Lecture18 - Advanced Operations Research Techniques IE316...

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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 18 Dr. Ted Ralphs 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 is called a feasible tree solution . Theorem 1. A flow vector is a basic solution if and only if it is a tree solution. 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 variablethis is an arc not in T . Adding this arc to T forms a cycle . To increase flow on the new arc, push 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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This note was uploaded on 08/06/2008 for the course IE 316 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture18 - Advanced Operations Research Techniques IE316...

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