15_Network_Simplex - 15.082 and 6.855J The Network Simplex...

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1 15.082 and 6.855J The Network Simplex Algorithm
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2 Calculating A Spanning Tree Flow 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 A tree with supplies and demands. (Assume that all other arcs have a flow of 0) What is the flow in arc (4,3)? See the animation.
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3 What would happen if the flows in non- tree arcs were not 0? 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 Suppose that non- tree arcs had a non- zero flow. How would this change the computations? 2 1 3
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4 What would happen if the flows in non- tree arcs were not 0? 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 Adjust the supplies/ demands. They will be interpreted as excesses and deficits . 2 1 3 0 4 2 6 The compute flows as in the previous method; e.g., what is the flow in (4,3)?
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5 What would happen if the flow were negative? 1 3 6 4 5 2 7 1 3 -6 -4 1 2 3 If the direction of (4,3) were reversed, the flow in (3,4) would be negative. -2 3 6 4 4 3 A spanning tree flow is guaranteed to satisfy the supply/demand constraints. It may violate an upper or lower bound. A spanning tree flow is called feasible if it satisfies its upper and lower bound. Otherwise, it is infeasible .
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6 Basic Flows A basis structure consists of a spanning tree T, a set L of arcs, and a set U of arcs, such that T L U = A. For each (i,j) L, x ij = 0. For each (i,j) U, x ij = u ij . The arc flows in T are selected so that each node satisfies its supply/demand constraint. The basis structure is feasible if the arc flows also satisfy the upper and lower bounds. It is possible for a basis structure to be infeasible. In fact, this is normally the case in the dual simplex algorithm.
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7 Another way of calculating flows in arcs 1 3 6 4 5 2 7 1 3 -6
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