Lecture17 - Introduction to Mathematical Programming IE406...

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Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 17 Dr. Ted Ralphs IE406 Lecture 17 1 Reading for This Lecture • Bertsimas 7.3-7.5 IE406 Lecture 17 2 Tree Solutions • From now on, we assume that ∑ i ∈ N b i = 0 and that G is connected. • A flow vector f is called a tree solution if it can be constructed by the following procedure: – Pick a set of n- 1 arcs T that form a tree when their direction is ignored. – Set f ij = 0 for every ( i, j ) ∈ T . – Use the flow balance equations ˜ Af = ˜ b to determine the values of the flow variables f ij , ( i, j ) ∈ T . • Note that the flow balance equations always have a unique solution. • A tree solution that also satisfies f ≥ is called a feasible tree solution . Theorem 1. A flow vector is a basic solution if and only if it is a tree solution. IE406 Lecture 17 3 Network Simplex Method • We now introduce a simple version of the simplex method for solving network flow problems . • We have already seen what basic solutions look like. • How do we change the basis? – Choose a nonbasic variable—this is an arc not in T . – Adding this arc to T forms a cycle . – To increase flow on the new arc, push θ units of flow around the cycle. – Let F be the set of forward arcs and B be the set of backward arcs....
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Lecture17 - Introduction to Mathematical Programming IE406...

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