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Lecture18

# Lecture18 - Advanced Operations Research Techniques IE316...

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Advanced Operations Research Techniques IE316 Lecture 18 Dr. Ted Ralphs

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IE316 Lecture 18 1 Reading for This Lecture Bertsimas 7.3-7.5
IE316 Lecture 18 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 arcs T of n - 1 arcs that form a tree when their direction is ignored. Set f ij = 0 for every ( i, j ) 6∈ 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 0 is called a feasible tree solution . Theorem 1. A flow vector is a basic solution if and only if it is a tree solution.

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IE316 Lecture 18 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. Then the new flow is ˆ f kl = f kl + θ, if ( k, l ) F, f kl - θ, if ( k, l ) B, f kl , otherwise .
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Lecture18 - Advanced Operations Research Techniques IE316...

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