Theory

Theory

Conjunctive fusion is an intersection of fuzzy sets. This kind of fusion assumes the information sources in full agreement: the sets associated to the value of parameter x must overlap widely.

Among the various types of conjunctive fusion, the minimum function corresponds to a cautious approach. The source assigning the lowest degree of possibility to the value of parameter x is assumed the best informed. Note that minima do not introduce a strengthening effect between sources when they provide the same information.

**Figure 24:** Conjunctive fusion.
$\begin{figure}\begin{center}\fbox{\begin{tabular}{cc}\subfigure[Total confli......[No conflict]{\epsfbox{c3-conj4.eps}} \\\end{tabular}}\end{center}\end{figure}$

Let us then assume we have two information sources noted source 1 and source 2. These sources give their opinion on the value of parameter x from the possibility distributions noted $\pi_1$ and $\pi_2$ . Conjunctive fusion using the operator ``minimum'' is the following:

$\begin{displaymath}\forall \omega \in \Omega, \quad\pi_{\text{conj}}(\omega) = \min ( \pi_1(\omega), \pi_2(\omega) )\end{displaymath}$

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Figure 24 shows how conjunctive fusion behaves using the operator ``minimum'' in several cases. The degree of possibility that element $\omega \in \Omega$ is the true value of x is denoted $\mu$ .

(a): When sources are in total conflict, the conjunctive fusion is no longer suitable. The hypothesis of basic conjunctive fusion ensures that sources agree and that the information given by them overlaps widely. It is then no longer possible, with the two sources, to find a common opinion. So fusion fails.
(b): Conflict is not total, but strong because the intersection between distributions given by each source is very small. As conjunctive fusion is strict, the result is limited to the small common area between the sets. It is not very plausible that the true value of parameter x can be found in this intersection area.
(c): Conflict between sources decreases. However, they do not agree widely. As in case (b), but to a lesser extent, intersection between two intervals gives only one interval where parameter x has little chance of being found.
(d): This case, where the sources agree widely, is the only case where conjunctive fusion is appropriate. The hypothesis of agreeing sources seems credible as they are often in agreement. Conjunctive fusion provides an acceptable possibility distribution.

We note that conjunctive fusion does not always provide a normalized possibility distribution. A distribution is normalized when $\exists \omega \in \Omega$ such that $\pi(\omega) = 1$ . Conjunctive fusion provides a normalized distribution only when the possibility distributions given by each source overlap widely. In any other case, either sub-normalized distribution (conflict between sources), or no distribution (no common opinions given by sources) is reached.

So, obtaining a sub-normalized distribution shows there is conflict between sources. The less the result is normalized, the higher the conflict and the less the assumption of agreeing sources is credible. If there is conflict between sources, two solutions are conceivable:

either the sources are still assumed to agree; one solution consists in renormalizing the result of conjunctive fusion in order to find another possibility distribution (i.e. such that $\exists \omega \in \Omega$ with $\pi(\omega) = 1$ ) ;
or one source is assumed to make a mistake, in which case one solution consists in using disjunctive fusion.

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