The given interval can be represented probabilistically by a uniform distribution as: p e ( x ) = { 1 x U e − x L e if x L e ≤ x ≤ x U e 0 otherwise (1) which means that the probability is positive and inversely proportional to the interval length if the value of x is in the given interval, and zero otherwise. In this case, probability represents the degree of uncertainty expressed by the interval length: narrow estimates will get a higher probability than wider ones, as they are less uncertain.

Otherwise, interval estimates can be modeled by fuzzy numbers in the possibilistic case, showing the degree of being member of the set, defined by the tuple: π e ( x ) = ( x L e − α , x L e , x U e , x U e + α ) (2) where the first and last members define the support, and the second and third members define the core of the fuzzy number, as shown in Figure 1. Inside the core, the fuzzy number takes one, and outside the support (defined by tunable α bandwidth) zero.

Notice that in this case, without restriction on the integral, the given interval defines the core of the fuzzy number, so the x values inside [ x L e , x U e ] take one in all the cases regardless the interval width.

The anomaly mentioned above, namely, that the probabilistic approach prioritizes narrow interval estimates but the possibilistic approach assigns all equal importance, increasingly prevails when the judgments of multiple experts are aggregated.

The aim of the aggregation is to summarize the judgments of multiple experts in one probability/possibility distribution. In case of the probabilistic approach, averaging the distribution functions representing individual opinions is a conventional aggregation technique: p aggr ( x ) = 1 N e ∑ e p e ( x ) (3) where N e marks the number of experts.

In this work, we use the average operator in case of the possibilistic approach as well, which avoids giving zero possibility everywhere along the domain in case of conflicting estimates (e.g., min operator), and also covering uninformatively large area by possibility value one if the estimates are well distributed, without showing preferences (e.g., max operator): π aggr ( x ) = 1 N e ∑ e π e ( x ) (4)

In the case of averaging possibility distributions, all expert judgments are considered with equal importance, as values within an interval estimate of each expert are considered with a possibility equal to one during the aggregation. Consequently, the mean curve (disregarding bandwidth) roughly represents the voting ratio for certain x values as shown on the right in Figure 2. On the other hand, the x values falling within narrow intervals are overemphasized in the probabilistic case, as they take higher probability than the values belonging to wide-interval responses as can be seen on the left in Figure 2.

Three expert judgments with different interval widths were modeled and aggregated in Figure 2. It can be noticed that x = 4.25 takes higher a probability but a lower possibility value than x = 2 . This is a key consequence of the narrow-interval prioritization effect of the probabilistic approach.

Additionally, it has to be mentioned that the aggregated curve in the probabilistic case is still a probability distribution (with integral equal to one). Meanwhile, the averaging in the possibilistic case results in a subnormal possibility distribution (with a maximum below one). If needed, it should be normalized for further calculations. ¹⁰

An advantage score is defined to express the performance difference between the aggregated probability and possibility distributions. In order to determine the extent to which the probabilistic approach outperforms the possibilistic, the accuracies of the gained aggregated distributions are compared. Their means are calculated (in the possibilistic case, it corresponds to the centroid defuzzification method) and their distance from the correct value ( x correct ) is evaluated: e p ( x ) = | E ( p aggr ( x ) ) − x correct | (5) e π ( x ) = | E ( π aggr ( x ) ) − x correct | (6) where e p and e π refer to the absolute errors of the mean of the aggregated probability and possibility distributions, respectively, and E ( · ) represents the expected value of the function in the argument.

The difference of errors defines the advantage score ( s adv ) as: s adv ( x ) = e π ( x ) − e p ( x ) f x (7)

The normalization factor f x aims to bring the error differences to a common ground, if the x variables are scaled differently; it can be, e.g., equal to the domain width of x . In this work, this normalization operation was eliminated ( f x = 1 ), as all variables were scaled equally in our case study. The s adv metric is positive if the probabilistic approach performs better than the possibilistic, and negative if the possibilistic approach shows more appropriate results.

Calculating correlation has two roles in this work. Firstly, it is used to characterize the relation between the interval length and accuracy of estimates about a variable, marked by r . Afterwards, having estimates about multiple variables, the correlation of this correlation and the advantage score belonging to each is calculated, denoted by R . The Pearson’s correlation coefficient is used for both cases.

The interval lengths and estimation errors of estimates about x from multiple experts are collected in d and e vectors, respectively: d = [ d 1 , d 2 , … , d N e ] (8) e = [ e 1 , e 2 , … , e N e ] (9) where d e = x U e − x L e , e = 1 , … , N e (10) and e e absolute errors are defined as the deviation of the middle of the interval (if a point estimate should be given, supposedly this would be that) from the correct value: e e = | x U e − x L e 2 − x correct | , e = 1 , … , N e (11)

The confidence-accuracy interdependence belonging to an x variable can be defined as the correlation between interval lengths ( d ) and estimation errors ( e ) of the estimates from multiple experts: r = cov ( d , e ) σ d σ e (12) where the standard deviations of d and e are denoted by σ d and σ e , respectively.

If r shows a strong positive correlation, that means narrower interval estimates belong to higher expertise level. However, if there is a significant negative correlation, then the interval width increases with the level of expertise, which aligns with the Dunning-Kruger effect.

Having estimates about multiple variables ( x i , i = 1 , … , N x ), we can calculate the confidence-accuracy correlation and advantage score for each, collected them as: r = [ r 1 , r 2 , … , r N x ] (13) s adv = [ s adv ( x 1 ) , s adv ( x 2 ) , … , s adv ( x N x ) ] (14)

Their relationship can also be described by a correlation coefficient ( R ) numerically: R = cov ( r , s adv ) σ r σ s adv (15)

In this work, we wanted to explore whether the distribution of answers on the Dunning-Kruger curve (characterized by r ) has any effect on the performance difference between the probabilistic and possibilistic approaches ( s adv ). This potential effect is quantified by R .

Interval estimates are available about nine key variables of a manual waste sorting system, including product purity, yields and feed composition, each with a percentage dimension. Thereby, their values are limited to between 0% and 100%. The data collection was performed in a classroom setting in which 10 students (representing experts) were involved. Ethical approval for this study was obtained from the Institutional Research Ethics Committee of the University of Pannonia (approval number: KEB 2/2024. (12.03.)).