## A Simple Proportional Conflict Redistribution Rule

**Authors:** Florentin Smarandache, Jean Dezert

One proposes a first alternative rule of combination to WAO (Weighted Average Operator) proposed recently by Josang,
Daniel and Vannoorenberghe, called Proportional Conflict Redistribution rule (denoted PCR1). PCR1 and WAO are particular cases of
WO (the Weighted Operator) because the conflicting mass is redistributed with respect to some weighting factors. In this first PCR rule,
the proportionalization is done for each non-empty set with respect to the non-zero sum of its corresponding mass matrix - instead of its
mass column average as in WAO, but the results are the same as Ph. Smets has pointed out. Also, we extend WAO (which herein gives
no solution) for the degenerate case when all column sums of all non-empty sets are zero, and then the conflicting mass is transferred
to the non-empty disjunctive form of all non-empty sets together; but if this disjunctive form happens to be empty, then one considers
an open world (i.e. the frame of discernment might contain new hypotheses) and thus all conflicting mass is transferred to the empty
set. In addition to WAO, we propose a general formula for PCR1 (WAO for non-degenerate cases). Several numerical examples and
comparisons with other rules for combination of evidence published in literature are presented too. Another distinction between these
alternative rules is that WAO is defined on the power set, while PCR1 is on the hyper-power set (Dedekind's lattice). A nice feature of
PCR1, is that it works not only on non-degenerate cases but also on degenerate cases as well appearing in dynamic fusion, while WAO
gives the sum of masses in this cases less than 1 (WAO does not work in these cases). Meanwhile we show that PCR1 and WAO do not
preserve unfortunately the neutrality property of the vacuous belief assignment though the fusion process. This severe drawback can
however be easily circumvented by new PCR rules presented in a companion paper.

**Comments:** 21 pages

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### Submission history

[v1] 8 Mar 2010

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