An alternative definition for fuzzy interval orders

A valued Ferrers relation for interval comparison

“…Hence when the comparison rules of semiorders or interval orders are used in real life problems some information is lost. For instance, consider three intervals x = [0,6], y = [1,5] and z = [5,10]. For interval orders, intervals x and y are indifferent but also x and z are indifferent.…”

“…The fuzzification of interval orders is already studied in a theoretical way where the researchers analyzed different possible ways to define them ( [5,7]). However a comparison rule of intervals satisfying such definitions is not proposed in the literature.…”

“…In the next section we introduce the basic notations, Section 3 is devoted to the definitions of valued interval orders where we put together some results of [5,7]. In Section 4 we present a valued outranking relation for the comparison of intervals and study its properties.…”

“…Condition (ii) is the most used one but other conditions are also suggested by different researchers as alternative definitions. Bufardi, Van de Walle and De Baets have analyzed relations between these definitions and defined equivalence classes (see [5,7]. Some new results on the approach of [7] can be found in [8]).…”

“…We present in the following a general scheme showing such relations between different definitions. For this purpose we put together in Figure 1 different results obtained by the three authors (see [5,7]) and add two counter examples on these results in order to complete the scheme.…”