lec14 - 15.053 April 5 2007 z The Minimum Cost Flow Problem...

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1 15.053 April 5, 2007 z The Minimum Cost Flow Problem
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To do. z Add an extra slide showing that the flows for the baseball elimination problem and how it leads to show that Tampa can win. z Add the matching problem from the previous lecture and match it up to the 0-1 matrix form. Then give the duality result in terms of eliminating the three popular stars. z check out who wants a piece of candy. 2
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Quotes of the day A process cannot be understood by stopping it. Understanding must move with the flow of the process, must join it and flow with it. -- Frank Herbert No question is so difficult to answer as that to which the answer is obvious. -- George Bernard Shaw taken from www.flowgo.com 3
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An Application The year is 2010, and there is incredible demand for the 15.053 live podcasts. MIT has set up three sites to handle the incredible load. The major demand for the podcasts are in London and China. There are some direct links from each of the three MIT sites, and the podcasts can be sent through two intermediate satellite dishes as well. Each node has a supply (or a demand) indicating how much should be shipped from (or to) the node. Each link (arc) has a unit cost of shipping flow, and a capacity on how much can be sent per second. What is the cheapest way of handling the required load? While the application to 15.053 podcasting is fanciful, this application is of great importance on the web. For example, the demand on CNN servers is very high. In order to meet this demand, it helps to have servers throughout the world with the content from CNN. This would be challenging and difficult for CNN to manage, and so it is handled by Akamai, a company founded by MIT students and faculty. 4
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Supplies, Demands, and Capacities 50 200 MIT1 London -400 satellite1 200 175 0 200 200 300 MIT2 150 0 satellite2 150 100 175 100 MIT3 75 China -200 Usually, one lists the capacities on the same diagram as the costs. But students have found this too much data to absorb in a single slide. And so we have one slide with supplies, demands and capacities. The next slide will give the arc costs. The node numbers are the supplies and demands. A positive number indicates a supply. A negative number indicates a demand. In a network flow problem, we have this constraint: The total flow leaving a node minus the total flow entering the node is the “supply” at the node. In this case, the total flow leaving the MIT node must be exactly 200. The total flow entering the London node (with the London Bridge) is exactly 400. In practical cases, there are examples where the total flow leaving a node can be less than the supply. We can deal with such cases through a transformation into an equivalent problem in which the above constraint on supply/demand is satisfied with equality. 5
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6 Costs and node numbers $ 0 $ .01 $.01 $.01 $.02 $ 0 $. 01 $. 01 $. 03 $. 04 1 2 3 4 5 6 7 satellite1 satellite2 MIT1 MIT2 MIT3 China London Here are the costs for the problem. they are not particularly realistic, but are just intended to illustrate a min cost network flow problem.
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This note was uploaded on 01/17/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.

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lec14 - 15.053 April 5 2007 z The Minimum Cost Flow Problem...

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