We have discussed how to aggregate two or more criteria as part of a
geographical objects ranking ( GIS and decision support (4): a tool for aggregating fuzzy criteria
with ArcMap
). The method employee requires the criteria be perceived by the decision maker
as being of equal relevance. In this article we will discuss the theoretical
bases to meet the uneven relevance criteria.
To determine the aggregation method to be used across two criteria, we
propose to the user to evaluate three situations:
- The first question (S1) proposes a very poor value for the first criterion and an
excellent value for the second criterion. - The second question (S2) proposes two average
values for both criteria. - The third question (S3) proposes an average value for the first criterion and an excellent value for the second .
The three replies (triplets) are, then, used as key input for a table of
aggregation methods including 50 possible replies (triplets).
One of the constraints of this method is that both criteria have equal
relevance, ie that the answers to the three questions are the same whichever
the criteria order. This is called Replies “symmetry”.
Two criteria have the same importance if the aggregation function is symmetrical,
i.e. if the answer to the three assessment questions is the same if we reverse
the order of the criteria.
For example, in the case of choosing a car, the consideration of the criteria
“colour and price”, we can build the first question in two different ways:
- a) a totally incompatible colour (E) and a fully compatible price (A) if we consider C1 = colour and C2 = price, or so
- (b) a completely incompatible price (E) and a completely compatible colour(A) if we consider C1 = price and C2 = colour .
If both criteria have the same relevance, the answer to this question
will be the same in both cases . Through this reply, the subjective way of
aggregating the two criteria ( conjunction or disjunction) or the underlying compromise
mechanism that the decision maker uses, will be considered.
By cons, if one of the two criteria is more relevant than the other, the
symmetry is not verified. In our example it would not be unusual to get
“good” as reply, if the price is our concern, and not the colour, and
“poor” if the colour satisfies us but not the price.
In this scenario, the aggregation operations table is no longer valid.
The concept of relevance for a criterion in relation to another has not been,
yet, solved. The meaning we give to this word is very variable according to the
decision makers or the situations.
Unlike equal relevance aggregation criteria, for which we can find the
development of the calculations in the current literature, a method to treat in
the case of uneven relevance criteria aggregation has to be developed.
Problem statement
How to enhance the list of Questions SI, S2, S3 with the lowest number
of new questions to determine:
- if
the function aggregation is symmetrical or not, and therefore if we can use the
aggregation operations table of equal relevance goals; - if the
function is not symmetrical , what is the weight relative of each criterion C1
and C2?
Suggested solution.
We have S1 (E, A), S2 (C, C), S3 (C, A). We suggest to add S4 (A, E),
that is to say the symmetrical question to S1 comprising a proposal fully
compatible with criterion C1 and another totally incompatible with criterion
C2.
All responses forming a doublet S1, S4 (AA, BB, CC, DD, EE) refers to equal
relevance criteria treatment.
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 what aggregation
operation can be used, before determining the weights to be applied.
Among the aggregation operations, min, max and the symmetrical sums, can
apply only on symmetrical criteria, and therefore they have to be eliminated ex
officio .
Among the average operations, only the arithmetic average can give a
result different from 0 in the case when one of the criteria is 0 (√ xy = 0 and
2xy / ( x + y ) = 0 if x = 0 or y = 0).
Therefore we will withhold the arithmetic average as aggregation operation
as follows
( Px.x + Py.y ) / ( Px + Py )
Px and Py being the respective weights for C1 and C2 criteria.
In the case of doublets (D, B) and (B, D) it is easy to demonstrate that
the weights have to be 3 and 1 for (D, B) and 1 and 3 for (B, D).
There are no other possible doublets (DC , DA , …) if Px and Py are
constant. The other doublets assume that Px = f (x) and Py = f (y).
We can conclude that the weighting of objectives is, only, necessary when
the number of classes is greater than three, and cannot apply, for example, to
a criterion that would be : good, average , poor . In this case it would always
be within the field of a symmetrical function .
In the case of n = 5, only the weighting factor 3-1 is usable in
order to keep on within the range of a human decision maker.
Practical solution
There are 25 possible combinations answers to question S4, S1
symmetrical. We have divided them in 5 groups :
- Replies confirming the equal relevance of both criteria: five combinations, the doublets AA, BB, CC, DD and EE. The aggregation formula is sought in the table of 50 triplets ignoring the S4 reply.
- Replies demonstrating the lack of consideration of a whole criterion: two combinations, the doublets AE and EA. The result of the
aggregation is directly the value of the criterion taken in account, ignoring the value of the other criterion. - The replies where the difference between S1 and S4 is 0,5 and which imply a different weight from the two criteria ( weight 1 and 3); six combinations , doublets AC, BD, CA, CE, DB and EC. We use the weights of 1 and 3 for the input criteria and the arithmetic average.
- Answers approaching an equal relevance criteria, but dubious. The difference between the answers is 0.25 : six combinations, the doublets AB, BA, BC, CB, DC and CD. The aggregation formula is sought in the table of 50 triples ignoring the S4 reply, then we shade the 0, 25 result obtained for the most relevant criterion.
- The replies that come close to not taking into account one of the two criteria. The gap between the two responses is 0 , 75 : six combinations , the doublets AD, BE, ED, DA, DE and EB. The result of the
aggregation will be the criterion considered as relevant, shadowed by 0, 25 for the less important criterion.
The following table details the 25 combinations
and the aggregation formulas used.
In the next article we will discuss an ArcMap command that allows performing these uneven relevance aggregation criteria.
