You can compute the maximum weight of an antichain, or more generally the maximum weight of a union of $k$ antichains, by reducing to the maximum flow problem. See for example a technical report by Congthe following technical report by Cong:
Computing Maximum Weighted K-Families and K-Cofamilies in Partially Ordered Sets. J. Cong. UCLA CS Dept Technical Report CSD-930014, May 1993.