# GIS and Decision Support (6): Theoretical Fundamentals (Part 2)

## Aggregation example.

We consider the aggregation of the criterion “bathymetry” as it is shown in Figure 2, and another criterion, for example, the substrate that is defined according to a particle size scale:

Fuzzy representation of the substrate criterion with 5 degrees of satisfaction

We propose the user to evaluate three situations

1)   How would you classify a site with a depth of 30 meters and a silt bottom?

oVery good       o Good         o Pretty good       o Poor       o Very bad

2)   How would you classify a site with a depth of 17 meters and a sandy bottom?

oVery good       o Good         o Pretty good       o Poor       o Very bad

3)   How would you classify a site with a depth of 17 meters and a silt bottom?

oVery good       o Good         o Pretty good       o Poor       o Very bad

The first question corresponds to C1 = E and C2 = A, that is to say that it proposes a very bad value for the first criterion (30m is beyond the 25m) and an excellent value for the second criterion.

The second question corresponds to C1 = C2 = C, ie it proposes two average values ​​for the two criteria.

The third question corresponds to C1 = C and C2 = A, ie it proposes a mean value for the first criterion and an excellent value for the second one.

A user can answer for example:

1)   very bad (the depth criterion seems to be crucial here),
2)   pretty good (average depth and average bottom represent average site)
3)   pretty good (for this user the depth criterion is more important, despite the fact that the substrate is very good, having a moderate depth makes the site a medium).

Therefore, we have R1 = E, R2 = C, R3 = 0, which corresponds to the third line of the table of aggregation operations where we find as operation: min ( u, v )

u is the value of the layer pixel “bathymetry [0, 1]” v is the value of the pixel on the layer “substrate [0,1], that is to say that the pixels have been classified with values ​​between 0 and 1 according to the criteria defined in the figures 1 and 3.

The aggregation operation is, therefore, to keep for each pixel the minimum value of u and v This corresponds in an image processing software to the Overlaying function with creation of a composite layer with the minimum values ​​of the two layers.

Another user can answer, for example:

2. pretty good
3. good

For this user, the criteria are roughly equivalent starting from a certain average threshold, which means that for the third question his evaluation is halfway between the two partial criteria. Therefore, R1 = I, R2 = C, R3 = B, corresponding to the fourth line of the table of aggregation operations where there are two possible actions:

or,

If one wishes to refine the calculation, a fourth question would be necessary to decide the method of calculation. In most cases, one or the other can be used indifferently because the results are very close together.

2: Criteria of uneven importance

Two criteria have the same importance if the aggregation function is symmetrical, ie if the answer to the three evaluation questions is the same when the order of the criteria is reversed.
For example, for the bathymetry and substrate criteria one can construct the first question in two ways:

a) a totally incompatible bathymetry (E) and a completely compatible substrate (A) if we take C1 = bathymetry and C2 = substrate, or

b) a completely incompatible substrate (E) and a completely compatible bathymetry (A) if C1 = substrate and C2 = bathymetry.

If both criteria are equally important, the answer to this question will be the same in both cases. This response highlights the subjective way of aggregating the two criteria (conjunction or disjunction) or the underlying trade-off mechanism that the decision maker uses.
On the other hand, if one of the two criteria is of greater importance than the other, the symmetry will not be verified. For example, if the bathymetry is more important for the decision-maker than the substrate, he may answer mediocre (D) in the first case and good (B) in the second case. In this case, the aggregate operation table is no longer valid.

The concept of the importance for a criterion relative to another has been little understood until now. The meaning given to this word varies greatly depending on the decision-maker or the situation.
Unlike the aggregation of criteria of equal importance, for which we can find in the available literature the development of calculations, it will be necessary to develop a method for dealing with the aggregation of criteria of uneven importance.

Limitations of the problem. One must not confuse the uneven importance with the threshold of discrimination for a criterion.
By threshold of discrimination we mean the relation between the interval corresponding to a criterion and the global domain of variation of the values.

For example, if the depths vary between 0 and 30 m, settting as a criterion the fuzzy number (4, 10, 20, 26) is less selective than (11 , 12 , 14, 15 ). It is observed that, generally, the more a goal is considered important by a decision-maker, the more he will tend to define supports and narrow nuclei and, on the contrary, the less important he judges a criterion, the more he will tend to establish distant boundaries.
This is a way of conveying a certain idea of ​​the criteria relevance, but it remains in the realm of what we call equally important criteria.

Another common form of expressing the uneven importance of the criteria is the weighting of objectives: a weight is assigned to each objective and this weight is included in the aggregation operation.
This method has the disadvantage that it cannot be applied to anything else but numbers, and that the evaluation, a priori, of the weights is problematic.

Statement of the problem

How to enrich the list of Questions SI, S2, S3 with the smallest number of new questions to determine:

a) if the aggregation function is symmetrical or not, and therefore whether the aggregation operation table of equal importance for the goals, can be used;

b) if the function is not symmetrical, what is the relative weight of each criterion C1 and C2?
Proposed solution.

We have S1 ( E, A), S2 (C, C), S3 (C, A). We propose to add S4 (A, E), ie the symmetrical question to S1 including a proposition totally compatible with criterion C1 and another totally incompatible with criterion C2.

All the responses forming a doublet S1, S4 (AA, BB, CC, DD, EE) refer to the treatment of criteria of equal importance.

The doublets (A, E) and (E, A) correspond to a particular case where the weight of a criterion is equal to 0, the aggregation is not necessary because the result is equal to C1 in the case of ( A, E), or C2 in the case of (E, A).
For the other possible doublets it is necessary to determine which aggregation operation can be used, before determining the weights to apply.

Among the aggregation operations, min, max and symmetric sums can only be applied on symmetric criteria, and therefore they must be eliminated automatically.

Of the average operations, only the arithmetic mean can give a result other than 0 in the case where one of the criteria is 0 (√xy = 0 and 2xy / (x + y ) = 0 if x = 0 or y = 0).

We therefore retain the arithmetic mean as an aggregation operation in the form

(Px.x + Py.y) / ( Px + Py)

Px and Py being the respective weights of criteria C1 and C2.

In the case of doublets (D, B) and (B, D) it is easy to demonstrate that the weights must be 3 and 1 for (D, B) and 1 and 3 for (BD).

There are no other doublets possible (DC , DA , …) if Px and Py are constant. The other doublets assume that Px = f (x) and Py = f (y).
It can be concluded that the weighting of the goals is necessary only for a number of classes greater than three, and cannot be applied for example to a criterion that would be: good, average, bad. In this case we would always be in the domain of a symmetric function.

In the case of n = 5, as is the case in Table 1, only the weighting factor 3 – 1 is usable, if one holds to a reasoning close to the attitude of a decision maker.

Download the theoretical bases This document contains the two articles on the theoretical bases of the processing of geographic information with fuzzy logic.