# A New Generalization Of The Proportional Conflict Redistribution Rule Stable In Terms Of Decision

Arnaud Martin and Christophe Osswald October 29, 2018

## Abstract

In this chapter, we present and discuss a new generalized proportional conflict redistribution rule. The Dezert-Smarandache extension of the Demster-Shafer theory has relaunched the studies on the combination rules especially for the management of the conflict. Many combination rules have been proposed in the last few years. We study here different combination rules and compare them in terms of decision on didactic example and on generated data. Indeed, in real applications, we need a reliable decision and it is the final results that matter. This chapter shows that a fine proportional conflict redistribution rule must be **preferred for**
the combination in the belief function theory.

Keywords: **Experts fusion, DST, DSmT, generalized PCR, DSmH.**

## 1 Introduction

Many fusion theories have been studied for the combination of the experts opinions such as voting rules [24, 10], possibility theory [26, 6], and belief function theory [2, 14]. We can divide all these fusion approaches into four steps: the modelization, the parameters estimation depending on the model (not always necessary), the combination, and the decision. The most difficult step is presumably the first one. If both possibility and probability-based theories can modelize imprecise and uncertain data at the same time, in a lot of applications, experts can express their certitude on their perception of **the reality. As**
a result, probabilities theory such as the belief function theory is more adapted. In the context of the belief function theory, the Dempster-Shafer theory (DST)
[2, 14] is based on the use of functions defined on the power set 2Θ **(that is the**
set of all the disjunctions of the elements of Θ). Hence the experts can express their opinion not only on Θ but also on 2Θ **as in the probabilities theory. The**
extension of this power set into the hyper power set DΘ **(that is the set of all**
the disjunctions and conjunctions of the elements of Θ) proposed **by Dezert and** Smarandache [3], gives more freedom to the expert. This extension **of the DST**
is called Dezert-Smarandache Theory (DSmT).

This extension has relaunched the studies on the combination rules. The combination of multiple sources of information has still been an important subject of research since the proposed combination rule given by Dempster
[2]. Hence, many solutions have been studied in order to manage the conflict [25, 7, 9, 21, 22, 11, 12, 18, 8]. These combination rules are the **most of**