PRELIM 3 ORIE 3310/5310 April 14, 2009 Closed book exam. Justify all work. 1. A supply-demand network is a digraph G = ( V, E ) , for which V = S ∪ D ∪ I , with the disjoint subsets S, D, I denoting, respectively, supply , demand , and intermediate nodes. For each i ∈ S we are given a i >0 units of supply of some commodity and to each i ∈ D there is an associated demand, b i >0 units of the commodity. We assume that ∑ i ∈ S a i = ∑ i ∈ D b i . For this model the lower ±ow limit along any arc is 0 and there are arc capacities c ij ≥0 ∀ ( i, j ) ∈ E which limit the ±ow of the commodity along individual arcs a. (5) Give a mathematical programming formulation whose constraints (±ow conservation restric-tions and arc-±ow limitations) must be satis²ed by any feasible ±ow in G which meets the speci²ed demand from the existing supply. b. (10) Give a max-±ow model which can be used to determine whether there is a feasible ±ow in G which satis²es the speci²ed demand from the existing supply. c. (20) Show that if your model is feasible, then for any cut
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