The idea is here to redistribute the masses of the focal elements giving a partial conflict proportionally to the initial masses on these elements.

III. A GENERAL FORMULATION FOR COMBINATION RULES
In [18], [20] we can find two propositions of a general formulation of the combination rules. In the first one, Smets considers the combination rules from a matrix notation and find the shape of this matrix according to some assumptions on the rule, such as linearity, commutativity, associativity, etc. In the second one, a generic operator is defined from the plausibility functions.

A general formulation of the global conflict repartition have been proposed in [11], [19] for all X ∈ 2 Θ by:

$$m_{c}(X)=m_{\mathrm{Conj}}(X)+w(X)m_{\mathrm{Conj}}(\varnothing),$$
where $\sum_{X\in2^{\Theta}}w(X)=1.$ The problem is the choice of the weights $w(X).$
We have proposed also a parametrized PCR to decrease or increase the influence of many small values toward one large one. The first way is given by **PCR6f**, applying a function on each belief value implied in the partial conflict. Any nondecreasing positive function f defined on ]0, 1] can be used.

$$\begin{array}{c}{{m_{\mathrm{PCRgf}}(X)=m_{\mathrm{Conj}}(X)+\sum_{i=1}^{M}m_{i}(X)f(m_{i}(X))}}\\ {{\sum_{\begin{array}{l}{{\bigcap_{i=1}^{M-1}m_{\sigma_{i}(j)}(Y_{\sigma_{i}(j)})}}\\ {{\bigcap_{k=1}^{M-1}Y_{\sigma_{i}(k)}\cap X=\emptyset}}\\ {{(Y_{\sigma_{i}(1)},\ldots,Y_{\sigma_{i}(M-1)})\in(2^{\Theta})^{M-1}}}\end{array}}}\end{array}$$

The second way, given by **PCR6g** is to apply a similar function g on the sum of belief functions given to a focal element.

mPCR6g(X)=mConj(X)+X M i=1 mi(X)X M−1 ∩ k=1 Yσi (k)∩X=∅ (Yσi (1),...,Yσi (M−1))∈(2Θ)M−1  MY−1 j=1 mσi(j)(Yσi(j)) ! Y Yσi (j)=X 1lj>i!g   mi(X)+X Yσi (j)=X mσi(j)(Yσi(j)) ! Z∈{X,Yσi (1),...,Yσi (M−1)} g  X Yσi (j)=Z mσi(j)(Yσi(j)) + mi(X)1lX=Z  X 
where 1l is the characteristic function (1lX=Z is 1 if X = Z
and 0 elsewhere). Nevertheless, here also the problem is the choice of the functions f and g.

A. How to choose conjunctive and disjunctive rules?

We have seen that conjunctive rule reduces the imprecision and uncertainty but can be used only if one of the experts is reliable, whereas the disjunctive rule can be used when the experts are not reliable, but allows a loss of specificity.

Hence, Florea [15] proposes a weighted sum of these two rules according to the global conflict k = m**Conj**(∅) given for X ∈ 2 Θ, X 6= ∅ by:

$$m_{\rm Flo}(X)=\beta_{1}(k)m_{\rm Dis}(X)+\beta_{2}(k)m_{\rm Conj}(X),\tag{15}$$  where $\beta_{1}$ and $\beta_{2}$ can admit $k=\frac{1}{2}$ as symmetric weight:  $$\beta_{1}(k)=\frac{k}{1-\frac{k}{1-k}\frac{+}{k}k^{2}},\tag{16}$$ $$\beta_{2}(k)=\frac{1-k}{1-k+k^{2}}.$$
$$(14)$$

Consequently, if the global conflict is high (k near 1) the behavior of this rule will give more importance to the disjunctive rule. Thus, this rule considers the global conflict coming from the non-reliability of the experts.

In order to take into account the weights more precisely in each partial combination, we propose the following new rule. For two basic belief assignments m1 and m2 and for all X ∈ 2 Θ we have:

$X\in\mathbb{Z}^{-}$ we have:  $$\begin{array}{rcl}m_{\rm Mix}(X)&=&\sum_{A\cup B=X}\delta_{1}m_{1}(A)m_{2}(B)\\ &&+\sum_{A\cap B=X}\delta_{2}m_{1}(A)m_{2}(B).\end{array}\tag{17}$$  Of course, if $\delta_{1}=\beta_{1}(k)$ and $\delta_{2}=\beta_{2}(k)$ we obtain the 
Florea's rule. In the same manner, if δ1 = 1−δ2 = 0 we obtain the conjunctive rule and if δ1 = 1−δ2 = 1 the disjunctive rule.

If δ1(A, B) = 1−δ2(A, B) = 1lA∩B=∅ we retrieve the Dubois and Prade's rule and the partial conflict can be considered, whereas the rule (15).

The choice of δ1 = 1 − δ2 can be done by a dissimilarity such as:

$$\delta(A,B)=1-{\frac{|A\cap B|}{\operatorname*{min}(|A|,|B|)}},$$
$$(18)$$

where |A| is the cardinality of A. Note that is not a distance nor a proper dissimilarity (*e.g.* δ(A, B**) = 0** does not imply A = B). We can also take for δ2, the Jaccard distance given by:

$$d(A,B)={\frac{|A\cap B|}{|A\cup B|}},$$
$$(19)$$

used by [23] on the belief functions. Thus, if we have a partial conflict between A and B, |A ∩ B| = 0 and the rule transfers the mass on A ∪ B. In the case A ⊂ B (or the contrary),
A ∩ B = A and A ∪ B = B, so with δ the rule transfers the mass on A and with d on A and B according to the rate |A|/|B| of the cardinalities. In the case A ∩ B 6= **A, B** and ∅,
the rule transfers the mass on A ∩ B and A ∪ B according to δ and d.