16networksimplexalgorithm

# 16networksimplexalgorithm - 15.082 and 6.855J The Network...

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1 15.082 and 6.855J The Network Simplex Algorithm

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2 Calculating A Spanning Tree Flow A tree with supplies and demands. (Assume that all other arcs have a flow of 0) 1 3 6 4 5 2 7 1 -6 -4 1 3 3 What is the flow in arc (4,3)? 2 See the animation.
3 What would happen if the flows in non- tree arcs were not 0? 1 3 6 4 5 2 7 1 -6 -4 1 3 Suppose that non- tree arcs had a non- zero flow. How would this change the computations? 3 1 3 2 2

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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 3 Adjust the supplies/demands. They will be interpreted as excesses and deficits . 2 2 1 3 4 2 6 The compute flows as in the previous method; e.g., what is the flow in (4,3)? 0
5 What would happen if the flow were negative? If the direction of (4,3) were reversed, the flow in (3,4) would be negative. 1 3 6 4 5 2 7 1 3 -6 -4 1 3 -2 3 64 43 A spanning tree flow is guaranteed to satisfy the supply/demand constraints. It may violate an upper or lower bound. 2 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.
7 Another way of calculating flows in arcs Case 1. If (i,j) is not in the tree, then x ij = 0.

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## This note was uploaded on 03/15/2010 for the course IE 505 taught by Professor Yok during the Spring '10 term at Galatasaray Üniversitesi.

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16networksimplexalgorithm - 15.082 and 6.855J The Network...

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