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Min cost network flow-2

# Min cost network flow-2 - Minimum Cost Flow Problem Lecture...

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Minimum Cost Flow Problem Lecture Notes Problem Setup Let x ij  be the flow on arc (i,j).  Minimize the cost of sending flow s.t.  Flow into of i - Flow out of i = b i  , for every node i;                  lij   x ij    u ij   , for every arc (i,j) .     Theorem (Integrality theorem for min cost flow)  If a flow network has capacities and balances which are all integer valued and there  exists some feasible flow in the network, then there is a minimum cost feasible flow with  an integer valued flow on every arc. Minimum Cost Flows Application shortest paths : lower bound on the arc = 0, upper bound on the arc = 1  Demands/supplies on the nodes are 1 or -1 Costs on the arcs are length of the arc transportation problem Example: Three factories can supply any of six customers with a particular product. The demand for this product from each of the customers 1, 2, 3, 4, 5 and 6 is 40, 35, 25, 20, 60 and 30 tons respectively. Maximum production at factories A, B and C is 60, 70 and 80 tons respectively. The variable production cost per ton is 11.3, 11.0 and 10.8 (£) at factories A, B and C respectively and the transportation cost per ton from each factory to each customer is as shown below. Transportation cost (£) per ton to customer

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1 2 3 4 5 6 From factory A 1.5 1.8 3.1 4.2 2.5 3.0 B 2.2 4.6 3.5 2.4 1.8 4.0 C 3.6 4.8 1.6 4.4 2.8 2.0
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Min cost network flow-2 - Minimum Cost Flow Problem Lecture...

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