### 3.3 Modeling of Preferences

The preferences can be modeled using the notion of binary relation or the
notion of utility function or criterion function.

Here we use the binary relation, which uses the definition:

*i R j* = *i* is at least as preferable
as *j*, *i* ³ *j*

The following three fundamental situations can be modeled with the help of
single binary relationships:

(i) * i R j* and not *j R i* Þ *i *is preferred to *j*:— a
situation which we denote as:

* i j *: *i **® j*

(ii) *i R j* and *j R i* Þ *i* and *j* are indifferent *i* « *j*

(iii) not *i R j* and not* j R i* Þ *i* and *j* are incomparable (or
not compared) *i*. .*j*

All the evaluators *e*_{i} give a rank–number r _{i}(*a*_{j}) = r _{ij} to all the alternatives. The
data are provided in the form:

*r *_{i}
(*a*_{1}), r _{i}
(*a*_{2}), . . . r _{i}
(*a*_{n})

In the case of strict preference, the sequence numbers in *A*_{i}
are attributed to the alternatives selected in the evaluations, and (r _{i} + 1 + *n*/2) to the
non–selected ones. This can be written as follows:

- for
*a*_{j} Î *A*_{j} : r _{i}
(*a*_{j1}) = 1, r _{i} (*a*_{j2}) = 2, .
. . , r _{i} (*a*_{j}_{r}_{ i}) = r _{i},
- for
*a*_{j} Ï *A*_{j} :

In the case of weak preference, it is assumed that all the selected
alternatives are at the same level of preference, therefore the *A*_{i}’s
are not actually ordered. Here the transformation rule is:

- if a
_{j} Î A_{j} :
- if a
_{j} Ï A_{j} :

As a result of these transformations, the preference data for further
analysis are in the form: