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 chapter we propose five versions of a Proportional Conflict Redistribution rule (PCR) for information fusion together with several examples.

In this chapter, we present two applications in information fusion in order to evaluate the generalized proportional conflict redistribution rule presented in chapter [7]. Most of the time the combination rules are evaluated only on simple examples. We study here different combination rules and compare them in terms of decision on real data. Indeed, in real applications, we need a reliable decision and it is the final results that matter. Two applications are presented here: a fusion of human experts opinions on the kind of underwater sediments depicted on a sonar image and a classifier fusion for radar targets recognition.

In this paper we propose a new family of Belief Conditioning Rules (BCR) for belief revision. These rules are not directly related with the fusion of several sources of evidence but with the revision of a belief assignment available at a given time according to the new truth (i.e. conditioning constraint) one has about the space of solutions of the problem.

This presentation may look like a glossary of the fusion rules and we also introduce new ones presenting their formulas and examples: Conjunctive, Disjunctive, Exclusive Disjunctive, Mixed Conjunctive-Disjunctive rules, Conditional rule, Dempster's, Yager's, Smets' TBM rule, Dubois-Prade's, Dezert-Smarandache classical and hybrid rules, Murphy's average rule, Inagaki-Lefevre-Colot-Vannoorenberghe Unified Combination rules [and, as particular cases: Iganaki's parameterized rule, Weighting Average Operator, minC (M. Daniel), and newly Proportional Conflict Redistribution rules (SmarandacheDezert) among which PCR5 is the most exact way of redistribution of the conflicting mass to non-empty sets following the path of the conjunctive rule], Zhang's Center Combination rule, Convolutive x-Averaging, Consensus Operator (Josang), Cautious Rule (Smets), α-junctions rules (Smets), etc. and three new T-norm & T-conorm rules adjusted from fuzzy and neutrosophic sets to information fusion (TchamovaSmarandache). Introducing the degree of union and degree of inclusion with respect to the cardinal of sets not with the fuzzy set point of view, besides that of intersection, many fusion rules can be improved.

This chapter may look like a glossary of the fusion rules and we also introduce new ones presenting their formulas and examples.

We present a new methodology for decision-making support based on belief functions thanks to a new theoretical canonical decomposition of dichotomous basic belief assignments (BBAs) that has been developed recently. This decomposition based on proportional conflict redistribution rule no 5 (PCR5) always exists and is unique. This new PCR5-based decomposition method circumvents the exponential complexity of the direct fusion of BBAs with PCR5 rule and it allows to fuse quickly many sources of evidences. The method we propose in this paper provides both a decision and an estimation of the quality of the decision made, which is appealing for decision-making support systems.