A weak order is a poset P = (V, ≺) that can be assigned a real-valued function f : V → R so that a ≺ b in P if and only if f(a) < f(b) 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 (a ǁ b). When P is not a weak order, it is not possible to resolve ties as fairly. The weak discrepancy of a poset, introduced in Trenk (1998) as the weakness 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.