masses and cardinalities

zθ6 0 ⊞ 0.001
=

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

′′
whence

zθ6 = 0.001  [0.302943, 1.**20385]**
= [0.000303, 0.**001204]**
y
′′
θ8 = [0.195250, 0.330250] [0.302943, 1.**20385]**
= [0.0591496, 0.**3975714]**
The focal element f3 = θ4 ∪ θ8 **which is also equal to** θ1 ∪ θ2 ∪ θ3 ∪ θ5 ∪ θ6 ∪ θ7 **is redistributed back to** θ1, θ2, θ3, θ5, θ6 and θ7 **directly proportionally to their** corresponding masses and cardinalities

 $=\dfrac{\overbrace{0\boxplus0.001}^{-16}}{0.001}=\dfrac{\overbrace{0\boxplus0.001}^{-17}}{0.001}\\ =\dfrac{m_{P C R6}(\overbrace{\theta_4\cup\theta_8}^{(\theta_4\cup\theta_8)}}{0.006}\\ =\dfrac{[0.106767,0.226620]}{0.006}\\ =[17.7945,37.770]$

![0_image_3.png](0_image_3.png)

wθ1 0 ⊞ 0.001

![0_image_2.png](0_image_2.png)

![0_image_1.png](0_image_1.png)

Since all are equal, we get

wθ1 = wθ2 = wθ3 = wθ5 = wθ6 = wθ7
$$\begin{array}{l l}{{\mathrm{~\tilde{~}~}}}&{{=0.001\left[\right.\left[17.7945,37.770\right]}}\\ {{\mathrm{~\tilde{~}~}}}&{{=\left[0.0177945,0.03777\right]}}\end{array}$$
The total ignorance It = θ1∪θ2∪θ3∪θ4∪θ5∪θ6∪θ7∪θ8
is redistributed back to all eight elements of the frame
Θ directly proportionally to their corresponding masses
and cardinalities
$$\frac{v_{\theta_{1}}}{0\boxplus0.001}$$
= $ \frac{v_{\theta_2}}{0\boxplus0.001}=\frac{v_{\theta_3}}{0\boxplus0.001}=\frac{v_{\theta_4}}{0\boxplus0.001}\\ =\frac{v_{\theta_5}}{0\boxplus0.001}=\frac{v_{\theta_6}}{0\boxplus0.001}=\frac{v_{\theta_7}}{0\boxplus0.001}\\ =\frac{v_{\theta_8}}{[0.194250,0.329250]\boxplus0.001}\\ =\frac{m_{PCR6}(I_t)}{[0.194250,0.329250]\boxplus0.008}\\$. 

=
[0.100351, 0.**236255]**
[0.202550, 0.**337250**
= [0.297557, 1.**16813]**
whence

Page 22
$${\boldsymbol{u}}\times{\boldsymbol{1}}$$
$$\operatorname{SS}13]$$
$=\;0$. 
$\equiv$ . 
vθ1 = vθ2 = vθ3 = vθ4 = vθ5 = vθ6 = vθ7
= 0.001  [0.297557, 1.**16813]**
= [0.000298, 0.**001168]**
vθ8 = [0.195250, 0.330250]  [0.297557, 1.**16813]**
= [0.058098, 0.**385775]**
The imprecise DSmP probabilities are computed by DSmPǫ,P CR6(θ1) = wθ1 ⊞ vθ1 DSmPǫ,P CR6(θ2) = wθ2 ⊞ vθ2 DSmPǫ,P CR6(θ3) = wθ3 ⊞ vθ3 DSmPǫ,P CR6(θ4) = xθ4 ⊞ vθ4 DSmPǫ,P CR6(θ5) = wθ5 ⊞ vθ5 DSmPǫ,P CR6(θ6) = zθ6 ⊞ wθ6 ⊞ vθ6 DSmPǫ,P CR6(θ7) = wθ7 ⊞ vθ7 DSmPǫ,P CR6(θ8) = y
′
θ8 ⊞ y
′′
θ8 ⊞ vθ8 which are summarized9**in Table 61 below.**

| Singletons   | DSmPǫ,P CR6(.)   |
|--------------|------------------|
| θ1           | [0.0181,0.0389]  |
| θ2           | [0.0181,0.0389]  |
| θ3           | [0.0181,0.0389]  |
| θ4           | [0.0008,0.0028]  |
| θ5           | [0.0181,0.0389]  |
| θ6           | [0.0184 0.0402]  |
| θ7           | [0.0181,0.0389]  |
| θ8           | [0.2072,1]       |

Table 61: Imprecise DSmPǫ of m0 ⊕ m1 ⊕ m2 ⊕ m3 for Table 59.

$\quad\quad\bullet\quad\bullet\quad\bullet$ 4. 
- Answer to Q1: **As we have shown, it is possible to fuse imprecise bba's with PCR6, and PCR5**
too (see [6], Vol. 2) to get an imprecise result for decision-making support under uncertainty and imprecision. It is also possible to compute the exact imprecise values of DSmP if necessary. According to our analysis and our results, and using either the max of Bel, the max of Pl of the max of DSmP
criterion, the decision will be to evacuate the building B**. Of course, a similar analysis can be done**
to answer to the question Q2 when working with imprecise bba's, and for for computing imprecise BetP values as well.

## 4 Qualitative Approach

In this section we just show how the fusion and decisionmaking can be done using qualitative information expressed with labels. In our previous examples the quantitative baa's have been defined ad-hoc in satisfying 9Actually for θ8, one gets with exact calculus of imprecision DSmP**ǫ,P CR**6(θ8) = [0.2072, 1.3281], but since a probability cannot be greater than 1, the upper bound of imprecision interval has been set to 1.