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networksimplexmethod

# networksimplexmethod - Network Simplex Method Fatme...

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Network Simplex Method Fatme Elmoukaddem Jignesh Patel Martin Porcelli

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Outline Definitions Economic Interpretation Algebraic Explanation Initialization Termination
Transshipment Problem Find the cheapest way to ship prescribed amounts of a commodity from specified origins to specified destinations through a transportation network

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Network A network is a collection of nodes connected by arcs Each node has a demand for the commodity Nodes that are sources of the commodity have a negative demand The sum of all the demands is zero Each arc has a cost to ship a unit of commodity over it
Example 1 2 3 4 5 -5 1 3 3 -2 5 4 7 3 9 1 5

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Schedule A schedule describes how much of the commodity is shipped over each arc Requirements The amount entering a node minus the amount leaving it is equal to its demand The amount shipped over any arc is nonnegative
Example 1 2 3 4 5 -5 1 3 3 -2 1 3 1 0 0 0 2

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LP Formulation Let c be a row vector and x a column vector indexed by the set of arcs c ij is the cost of shipping over ij x ij is the amount to ship over ij Let b be a column vector indexed by the set of nodes b i is the demand at i
Example 1 2 3 4 5 -5 1 3 3 -2 5 4 7 3 9 1 5 [ ] 5 1 9 3 7 4 5 = c = 54 35 25 23 14 13 12 x x x x x x x x [ ] 2 3 3 1 5 - - = b

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LP Formulation ( 29 ( 29 0 0 = = - = i i i ij ij ji ji ij ij ij ij b b x x i x ij x c to subject minimize cx
LP Formulation (2) Let A be the matrix indexed by the set of nodes x the set of arcs A i,jk is either -1 if i = j 1 if i = k 0 otherwise A is known as the incidence matrix of the network

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Example 1 2 3 4 5 -5 1 3 3 -2 5 4 7 3 9 1 5 - - - - - - - 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 5 4 3 2 1 54 35 25 23 14 13 12
LP Formulation (2) ( 29 0 0 = = i i ij b x ij b Ax cx to subject minimize

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Tree Solution A spanning tree of a network is a network containing every node and enough arcs such that the undirected graph it induces is a tree A feasible tree solution x associated with a spanning tree T is a feasible solution with x ij = 0 if ij is not an arc of T
Network Simplex Method Search through feasible tree solutions to find the optimal solution Has a nice economic interpretation

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Given a spanning tree T and an associated feasible tree solution x Imagine you are the only company that produces the commodity What price should you sell the commodity for at each node? Assume that you ship according to
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networksimplexmethod - Network Simplex Method Fatme...

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