Minimal representation of a semiorder

In a multicriteria decision problem it may happen that the preference of the decision-maker on some criterion is modeled by means of a semiorder structure. If the available information is qualitative, one often needs a numerical representation of the semiorder. We investigate the set of representations of a semiorder and show that, once a unit has been fixed, there exists a minimal representation. This representation can be calculated by linear programming and exhibits some interesting properties: all values are integer multiples of the unit and are as scattered as possible in the sense that, in the set of all representations contained in the same bounded interval, the minimal representation is a representation for which the minimal distance between two values is maximal. The minimal representation can also be interpreted as a generalisation of the rank function associated to linear orders. © 1990 Kluwer Academic Publishers.