13_Min_Cost_Flows_1 - 1 15.082 and 6.855J April 3 2003...

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Unformatted text preview: 1 15.082 and 6.855J April 3, 2003 Introduction to Minimum Cost Flows 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. 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 b(1) = 1 b(6) = -1 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?) 4 Find the Maximum Flow from s to t s 1 2 t 10 , 8 8, 7 1, 1 10, 6 6, 5 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. 13 The optimal solution in the corresponding minimum cost flow problem will send as much flow in (t,s) as possible. 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? 6 A Network Representation Plants Warehouses Retailers 1 2 3 4 5 6 7 190 310 100 400 180 Demands 1 2 3 4 5 6 7 400 1 2 3 4 5 6 7 400 1 2 3 4 5 6 7 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 demand arcs a c b 1’ 8 Purchase arcs In any period of the seven periods, one can purchase napkins, at a cost of a cents per napkin....
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13_Min_Cost_Flows_1 - 1 15.082 and 6.855J April 3 2003...

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