
## 4 Quasi-Normalization Of Qm(.)

There is no way to define a normalized qm(.**), but a qualitative quasi-normalization [14, 15] is nevertheless**
possible when considering equidistant linguistic labels because in such case, qm(Xi) = Li**, is equivalent to a** quantitative mass m(Xi) = i/(n **+ 1) which is normalized if**

$$\sum_{X\in D^{\Theta}}m(X)=\sum_{k}i_{k}/(n+1)=1$$
but this one is equivalent to
$$\sum_{X\in D^{\Theta}}q m(X)=\sum_{k}L_{i_{k}}=L_{n+1}$$

In this case, we have a qualitative normalization**, similar to the (classical) numerical normalization. But, if the**
previous labels L0, L1, L2, . . ., Ln, Ln+1 from the set L are not equidistant, so the interval [0, **1] cannot be**
split into equal parts according to the distribution of the labels, then it makes sense to consider a qualitative quasi-normalization**, i.e. an approximation of the (classical) numerical normalization for the qualitative masses** in the same way:

$$\sum_{X\in D^{\Theta}}q m(X)=L_{n+1}$$

In general, if we don't know if the labels are equidistant or not, we say **that a qualitative mass is quasi-normalized**
when the above summation holds. In the sequel, for simplicity, one assumes to work with quasi-normalized qualitative basic belief assignments.

From these very simple qualitative operators, it is thus possible to extend directly the DSmH fusion rule for combining qualitative basic belief assignments by replacing classical addition and multiplication operators on numbers with those for linguistic labels in DSmH formula. In a similar way, it is also possible to extend PCR5 formula as shown with detailed examples in [14] and in section 6 of this paper. In the next section, we propose new qualitative-enriched (qe) operators for dealing with enriched linguistic labels which mix the linguistic value with either quantitative/numerical supporting degree or qualitative supporting degree as well.

The direct qualitative discounting (or reinforcement) is motivated by the fact that in general human experts provide more easily qualitative values than quantitative values when analyzing complex situations.

In this paper, both quantitative enrichments and qualitative enrichments **of linguistic labels are considered**
and unified through same general qe**-operators. The quantitative enrichment is based directly on the percentage** of discounting (or reinforcement) of any linguistic label. This is what we call a Type 1 of enriched labels. The qualitative enrichment comes from the idea of direct qualitative discounting (or reinforcement) and constitutes the Type 2 of enriched labels.

## 5 Qe**-Operators**

We propose to improve the previous q-operators for dealing now with enriched qualitative beliefs provided **from** human experts. We call these operators the qe**-operators. The basic idea is to use "enriched"-linguistic labels**
denoted Li(ǫi), where ǫi can be either a numerical supporting degree in [0, ∞**) or a qualitative supporting degre**
taken its value in a given (ordered) set X of linguistic labels. Li(ǫi**) is called the qualitative enrichment**5 of Li. When ǫi ∈ [0, ∞), Li(ǫi) is called an enriched label of Type 1, whereas when ǫi ∈ X, Li(ǫi**) is called an**
enriched label of Type 2. The (quantitative or qualitative) quantity ǫi **characterizes the weight of reinforcing or**
discounting expressed by the source when using label Li for committing its qualitative belief to a given proposition A ∈ GΘ**. It can be interpreted as a second order type of linguistic label which includes both the linguistic**
value itself but also the associated degree of confidence expressed by the source. The values of ǫi **express the**
expert's attitude (reinforcement, neutral, or discounting) to a certain proposition when using a given linguistic label for expressing its qualitative belief assignment.