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Overlap functions are a class of aggregation functions that measure theoverlapping degree between two values. Interval-valued overlap functions weredefined as an extension to express the overlapping of interval-valued data, andthey have been usually applied when there is uncertainty regarding theassignment of membership degrees. The choice of a total order for intervals canbe significant, which motivated the recent developments on interval-valuedaggregation functions and interval-valued overlap functions that are increasingto a given admissible order, that is, a total order that refines the usualpartial order for intervals. Also, width preservation has been considered onthese recent works, in an intent to avoid the uncertainty increase andguarantee the information quality, but no deeper study was made regarding therelation between the widths of the input intervals and the output interval,when applying interval-valued functions, or how one can control suchuncertainty propagation based on this relation. Thus, in this paper we: (i)introduce and develop the concepts of width-limited interval-valued functionsand width limiting functions, presenting a theoretical approach to analyze therelation between the widths of the input and output intervals of bivariateinterval-valued functions, with special attention to interval-valuedaggregation functions; (ii) introduce the concept of $(a,b)$-ultramodularaggregation functions, a less restrictive extension of one-dimension convexityfor bivariate aggregation functions, which have an important predictablebehaviour with respect to the width when extended to the interval-valuedcontext; (iii) define width-limited interval-valued overlap functions, takinginto account a function that controls the width of the output interval; (iv)present and compare three construction methods for these width-limitedinterval-valued overlap functions.

Towards interval uncertainty propagation control in bivariate  aggregation processes and the introduction of width-limited interval-valued  overlap functions