  This paper considers the arbitrary-proportional finite-set-partitioning
problem which involves partitioning a finite set into multiple subsets with
respect to arbitrary nonnegative proportions. This is the core art of many
fundamental problems such as determining quotas for different individuals of
different weights or sampling from a discrete-valued weighted sample set to get
a new identically distributed but non-weighted sample set (e.g. the resampling
needed in the particle filter). The challenge raises as the size of each subset
must be an integer while its unbiased expectation is often not. To solve this
problem, a metric (cost function) is defined on their discrepancies and
correspondingly a solution is proposed to determine the sizes of each subsets,
gaining the minimal bias. Theoretical proof and simulation demonstrations are
provided to demonstrate the optimality of the scheme in the sense of the
proposed metric.
