- **Frame related with the position of a driver of a car**
w.r.t the given building B:
Θ3 = {B = near building, B¯ **= not** B}
The underlying models of marginal frames are based on the following very reasonable assumptions:

- Assumption 1**: We assume naturally** A ∩ A¯ =
∅ **(avoiding Shrdinger's cat paradox). If working** only with the frame of people Θ3**, the marginal** bba's must be defined on the power-set
$2^{\Theta_1}=\{\emptyset_1,A,\bar{A},A\cup\bar{A}\}$. 
- Assumption 2**: We assume also that** V ∩V¯ = ∅ so that the marginal bba (if needed) must be defined on the power-set

$$2^{\Theta_{2}}=\{\emptyset_{2},V,\bar{V},V\cup\bar{V}\}$$

- Assumption 3**: We assume also that** B∩B¯ = ∅ so that the marginal bba (if needed) must be defined on the power-set
2 Θ3 = {∅3, B, **B, B** ¯ ∪ B¯}
This modeling is disputable since the notion of closeness/"near" is not clearly defined and we could prefer to work on D
Θ3 = {∅3, B ∩ B, B, ¯ **B, B** ¯ ∪ B¯}
The emptyset elements have been indexed by the index of the frame they are referring to for notation convenience and avoiding confusion.

## 2.2 Joint Frame And Its Model

Since we need to work with all aspects of available information, we need to define a common joint frame to express all what we have from different sources of information. The easiest way for defining the joint frame, denoted Θ, is to consider the classical Cartesian (cross) product space and to work with propositions (a Lindenbaum-Tarski algebra of propositions)
since one has a correspondence between sets and propositions [5, 6], i.e.

Θ = Θ1 × Θ2 × Θ3 which consists of the following 8 triplets elements

$$\begin{array}{c}{{\Theta=\{\theta_{1}=(\bar{A},\bar{V},\bar{B}),\theta_{2}=(A,\bar{V},\bar{B}),}}}\\ {{\theta_{3}=(\bar{A},V,\bar{B}),\theta_{4}=(A,V,\bar{B}),}}\\ {{\theta_{5}=(\bar{A},\bar{V},B),\theta_{6}=(A,\bar{V},B),}}\\ {{\theta_{7}=(\bar{A},V,B),\theta_{8}=(A,V,B)\}}}\end{array}$$

We define the union ∪ , intersection ∩ **as componentwise operators in the following way:**

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(x1, x2, x3) ∪ (y1, y2, y3) , (x1 ∪ y1, x2 ∪ y2, x3 ∪ y3) (x1, x2, x3) ∩ (y1, y2, y3) , (x1 ∩ y1, x2 ∩ y2, x3 ∩ y3)
The complement X¯ of X **is defined in the usual way by**

$\phi$. 
$$y_{2},y_{3})\equiv(x_{1}\cup y_{3})$$
, ) $\triangle$ ($x\cdot\bigcap x$). 
$\bar{X}=\overline{\left(x_{1},x_{2},x_{3}\right)}\triangleq I_{t}\setminus\{X\}$.  
where It **is the total ignorance (i.e. the whole space of**
solutions) which corresponds to the maximal element defined by It = (It1, It2, It3), where Iti **is the maximal**
(ignorance) of Θi, i = 1, 2, **3. The minimum element**
(absolute empty proposition) is ∅ = (∅1, ∅2, ∅3**), where**
∅i**is the minimum element (empty proposition) of** Θi**. We also define a relative minimum element in**
S
Θ1×Θ2×Θ3 as follows: ∅r = (x, y, z**), where at least**
one of the components x, y, or z **is a minimal element**
in its respective frame Θi**. A general relative minimum** element ∅gr **is defined as the union/join of all relative**
minima (including the absolute minimum element). Similarly to the relative and general relative minimum we can define a relative maximum and a general relative maximum, where the empty set in the above definitions is replaced by the total ignorance. Whence the super-power set (S
Θ, ∩, ∪, ¯ , ∅, It**) is equivalent to**
Lindenbaum-Tarski algebra of propositions. For example, if we consider Θ1 = {x1, x2} **and Θ**2 = {y1, y2} **satisfying both Shafer's model, then Θ = Θ**1 × Θ2 = {(x1, y1),(x1, y2),(x2, y1),(x2, y2)}**, and one has:**

$$\begin{array}{l}{{\varnothing=(\varnothing_{1},\varnothing_{2})}}\\ {{\varnothing_{r1}=(\varnothing_{1},y_{1})}}\\ {{\varnothing_{r2}=(\varnothing_{1},y_{2})}}\\ {{\varnothing_{r3}=(\varnothing_{1},y_{1}\cup y_{2})}}\\ {{\varnothing_{r4}=(x_{1},\varnothing_{2})}}\\ {{\varnothing_{r5}=(x_{2},\varnothing_{2})}}\\ {{\varnothing_{r6}=(x_{1}\cup x_{2},\varnothing_{2})}}\end{array}$$

and thus

$\emptyset_{gr}=\emptyset\cup\emptyset_{r1}\cup\emptyset_{r2}\cup\ldots\cup\emptyset_{r6}$.  
Based on definition of joint frame Θ with operations on its elements, we need to choose its underlying model (Shafer's, free or hybrid model) to define its fusion space where the bba's will be defined on. According to the definition of absolute and relative minimal elements, we then assume for the given VBIED problem that Θ
satisfies Shafer's model, i.e. all (triplets) elements θi ∈
Θ are exclusive, so that the bba's of sources will be defined on the classical power-set 2Θ.