3.4 Constructing a Fuzzy Relation from Preference Data

The evaluations, corresponding to the rows in the above matrix generally do not rank–order the alternatives unanimously. Therefore the preference relation, which can be derived from the matrix can first be modeled by a fuzzy relation.

For each evaluation e_i an individual preference relation is given implicitly in and it can be represented by the matrix.

in which,

Depending on the preference type used, is equivalent to the inequality

• r _ik < r _i (strict preference)

or

• r _ik ≤ r _i (weak preference).

Before constructing the fuzzy relations we define a weight vector, whose purpose would be to characterize the importance, relevance or credibility of the evaluations.

If no weighting is needed in practical applications, the programs assign a weight by default taking the value of 1.

From the individual preference relations matrices, and using the weights, we construct the matrix representing a fuzzy relation on the set of alternatives A:

Each component of R can be interpreted as the credibility of the proposition is preferred to " in a global sense, and without referring to the single evaluation.

As a consequence of the construction of R the following interpretation is correct (note that 0 ≤ ≤ 1):

• = 1 implies that is true in all the evaluations,
• = 0 implies that is false in all the evaluations
• 0 < < 1 implies that

is true in a certain portion of the evaluations.

A relation represented by the matrix R, obtained according to the construction rules outlined above, has the following main characteristics:

• Fuzzyness

Non–fuzzy : if = 0 or = 1 for k, = 1, 2, . . . , n;
Fuzzy : otherwise;

• Reflexivity

Reflexive : if = 1 for k = 1, . . . , n
Anti–reflexive : if = 0 for k = 1, . . . , n
Irreflexive otherwise

• Symmetry

Symmetric : if = for k, = 1, . . . , n;
Anti–symmetric : if ¹ 0 implies = 0 for all k ¹ 1;
Asymmetric : otherwise

• Trichotomy

Trichotome (normalized) : if + = 1 for k, = 1, . . ., n and k ¹ 1;
Non–trichotome (nor–normalized) : otherwise.

From a non–normalized matrix a normalized one can be obtained, using the following transformation:

if k ¹ 1 and + ¹ 0, otherwise

For a global description of the evaluations represented by the matrix R, three indicators can be computed:

1. Coherence Index, which is defined by the expression:

Obviously, the value of C depends on the order of the rows and columns in R (i.e. on the order of the alternatives in A). The value of C ranges between –1 and +1 (inclusive).

– 1 ≤ C ≤ 1

An order–independent modification of C is the absolute coherence index, which is defined as:

For C_a, the inequalities C ≤ C_a and 0 ≤ C_a ≤ 1 hold so C_a is an upper bound for C. Indices C and C_a are indicators of unanimity in the preference data.

It can be easily seen that C = 1 implies full coherence and C_a = 0 implies full lack of coherence. The value –1 of the Index C can be interpreted as an order of alternatives opposite to the order defined by the fuzzy relation.

1. Intensity Index, which is defined by the expression:

This index can be interpreted as an average credibility level of the statements "a_k is preferred to a_ℓ" or "a_ℓ is preferred to a_k". In general, its value –1 £ I £ 2, which in the case of strict preference , and therefore 0 £ I £ 1. Here I = 1 implies a normalized relation – which is a generalization of trichotomy – and means that in all the preference data one of the statements or is valid for all the pairs of alternatives. The index I can be interpreted as average credibility level of the propositions or .

2. Dominance Index, which is defined by the expression

It is also an order – dependent index, and –1 £ D £ 1 holds for it.

As in the case of coherence index, we can define the order–independent absolute dominance index:

for the value of which the inequalities D £ D_a and 0 £ D_a £ 1 are valid. The index D (and its upper bound D_a) indicate the average difference between the credibilities of the propositions S_kℓ and of their opposite propositions S_ℓk.

It may be pointed out that C, I, D and C_a, I, D_a are not independent of one another. Their relationships are represented by the equations:

C × I = D

C_a × I = D_a

It may also be pointed out that the order–dependent C and D indices can have negative values if the actual order of alternatives (order of rows and columns in R) is rather close to an opposite order than to the order corresponding to the general tendency in the data. An order which maximizes C is called the tendency in the relational structure represented by R.