A weak order is a poset
= (V, ≺) that can be assigned a real-valued function
→ R so that
in P if and only if
Bogart (1990). Thus, the elements of a weak order can be ranked by a function that respects the ordering ≺ and issues a tie in ranking between incomparable elements (
is not a weak order, it is not possible to resolve ties as fairly. The
of a poset, introduced in Trenk (1998) as the
of a poset, is a measure of how far a poset is from being a weak order [Gimbel and Trenk (1998); Tanenbaum, Trenk, & Fishburn (2001)]. In Shuchat, Shull, and Trenk (2007), the problem of determining the weak discrepancy of a poset was formulated as an integer program whose linear relaxation yields a fractional version of weak discrepancy given in Definition 1 below.