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Lecture19 - Advanced Operations Research Techniques IE316...

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Advanced Operations Research Techniques IE316 Lecture 19 Dr. Ted Ralphs
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IE316 Lecture 19 1 Reading for This Lecture Bertsimas 7.4-7.5
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IE316 Lecture 19 2 Negative Cost Cycle Algorithm Recall the negative cost cycle algorithm from last lecture. The basic algorithm consisted of iteratively saturating unsaturated negative cost cycles. Questions to be Answered How do we start the algorithm (find a feasible flow)? How do we find unsaturated, negative cost cycles? If there are no unsaturated negative cost cycles, is the current solution optimal? Is the algorithm guaranteed to terminate?
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IE316 Lecture 19 3 The Residual Network We are only interested in arcs on which flow can be pushed (forwards or backwards). Corresponding to a given flow f , we build a residual network . For each arc ( i, j ) in the original network, we introduce two arcs into the residual network: an arc ( i, j ) with capacity u ij - f ij and cost c ij , and an arc ( j, i ) with capacity f ij and cost - c ij . We delete arcs that have zero capacity from the network. Flow on the arc ( j, i ) in the residual network can be interpreted as a reduction in the flow on arc ( i, j ) in the original network.
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IE316 Lecture 19 4 Interpreting the Residual Network Consider a feasible flow f and another feasible flow f + ¯ f . The flow increment ¯ f can be associated with a flow ˜ f in the residual network.
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