
A′ B′ C′ D′ **P IC**(.)

P rBel(.) NaN NaN 0.3000 0.7000 NaN

P rNP l(.) 0.2728 0.1818 0.1818 0.3636 0.0318 CuzzP(.) 0.2000 0.1333 0.2667 0.4000 0.0553 BetP(.) 0.2084 0.1250 0.2583 0.4083 0.0607 P raP l(.) 0.1636 0.1091 0.3091 0.4182 0.0872 P rP l(.) 0.2035 0.0848 0.2404 0.4713 0.1124 P rHyb(.) 0.1339 0.0583 0.2656 0.5422 0.1928

DSmPǫ=0.001(.) 0.0025 0.0017 0.2996 0.6962 0.5390

TABLE XI

RESULTS FOR EXAMPLE 8.

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

## D. Example 9 (Free Dsm Model)

We consider the free DSm model depicted on Figure 2 with the input masses given in Table XII. To apply Sudano's and Cuzzolin's mappings, one works on the refined frame Θref =
{A′, B′, C′, D′, E′, F′, G′} where the elements of Θref are exclusive (assuming such refinement has a physically sense) according to Figure 2. This refinement step is not necessary when using **DSmP** since it works directly on DSm free model. The PIC values obtained with the different mappings are given in Table XIII. One sees that **DSmP**ǫ→0 provides here again the best results in term of PIC.

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

Fig. 2. Free DSm model for a 3D frame for example 9.

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

## Ix. Extension Of Dsmp For Qualitative Belief

A. Qualitative belief assignment qm(.)
In order to compute directly with words (linguistic labels),
Smarandache and Dezert have defined in [7] a *qualitative* basic belief assignment qm(.) as a mapping function from

Transformations **P IC**(.)

P rBel(.) NaN

P rNP l(.) 0.0414 CuzzP(.) 0.0621 P raP l(.) 0.0693 BetP(.) 0.1176 ![0_image_0.png](0_image_0.png)

DSmPǫ=0.001(.) 0.8986

TABLE XIII

RESULTS FOR EXAMPLE 9.

GΘ into a set of linguistic labels L = {L0**,L, L** ˜n+1} where L˜ = {L1, · · · , Ln} is a finite set of linguistic labels and where n ≥ 2 is an integer. For example, L1 can take the linguistic value "poor", L2 the linguistic value "good", etc.

L˜ is endowed with a total order relationship ≺, so that L1 ≺ L2 **≺ · · · ≺** Ln. To work on a true closed linguistic set L under linguistic operators, L˜ is extended with two extreme values L0 = Lmin and Ln+1 = Lmax, where L0 corresponds to the minimal qualitative value and Ln+1 corresponds to the maximal qualitative value, in such a way that L0 ≺ L1 ≺ L2 **≺ · · · ≺** Ln ≺ Ln+1, where ≺ means inferior to, or less
(in quality) than, or smaller than, etc.

## B. Operator On Qualitative Labels

From the extension of the isomorphism between the set of linguistic equidistant labels and a set of numbers in the interval
[0, 1], one can built exact operators on linguistic labels which makes possible the extension all the quantitative fusion rules and probabilistic transformations into their qualitative counterparts [3]. We briefly remind the main qualitative operators
(or q-operators for short) on linguistic labels:
- q-addition:

$$L_{i}+L_{j}={\begin{cases}L_{i+j}&{\mathrm{if~}}i+j<n+1,\\ L_{n+1}=L_{\operatorname*{max}}&{\mathrm{if~}}i+j\geq n+1.\end{cases}}$$
$$(14)$$

The q-addition is an extension of the addition operator on equidistant labels which is given by Li +Lj = i n+1 +
j n+1 =
i+j n+1 = Li+j .

- q-subtraction:

$$L_{i}-L_{j}={\begin{cases}L_{i-j}&{\mathrm{if}}\quad i\geq j,\\ -L_{j-i}&{\mathrm{if}}\quad i<j.\end{cases}}$$
$$(15)$$
where $-L=\{-L_{1},-L_{2},\ldots,-L_{n},-L_{n+1}\}$. The $q$-
where −L = {−L1, −L2**, . . . ,** −Ln, −Ln+1}. The qsubtraction is justified since when i ≥ j, one has with equidistant labels Li − Lj = i n+1 −j n+1 =
i−j n+1 .

- q-multiplication13:

$=\;\hdots$ . 
$$(16)$$
$$L_{i}\cdot L_{j}=L_{[(i\cdot j)/(n+1)]}.$$
13The q-multiplication of two linguistic labels defined here can be extended directly to the multiplication of n > 2 linguistic labels. For example the product of three linguistic label will be defined as Li · Lj · Lk =
L[(i·j·k)/(n+1)(n**+1)]**, etc.