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.
