14mincostflows - 15.082 and 6.855J April 3, 2003...

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1 15.082 and 6.855J April 3, 2003 Introduction to Minimum Cost Flows
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2 The Minimum Cost Flow Problem u ij = capacity of arc (i,j). c ij = unit cost of shipping flow from node i to node j on (i,j). x ij = amount shipped on arc (i,j) Minimize (i,j) A c ij x ij j x ij - k x ki = b i for all i N. and 0 x ij u ij for all (i,j) A.
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3 Find the shortest path from node 1 to node 6 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 0 0 b(6) = -1 b(1) = 1 0 0 The optimal flow is to send one unit of flow along 1-2-5-6. This transformation works so long as there are no negative cost cycles in G. (What if there are negative cost cycles?)
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4 Find the Maximum Flow from s to t b(i) = 0 for all i; add arc (t,s) with a cost of -1 and large capacity. The cost of every other arc is 0. s 1 2 t 10 , 8 8, 7 1, 1 10, 6 6, 5 13 The optimal solution in the corresponding minimum cost flow problem will send as much flow in (t,s) as possible.
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5 Transshipment Problems Plants with given production capabilities for a product. One can ship directly from the plants to retailers, or from plants to warehouses, and then from warehouses to retailers. There is a given demand for each retailer. Costs of shipment are given. What is the minimum cost method for satisfying demands?
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6 A Network Representation Retailers 1 2 3 4 5 6 7 190 310 100 400 180 Demands Plants Warehouses
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7 The Caterer Problem Demand for d i napkins on day i for i = 1 to 7 (so, j [1. .7]). Cost of new napkins: a cents each, 2-day laundry: b cents per napkin 1-day laundry: c cents per napkin. Minimize the cost of meeting demand. 1 2 3 4 5 6 7 2’ 3’ 4’ 5’ 6’ 7’ clean dirty 0 demand arcs a c b 1’
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8 Purchase arcs In any period of the seven periods, one can purchase napkins, at a cost of a cents per napkin. clean napkins 1 2 3 4 5 6 7 0 a
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9 Demand Arcs You must use d i napkins on day i dirty napkins 1 2 3 4 5 6 7 0 a lower bound on flows 1’ d 1 2’ d 2 3’ d 3 4’ 5’ 6’ d 4 d 5 d 6 7’ d 7
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10 The rest of the arcs
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14mincostflows - 15.082 and 6.855J April 3, 2003...

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