalities (in
    red)
  • those that match   rather well to the criterion: resulting values ​​between 0.6 and 0.8   : 16   common (in dark orange)
  • those that correspond moderately   to the criterion: resultant values ​​between 0.4 and 0.6: 16   common (in light orange)
  • those that match   rather badly   the criterion: result values ​​between 0.2 and 0.4:
    18 common (in yellow   dark)
  • those that do not match the criterion: Resulting values ​​between 0.0 and 0.20:
    170 communes (in light yellow)

If we apply the   belonging function to our
Finistère municipalities to the category of average surface municipalities with
limits between 2000 ha – 2500 ha – 3000 ha and 4000 ha, we obtain the following
result:

Each municipality has a resultant value between 0 and 1.

We have grouped the municipalities into 5 classes according to the population:

  • those which correspond very well
    to the criterion: resultant values ​​between 0.8 and 1.0: 46   common (in red)
  • those that match   rather well to the criterion: resulting values ​​between 0.6 and 0.8   : 14 common (in dark orange)
  • those that correspond moderately the criterion: resulting values ​​between 0.4 and
    0.6: 10   common (in light orange)
  • those that match   rather badly the criterion: resulting values ​​between 0.2 and 0.4: 6   common (in yellow   dark)
  • those that do not match the criterion:
    resulting values ​​between 0.0 and 0.20: 207   common (in light yellow)

Now let’s cross the two fuzzy numbers (again, we’ll discuss the tools in
the following article):

The aggregation results of the two fuzzy
numbers: area and population is as follows

Each municipality has a resultant value between 0 and 1.

We have grouped the communes into 5 classes:

  • those which correspond very well
    to both criteria: resultant values ​​between 0.8 and 1.0: 15   common (in red)
  • those that match   rather well to both criteria: resulting values ​​between 0.6 and 0.8   : 6   common (in dark orange)
  • those that correspond moderately
    to both criteria: resultant values ​​between 0.4 and 0.6: 113 common (in light
    orange)
  • those that match   rather poorly to both criteria: resultant values ​​between 0.2 and 0.4:
    19   common (in yellow   dark)
  • those that do not match both
    criteria: resulting values ​​between 0.0 and 0.28   : 130   common (in light yellow)

If we compare the two types of approach, considering that for the fuzzy
approach an 80% membership can be considered very good, we get:

Approach

 
Population
 
Area
  Crossing

Boolean

    54     26    
10
Fuzzy  
  63
 
  46
 
15

Of course, both municipalities, Rédené and Plouenan, which initially
posed a problem for us, are included in the 15 municipalities selected through
the fuzzy treatment. Rédené has a membership of 0.93 and Plouenan of 0.92.

The classification of geographical entities

In this example we have used two criteria to “classify” our
municipalities.

The classification of objects according to several criteria is a common
operation in everyday life. When you buy a product you take into account the
degree of   satisfaction given by its price, its lifetime, its
“standing” …

For each criterion we define ourselves “fuzzy” functions
(price   between X and Y euros, up to Z maximum) or “fuzzy
classifications “, for example the” standing “(ie: bad, average,
good, high, very   above).

We make our choices by crossing the values ​​of the different variables
taken into account and obtaining a ranking of the different products according
to the degree of   overall satisfaction.

Consider the simple case of crossing two criteria to which five values
​​are attributed   for satisfaction: bad, rather bad, average, rather
good, good. Each object will have   as  result a degree of satisfaction coded on these
same five values.

If we are looking for a vehicle based on its resistance and price characteristics,
for example, we will find very resistant vehicles, so maximum satisfaction
 for the first criterion, but whose price is a little above what we want,
  therefore average satisfaction fort the second criterion.

What is the resulting value of the crossing? In fact there is not a
single result value, but many, depending on who makes the choice. Some will do
a mental average of both and will give a “pretty good” rating, for
  others the price   will prevail and classify this vehicle as
“medium”, others   Finally, they will be more sensitive to the
resistance criterion and will classify the vehicle as   “Good”.

The example becomes even clearer if we consider a very resistant vehicle
but   very expensive (complete satisfaction of one criterion and complete
dissatisfaction of the other).   Will this scenario give an “average”,
“rather bad” or “bad” value?

GIS tools based on classical logic work on the principle of minimum
value. The result of the crossing is the smallest value of the two criteria,
which will be coded only as 0 or 1. If one of the two criteria is not satisfied
in a pair of values ​​1-0, the resultant crossing will be 0.

The use of a flexible spatial analysis tool makes it possible to
determine the function of   crossing used by the operator. This step is
simply to ask   the operator the result of three crosses: Very good –
medium, medium – medium, and   Very good – bad.

The result of this test allows you to choose a function among the 50
functions of possible crossings  when taking into account 5 degrees of
satisfaction (Theory   of possibilities, Applications to the
representation of computer skills,   D. DUBOIS and H. PRADE, Masson 1988).

In the following articles we will discuss two tools developed for ArcGis
that will allow you to perform all these operations.

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