Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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3) arcs. We represent with (i, j) the arc that joins i to j. Assume that for any pair of nodes there is at most one arc joining them. Consider the following logistics problem involving shipments of a certain product: Let G = (N, A) be a graph where N is the set of nodes and A is the set of Max. Eis)ea bijaij EKEN+ (1) "ik – Eken-(1) Tki = d;, i E V, 0 < xij < Cij, (i, j) E A, s.t. where: is the amount of product that is sent from node i to node j. • Xij bij is the profit of sending one unit of product through arc (i, j). • d; is the offer (if positive) or demand (if negative) of node i, an integer value. • Cij is the maximum amount of product that can circulate through arc (i, j), a nonnegative integer value. • N+(i) = {k E N / (i, k) E A}. • N-(i) = {k € N / (k, i) E A}. . Σεn d 0. Prove that this formulation has an integer optimal solution ã, that is, ñij E Z+ v(i,j) e A.
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Transcribed Image Text:3) arcs. We represent with (i, j) the arc that joins i to j. Assume that for any pair of nodes there is at most one arc joining them. Consider the following logistics problem involving shipments of a certain product: Let G = (N, A) be a graph where N is the set of nodes and A is the set of Max. Eis)ea bijaij EKEN+ (1) "ik – Eken-(1) Tki = d;, i E V, 0 < xij < Cij, (i, j) E A, s.t. where: is the amount of product that is sent from node i to node j. • Xij bij is the profit of sending one unit of product through arc (i, j). • d; is the offer (if positive) or demand (if negative) of node i, an integer value. • Cij is the maximum amount of product that can circulate through arc (i, j), a nonnegative integer value. • N+(i) = {k E N / (i, k) E A}. • N-(i) = {k € N / (k, i) E A}. . Σεn d 0. Prove that this formulation has an integer optimal solution ã, that is, ñij E Z+ v(i,j) e A.
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