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.
Approach assumption.
The objective associated with a criterion
(information layer) will be described as a fuzzy set. The values of the
pixels for the layer located in the core will, therefore, be perfectly
compatible with the goal, while the values located outside the support are, completely,
incompatible.
If we use only two values categories, for example favourable and unfavourable,
we will have for Bathymetry the following representation of the goal.
Even if the estimation of a mathematical function linking the depth to
the adequacy of the site for the oyster culture cannot be performed in an exact
way, the shape of the curve makes it possible to express certain behaviours of
the decision-maker. This is why; in general, it is preferable to use a discrete
notation scale, usually comprising 5 levels, maximum 7, according to the
decision-maker’s perception threshold.
A simple way is to, linguistically, express compatibility levels between goal
and evaluation, and then project them to [0 , 1 ] using the following table:
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Linguistic Appreciation | Level of compatibility consequence – objective | Digital convention in [0,1] | Ordinary Convention |
Very well |
Fully compatible |
1 | AT |
Good |
Rather compatible |
0.75 | B |
Pretty good |
Moderately compatible |
0.5 | C |
Poor |
Weakly compatible |
0.25 | D |
Very bad |
Incompatible | 0 | E |
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Representing the criterion using a fuzzy
interval allows for a more convenient and abundant info representation. Indeed,
the decision-maker must provide a desired value, for example, of bathymetry. He
must establish an interval but the question necessarily arises: should he fix
this interval by being pessimistic and, thus, establishing distant boundaries,
or being optimistic and thus tightening the limits?
The fuzzy interval makes it possible to have both representations at once: the
pessimistic interval will be the support and the optimistic interval the
nucleus.
For example: if the decision maker considers that it is impossible to raise
oysters at a depth less than 4m and more
than 25m, but the optimum depths are between 8 and 12m, we will have as
objective the following fuzzy interval:
The criteria aggregation
We will consider the case of a pair of criteria. The generalization in
the case of n criteria where n> 2 is presented in another document.
Two scenarios must be considered:
- two criteria of equal importance;
- two criteria of unequal importance.
1: Criteria of equal importance.
Two criteria of equal importance can be crossed according to the
all-or-nothing principle or by introducing nuances. The principle of all or
nothing excludes any compromise between the two criteria and results in two
aggregation operations: conjunction or disjunction. The conjunction is used in
the case where one wishes the simultaneous satisfaction of the two criteria
(the “and” logic). That is, the overall assessment can only be better
than the worst of the partial evaluations.
Example: aggregation of the substrate and productivity criteria. If the
decision-maker’s attitude implies the simultaneous satisfaction of the two
criteria, this means that if the substrate is moderately favourable and the
productivity is very favourable, the result of the aggregation of the two
criteria will be the most unfavourable of the two, that is to say moderately favourable.
Disjunction is used in cases where the criteria are redundant (the logical “or”). That is, the overall assessment will be equal to the best of the partial evaluations.
Example: aggregation of “water quality” and “productivity” criteria. If the decision-maker’s attitude implies a redundancy of these two criteria, it means that if the quality of the water is average and the productivity is very good, the result of the aggregation will be the most favourable of the two, that is to say “very good”.
A third attitude of the decision maker leaves aside all or none to introduce nuances into the aggregation. If the objectives become nuanced, the compromise between the two criteria becomes one of the natural attitudes of the decision maker.
The compromise results in the fact that the overall assessment is at an intermediate level between the partial evaluations. Taking the example of water quality and productivity, if one has average quality and excellent productivity, the result will be; for example, “good”.
On fuzzy sets, this type of set-up operation is performed using two families of aggregation operations: symmetric sums and parametric medians.
Procedure for determining the aggregation
operation.
In the case where two objectives are aggregated, there is a simple
procedure for determining the type of operation to be performed. It consists in
proposing to the decision-maker three typical situations and asking him to
evaluate them. Considering the three answers given, we search in a functions catalogue
the one that better fits to the wishes of the decision maker.
The three typical situations (Si, S2, S3) are chosen according to the two
criteria (C1, C2) so that:
- S1 is incompatible (Note E or 0)
with C1, but fully compatible (note A or 1) with C2; - S2 is moderately compatible (note
C or 0.5) with the two objectives C1 and C2: - S3 is moderately compatible (note
C or 0 , .5) with C1 and fully compatible (note A or 1) with C2.
We obtain three answers (RI, R2, R3) and we search
for the aggregation operation in the following table.
This table is not exhaustive and concerns only to the most common
answers. In reality, the set of possible answers has 50 triplets. These
triplets must however respect the following constraints:
1) R3 ≥ max (R1, R2), the evaluation of a situation that completely satisfies
criterion 2 and moderately criterion 1 must be at least equal to the best
evaluation of the other two situations (R1 and R2), where number one does not
satisfy the first criterion at all and the other satisfies only moderately the
two criteria;
2) R3≥ note C or 0.5, the total satisfaction of the second criterion cannot
bring down the overall satisfaction below the level of satisfaction of the
first criterion;
3) the aggregation function must be symmetrical, ie the objectives are of equal
importance and can therefore be interchanged in the aggregation process. (Caution:
to say that the objectives are of equal importance does not imply that they are
of the same critical nature, see below: objectives of unequal importance).
To be discussed in the following article …
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