Abstract: The double-quantification with precision and grade acts as a novel project in the approximate space．By the Cartesian-Product combination of quantitative information,this paper aimed to explore a double-quantitative boundary and its algorithms based on the variable precision upper approximation and grade lower approximation．First,a double-quantitative expansion-model was naturally constructed by the two approximations,and the double-quantitative expansion-boundary was correspondingly defined．Then,the double-quantitative semantics was analyzed for the boundary,and its precise description and mathematical properties were obtained,in order to calculate the boundary.The approximation-set algorithm and information-granule algorithm were proposed,analyzed and compared.The information-granule algorithm has more advantages on the space complexity．Finally,a medical example was provided to illustrate the boundary and its algorithms．The boundary expands the Pawlak-boundary,and makes the complete and fine double-quantitative descriptions for partial uncertainty,thus,it has a great significance for the uncertainty analyses and applications with respect to the double-quantification.

Key words: Rough set,Granular computing,Uncertainty,Double-quantification,Boundary