15 2 2 4 3 5 4 10 1 1 5 2 4 7 10 6 3 6 3 5 6 5 1 1 2 7 3 6 4 5 1 2 1 7 3 2 6 7 3 0 6 0 4 4 5 What is the flow in arc 4 3 What is the flow in arc 5 3 1 1 6 1 2 6 0 6 1 4 4 2 0 1 1 7 3 3 3 2 7 3 3 4 4 2 0 6 2 0 5 5 0 0 0 5 3 2 What is the flow in arc 3 2 What is the flow in arc 2 6 1 1 6 2 1 1 6 7 3 2 7 3 6 1 3 1 4 2 3 4 6 2 6 1 3 0 3 2 6 0 1 3 0 6 4 3 4 2 7 3 4 3 2 0 5 1 1 6 4 4 3 7 3 6 0 5 0 What is the flow in arc 1 2 4 3 2 3 2 1 1 0 6 4 4 3 What is the flow in arc 7 1 3 2 0 5 0 2 2 0 6 0 0 5 3 1 1 Note there are 4 two different ways 3 of calculating the 6 2 7 3 flow on 1 2 and 4 6 both ways give a 0 flow of 4 Is this a 1 3 6 coincidence 4 2 3 0 4 5 0 2 3 3 1 3 1 7 4 3 6 4 5 4 1 2 1 7 1 3 2 2 6 7 7 0 3 6 5 4 1 3 2 5 2 3 6 3 5 7 5 1 2 3 4 2 2 5 1 3 4 4 2 6 4 5 3 2 1 1 2 2 7 7 3 6 4 5 2 4 0 Here is a spanning tree with arc costs How can one choose node potentials so that reduced costs of tree arcs are 0 1 5 6 2 7 3 4 3 2 6 1 4 5 6 2 7 3 3 6 1 4 What is the node potential for 2 5 0 0 1 1 5 5 6 2 2 7 3 6 2 5 What is thenode potential for 7 6 1 4 5 What is the potential for node 3 0 0 1 5 5 3 2 2 4 1 6 2 7 6 2 3 2 6 1 5 5 5 6 4 3 6 4 3 1 4 6 2 4 3 5 5 7 3 One can set p1 arbitrarily We will let p1 0 4 2 5 There is a redundant constraint in the minimum cost flow problem 1 2 What is the potential for node 6 4 7 6 4 6 1 3 1 5 What is the potential for node 4 5 0 0 1 1 5 5 6 2 3 2 2 4 7 6 2 3 4 2 6 1 3 5 5 6 1 2 5 What is the potential for node 5 4 0 Node potentials Original costs 4 4 7 4 6 1 3 5 2 5 1 7 6 2 3 2 3 4 Flow on arcs 1 4 2 6 3 2 4 7 2 3 6 3 5 0 1 4 3 4 3 5 5 1 7 2 6 3 4 5 2 3 4 6 6 1 3 1 4 7 4 These are the node potentials associated with this tree They do not depend on arc flows nor on costs of non tree arcs 1 Flow on arcs Reduced costs 1 2 6 3 3 2 4 7 1 6 0 3 5 6 1 2 3 4 7 6 5 7 1 2 3 4 1 7 2 6 3 4 5 7 6 5 1 2 3 4 7 6 5 8 9 MIT OpenCourseWare http ocw mit edu 6 251J 15 081J Introduction to Mathematical Programming Fall 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms