lec15 - 15.053 Tuesday, April 10 The Network Simplex Method...

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1 15.053 Tuesday, April 10 The Network Simplex Method for Solving the Minimum Cost Flow Problem
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Quotes of the day I think that I shall never see A poem lovely as a tree. -- Joyce Kilmer Knowing trees, I understand the meaning of patience. -- Unknown Every man has his price. -- Robert Walpole 2
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A Minimum Cost flow Problem 1 $ 0 200 -400 6 MIT1 2 4 $ .01 7 $. 01 London 0 $. 01 $.01 300 satellite1 MIT2 5 $.01 0 $. 03 $. 04 $.02 satellite2 100 -200 3 $ 0 China MIT3 Here is the min cost assignment problem from the previous lecture. 3
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We will be treating the min cost flow problem in which there are no capacities. The algorithm that we present here readily generalizes to the problem in which there are capacities. But we do not include the generalization here because it adds one extra level of complexity, and it is difficult to follow the first time around. 4 The Minimum Cost Flow Problem Minimize the cost of sending flow s.t. Flow out of i - Flow into i = b i x ij 0 for all (i,j) A z Directed Graph G = (N, A) . Node set N , arc set A ; Lower bound of 0 on arc (i,j) No upper bounds on arc flows in this lecture Cost c ij on arc (i,j) Supply/demand b i for node i . (Positive indicates supply)
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5 LP Formulation 1 1 1 1 , 1 0 = = = = = = ∑∑ n n ij ij i j n n ij ki i j k ij cx x x b i n x Minimize subject to Here is an LP formulation of the generic min cost flow problem with no capacities.
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The Network Simplex Algorithm z We will present the simplex algorithm as it applies to the min cost flow problem z NO TABLEAUS (except next two slides) We compute the primal solution directly on the network We compute the simplex multipliers directly on the network We compute reduced costs directly on the network. The Network simplex algorithm provides another opportunity to visualize the simplex algorithm. In this case, one can visualize the algorithm in multiple dimensions (that is lots of variables), as opposed to the lectures on geometry where we were restricted to two or three dimensions (that is, two or three variables). We will not be using tableaus. Nevertheless, we will still compute basic feasible solutions as well as reduced costs. To carry out these computations, we will work directly on the network. 6
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7 x 14 x 23 x 32 x 34 x 42 x 12 RHS 1 0 0 0 0 1 -1-1 = = = = Formulating a min cost flow problem 1 2 4 3 z x ij = flow in (i, j) z arc costs c ij z no arc capacities today z node supply/demands b i (, ) Minimize ij ij ij A cx x ij 0 for all arcs (i, j) A b(1) b(2) b(3) b(4) To recall how to compute prices, it helps to write a minimum cost flow problem as a linear program. 0 0 0 0 0 0 0 0 1 1 1 1 - 1 - 1 - 1 - 1 - 1 - 1
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8 Prices and Pricing Out 1 2 4 3 x 14 x 23 x 32 x 34 x 42 x 12 RHS b(1) b(2) = = = = 0 Prices Reduced Costs of the Arcs x 23 x ij x 12 c 12 -(y 1 ×1) 2 ×-1) c 12 –y 1 + y 2 b(3) b(4) c 12 z 1 -1 0 0 -1 c 14 c 23 c 32 c 34 c 42 y 1 y 2 y 3 y 4 We use the usual rule for pricing out. Note that we have used -1 as the coefficient of z in the z-row, and so the costs are all original costs.
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lec15 - 15.053 Tuesday, April 10 The Network Simplex Method...

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