IV. CLASSICAL QUALITATIVE COMBINATION RULES
The classical qualitative combination rules are direct extensions of classical quantitative rules presented in previous section.

Since the formulas of qualitative fusion rules are the same as for quantitative rules, they will be not reported in this section. The main difference between quantitative and qualitative approaches lies in the addition, multiplication and division operators one has to use. For quantitative fusion rules, one uses addition, multiplication and division operators on numbers while for qualitative fusion rules one uses the addition, multiplication and division operators on linguistic labels defined as in section II-C.1. Example 8: Below is a very simple example used to show how classical qualitative fusion rules work. Let's consider the following set of linguistic labels L = {Lmin = L0, L1, L2, L3, L4, L5, Lmax = L6} and let's assume Shafer's model for the frame Θ = {**A, B**} we want to work on. In this example, we consider only two experts providing the qualitative belief assignments (masses) qm1(.) and qm2(.) such that:

![0_image_0.png](0_image_0.png)

The qualitative belief assignments qm1(.) and qm2(.) have been chosen quasi-normalized since L1 + L3 + L2 = L6 = Lmax and respectively L4 + L1 + L1 = L6 = Lmax.

- **Qualitative Conjunctive Rule** (QCR): This rule provides qm∧(.) according following derivations:
qm∧(A) = qm1(A)qm2(A) + qm1(A)qm2(A ∪ B) + qm2(A)qm1(A ∪ B)

 $={L}_{1}{L}_{4}+{L}_{1}{L}_{1}+{L}_{4}{L}_{2}={L}_{\frac{1\cdot4}{6}}+{L}_{\frac{1\cdot1}{6}}+{L}_{\frac{4\cdot2}{6}}={L}_{\frac{4+1+8}{6}}={L}_{\frac{13}{6}}$
$$=L_{1}L_{4}$$
$$q m_{\wedge}(B)=q m_{1}(B)q m_{2}(B)+q m_{1}(B)q m_{2}(A\cup B)+q m_{2}(B)q m_{1}(A\cup B).$$
 $$=L_3L_1+L_3L_1+L_1L_2\\ =L_{\frac{3\cdot1}{6}}+L_{\frac{3\cdot1}{6}}+L_{\frac{1\cdot2}{6}}\\ =L_{\frac{3\cdot2+3+2}{6}}\\ =L_{\frac{6}{6}}$$  . 

$\frac{1}{2}$  4. 

qm∧(A ∪ B) = qm1(A ∪ B)qm2(A ∪ B) = L2L1 = L2·1
6= L2
6
qm∧(A ∩ B) = qm1(A)qm2(B) + qm2(B)qm1(B) = L1L1 + L4L3 = L1·1
6+ L4·3
6= L**1+12**
6= L13
6
We see that not approximating the indexes (*i.e.* working with refined labels), the quasi-normalization of the qualitative conjunctive rule is kept:

$L_{\frac{13}{6}}+L_{\frac{8}{6}}+L_{\frac{2}{6}}+L_{\frac{13}{6}}=L_{\frac{36}{6}}=L_6=L_{\max}$. 
But if we approximate each refined label, we get:

$$[{\frac{1}{6}}]+[{\frac{1}{6}}]+[{\frac{1}{6}}]+[{\frac{1}{6}}]=22+21+20$$

```
L[
 13
 6
  ] + L[
      8
      6
       ] + L[
          2
          6
           ] + L[
               13
               6
                ] = L2 + L1 + L0 + L2 = L5 6= L6 = Lmax.

```

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