18_Generalized_Flows

18_Generalized_Flows - 15.082J and 6.855J Generalized Flows...

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1 15.082J and 6.855J Generalized Flows
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2 Overview of Generalized Flows Suppose one unit of flow is sent in (i,j). We relax the assumption that one unit arrives at node j. If 1 unit is sent from i, μ ij units arrive at j. μ ij is called the multiplier of (i,j) i j μ ij = 7 We will present: LP Formulation Two applications Generalized Network Simplex Algorithm
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3 LP Formulation of Generalized Flows ( , ) Minimize ij ij i j A c x j j :( , ) :( , ) subject to ( ) ij ji ji j i j A j i j A x x b i μ - = 0 for all ( , ) . ij ij x u i j A j x ij = amount of flow sent in (i,j) μ ij = multiplier of (i,j) b(i) = supply at node i c ij = unit cost of flow in (i,j) u ij = upper bound on flow in (i,j)
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4 Conversions of physical entities i j 4/1/03 4/1/04 μ ij = 1.05 (i,j) represents a 1 year investment in a CD. i j coal electricity μ ij = .4 (i,j) represents a conversion of coal into electricity
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5 Machine Scheduling i j job machine μ ij = 3 It takes 3 hours to make one unit of job i on machine j. x ij = proportion of product i made on machine j μ ij = number of hours to make product i on machine j d(i) = number of units of product i that need to be made. The total time available on machine j is u j
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6 Flows Along Directed Paths 1 μ 12 = 3 2 μ 23 = .5 3 μ 34 = 4 4 μ 45 = 1 5 Suppose that 1 unit is sent from node 1, that flow is conserved in 2, 3, and 4, arrives at node 5. 1 3 1.5 6 6 For a directed path P from i to j, if one unit of flow is sent from i, then the amount arriving at j is: ( , ) ( ) ij i j P P μ =
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7 Flows Along Non-directed Paths 1 μ 12 = 4 2 μ 23 = 2 3 μ 34 = 6 4 μ 45 = 4 5 Suppose that 1 unit is sent from node 1, that flow is conserved in 2, 3, and 4, arrives at node 5. 1 4 2 12 3 Let P be a path from i to j. ( , ) ( , ) ( ) / ij ij i j P i j P P μ = Forward arcs of P P = Backward arcs of P P = If one unit of flow is sent from i, then the amount arriving at j is: 1 -2 2 -3
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8 Flows Along Cycles 1 4 2 12 3 1.5 ( , ) ( , ) ( ) / ij ij i j W i j W W μ = 1 4 2 2 3 6 4 4 5 2 W Suppose 1 unit is sent around W starting and ending at node 1. If μ (W) 1, then the amount of flow arriving at node 1 is different then the amount leaving node 1. If μ (W) = 1, W is called a breakeven cycle . μ (W) = 1.5 1 -2 -3 2 -1.5
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9 Flows Along Cycles 1 4 2 12 3 1.5 s 4 2 2 3 6 4 4 5 2 W Suppose θ units are send around W starting and ending at node s. The net amount arriving at node 1 is: θ [ μ (W)- 1 ]. To create a “supply” of α at node s, send α/ [ μ (W)- 1 ] units of flow.
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10 On the LP for Generalized Flows ( , ) Minimize ij ij i j A c x j j :( , ) :( , ) subject to ( ) ij ji ji j i j A j i j A x x b i μ - = 0 for all ( , ) . ij ij x u i j A j The equality constraints have full row rank, which is n.
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18_Generalized_Flows - 15.082J and 6.855J Generalized Flows...

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