A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case

Aggregation functions with given super-additive and sub-additive transformations

“…Our main results in Theorems 1 -3 are related to the latest results of [10,11] that were based on the extra assumption of overrunning and underrunning some super-additive and sub-additive function, respectively. Namely, every function that overruns (underruns) a super-additive (sub-additive) function is automatically strictly super-additive (strictly subadditive), as it was noted in [10] in dimension one and in [11] in the multi-dimensional case.…”

“…We prove these results as corollaries of Theorems 1 and 2, in section 2. In the concluding section 3 we show that our results are, in a sense, best possible, as they fail to hold if the continuity or strict super-(or sub-) additivity assumptions are dropped, and we also discuss the relationship between our new conditions and the ones of [10,11].…”

“…Returning to the negative answer to our question, in [13] it was proved that if f and g are functions [0, ∞[ n → [0, ∞[ with zero value at the origin and f ≥ g, such that f is strictly directionally convex and g is sub-additive but not linear, then there is no aggregation function A on [0, ∞[ n such that A * = f and A * = g. This result was generalized in [10,11] by showing that strict directional convexity of f may be replaced by a weaker condition of f overrunning some super-additive function. In both cases, 'dual' versions of these results were also proved (assuming that, for a non-linear f , the function g is strictly directionally concave or underrunning some sub-additive function, with the obvious meaning of these concepts).…”

“…Since ∇ A ·x is a linear function, we have (∇ A ·x) * = ∇ A ·x, and so A * (x) ≥ ∇ A ·x for every x ∈ [0, ∞[ n . In conjunction with (10) …”

“…Further, we say that a function f on [0, ∞[ n overruns some super-additive function if there exists a super-additive function h on [0, ∞[ n such that f /h is strictly increasing in every coordinate on ]0, ∞[ n . The latter property is weaker than the former (and the two are not equivalent), as shown in [10,11].…”