Towards interval uncertainty propagation control in bivariate
  aggregation processes and the introduction of width-limited interval-valued
  overlap functions
  Overlap functions are a class of aggregation functions that measure the
overlapping degree between two values. Interval-valued overlap functions were
defined as an extension to express the overlapping of interval-valued data, and
they have been usually applied when there is uncertainty regarding the
assignment of membership degrees. The choice of a total order for intervals can
be significant, which motivated the recent developments on interval-valued
aggregation functions and interval-valued overlap functions that are increasing
to a given admissible order, that is, a total order that refines the usual
partial order for intervals. Also, width preservation has been considered on
these recent works, in an intent to avoid the uncertainty increase and
guarantee the information quality, but no deeper study was made regarding the
relation between the widths of the input intervals and the output interval,
when applying interval-valued functions, or how one can control such
uncertainty propagation based on this relation. Thus, in this paper we: (i)
introduce and develop the concepts of width-limited interval-valued functions
and width limiting functions, presenting a theoretical approach to analyze the
relation between the widths of the input and output intervals of bivariate
interval-valued functions, with special attention to interval-valued
aggregation functions; (ii) introduce the concept of $(a,b)$-ultramodular
aggregation functions, a less restrictive extension of one-dimension convexity
for bivariate aggregation functions, which have an important predictable
behaviour with respect to the width when extended to the interval-valued
context; (iii) define width-limited interval-valued overlap functions, taking
into account a function that controls the width of the output interval; (iv)
present and compare three construction methods for these width-limited
interval-valued overlap functions.
