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Lecture2 - Integer Programming IE418 Lecture 2 Dr Ted...

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Integer Programming IE418 Lecture 2 Dr. Ted Ralphs
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IE418 Lecture 2 1 Reading for This Lecture Wolsey Chapter 1 N&W Sections I.1.1-I.1.6
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IE418 Lecture 2 2 Piecewise Linear Cost Functions We can use binary variables to model arbitrary piecewise linear cost functions . The function is specified by ordered pairs ( a i , f ( a i )) and we wish to evaluate it at a point x . We have a binary variable y i , which indicates whether a i x a i +1 . To evaluate the function , we will take linear combinations k i =1 λ i f ( a i ) of the given functions values. This only works if the only two nonzero λ i s are the ones corresponding to the endpoints of the interval in which x lies.
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IE418 Lecture 2 3 Minimizing Piecewise Linear Cost Functions The following formulation minimizes the function. min k X i =1 λ i f ( a i ) s.t. k X i =1 = 1 , λ 1 y 1 , λ i y i - 1 + y i , i [2 ..k - 1] , λ k y k - 1 , k - 1 X i =1 y i = 1 , λ i 0 , y i ∈ { 0 , 1 } . The key is that if y j = 1 , then λ i = 0 , i = j, j + 1 .
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IE418 Lecture 2 4 Fixed-charge Problems In many instances, there is a fixed cost and a variable cost associated with a particular decision. Example : Fixed-charge Network Flow Problem We are given a directed graph G = ( N, A ) . There is a fixed cost c ij associated with “opening” arc ( i, j ) (think of this as the cost to “build” the link). There is also a variable cost d ij associated with each unit of flow along arc ( i, j ) . Consider an instance with a single supply node. * Minimizing the fixed cost by itself is a minimum spanning tree problem ( easy ). * Minimizing the variable cost by itself is a minimum cost network flow problem ( easy ). * We want to minimize the sum of these two costs ( difficult ).
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IE418 Lecture 2 5 Modeling the Fixed-charge Network Flow Problem To model the FCNFP, we associate two variables with each arc. x ij ( fixed-charge variable ) indicates whether arc ( i, j ) is open .
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