
## 5.2 Qualitative Belief Conditioning Rule No 2 (Qbcr2)

The second QBCR, denoted QBCR2, does a uniform redistribution of **masses, as follows:**

- transfer the mass of each element Y in D2 ∪ D3 to the largest element X in D1 **which is contained by** Y
(as QBCR1 does);
- if no such X element exists, then the mass of Y is uniformly redistributed to all subsets of A **whose**
(qualitative) masses are not L0 **(i.e. to all qualitative focal elements included in** A).

- if there is no qualitative focal element included in A, then the mass of Y **is transferred to** A.
The mathematical formula for QBCR2 is then given by:

- If X /∈ D1,
$$qm_{Q B C R2}(X|A)=L_{\min}\equiv L_{0}$$  $$qm_{Q B C R2}(X|A)=qm(X)+qS_{1}(X,A)+qS_{3}(X,A)+qS_{4}(X,A)$$
- If X ∈ D1,
$$\left(7\right)$$
6
where the addition operator involved in (8) corresponds to the addition operator on linguistic labels **defined in**
section 2 and where the qualitative summations qS1(X, A) is defined in (5), qS3(X, A) and qS4(X, A**) by:**

$$({\mathfrak{s}})$$
$$qS_{3}(X,A)\stackrel{{\triangle}}{{=}}\sum_{\begin{subarray}{c}Y\in D_{2}\cup D_{3}\\ Y\cap A=\emptyset\\ q_{F}\neq0\end{subarray}}\frac{qm(Y)}{q_{F}}$$  $$qS_{4}(X,A)\stackrel{{\triangle}}{{=}}\sum_{\begin{subarray}{c}Y\in D_{2}\cup D_{3}\\ Y\cap A=\emptyset\\ X=A,q_{F}=0\end{subarray}}qm(Y),$$
$$({\mathfrak{g}})$$

$$(10)$$

where qF , Card{Z|Z ⊂ A, qm(Z) 6= L0} **= number of qualitative focal elements of** A.

## Scalar Division Of Linguistic Label

For the complete derivation of (8) we need to define the scalar division of labels involved in (9). We propose the following definition:

$${\frac{L_{i}}{j}}\triangleq L_{[{\frac{i}{j}}]}$$
]**(11)**

```
for all i ≥ 0 and j > 0 where [ ij
                                      ] is the integer part of ij
                                                                  , i.e. the largest integer less than or equal to ij
                                                                                                                          . For
example, L5
            3 = L[
                    5
                     
                    3
                     
                     ] = L1, or L6
                                   3 = L[
                                           6
                                            
                                           3
                                            
                                            ] = L2, etc.

```

## 6 Examples For Qbcr1 And Qbcr2

Let's consider the following set of ordered linguistic labels L = {L0, L1, L2, L3, L4, L5, L6} **(for example,** L1, L2, L3, L4 and L5 **may represent the values:** L1 , very poor, L2 , poor, L3 , medium, L4 , good and L5 , very good, where the symbol , **means** by definition**). The addition and multiplication tables corresponds**
respectively to Tables 1 and 2.