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.
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Shuchat, A., Shull, R., Trenk, A.N. (2009). Fractional Weak Discrepancy of Posets and Certain Forbidden Configurations. In: Brams, S.J., Gehrlein, W.V., Roberts, F.S. (eds) The Mathematics of Preference, Choice and Order. Studies in Choice and Welfare. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79128-7_16
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