*3.3. Majority-Guided Fusion*

In essence, fusion rules, which focus and prioritise the consensus set—often also referred to as majority observation—fall under the category of majority-guided fusion. Majority-guided fusion is particularly sensible in cases in which information sources are known to produce consistent items. Possibility distributions deviating from the consensus set are then deduced to be faulty (unreliable) instead of giving useful information about the unknown value *v*.

With this in mind, Dubois' fusion rule (9) already satisfies as a majority-guided fusion rule because it ignores all inconsistent information items (although this fact is precisely one of the main points of criticism by Oussalah et al. [64]). In the specific case of assuming fully reliable sources and expecting consistency between items, it is reasonable to rely on simpler fusion rules; accordingly, it was proposed to use a purely conjunctive fusion rule [23]. Similarly simple are counting fusion functions; the result here is the alternative that most sources consider possible [5].

Estimation fusion rules, such as (13), favour the majority observation because of the averaging characteristic of the estimation operator F. A more complex majority-guided fusion rule, which is based on Yager's estimation fusion (13), was proposed by Glock et al. [67], the *majority-opinion-guided possibilistic fusion rule* (MOGPFR). The MOGPFR replaces both

the conjunctive fusion part G and the estimation operator F with the *Implicative Importance Weighted Ordered Weighted Averaging* (IIWOWA) operator. The IIWOWA operator, as proposed by [68], is an extension of the parent class of Ordered Weighted Averaging (OWA) operators [50]. An OWA operator allows weighting inputs with **w** = (*w*1, ... , *wn*), *wi* ∈ [0, 1], and ∑*<sup>i</sup> wi* = 1. Inputs *π<sup>i</sup>* are ordered in descending order. This results in aggregation <sup>1</sup> *<sup>n</sup>* ∑ *wi* · *π<sup>i</sup>* and allows the aggregation to be shifted between the minimum with **w** = (0, 0, ... , 1) and maximum **w** = (1, ... 0, 0). The MOGPFR is then defined as follows:

$$
\pi^{(\text{fu})}(\mathbf{x}) = \max\_{i} (\text{rel}\_{i}) \cdot \hat{\pi}^{(fu)}(\mathbf{x}) + 1 - \max\_{i} (\text{rel}\_{i})\text{, with}
$$

$$\hat{\pi}^{(fu)}(\mathbf{x}) = \left\{ \frac{\lambda\_{\text{IIWOWA}}\left(\mathbf{v}, \mathbf{w}\_{\text{P}'} \pi\_i(\mu^{(i)})\right)}{\lambda\_{\text{IWOWA}}\left(\mathbf{v}, \mathbf{w}\_{\text{m}'} \mu^{(i)}\right)} \right\};\tag{14}$$

in which *λ*IIWOWA(◦) denotes the IIWOWA operator, and *reli* is the reliability for each source. The MOGPFR specifically allows the control of fusion by (i) a reliability vector **v** = {*v*1, *v*2, ... , *vn*} with *vi* ∈ [0, 1], which discounts informations items and (ii) two weighting vectors, **w***<sup>p</sup>* and **w***m*, which control whether G and F are close to the minimum or maximum operator, respectively. The IIWOWA operator is defined only for inputs in [0, 1], which necessitates the fuzzification of *X* so that the possibility distributions become *π<sup>i</sup> μ*(*i*) .

The MOGPFR facilitates the prioritisation of information items belonging the majority observation. The importance values *vi* are determined by a distance function of *π<sup>i</sup>* to the majority set; the possibility distribution *π<sup>i</sup>* is discounted accordingly. The parameters **w***<sup>p</sup>* and **w***m* allow adapting fusion towards conjunctive and disjunctive behaviour. The benefit gained by the MOGPFR lies in its level of control through parametrisation.
