Lecture19

Lecture19 - Introduction to Mathematical Programming IE406...

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Unformatted text preview: Introduction to Mathematical Programming IE406 Lecture 19 Dr. Ted Ralphs IE406 Lecture 19 1 Reading for This Lecture • Papadimitriou and Steiglitz, Chapters 5 and 6. IE406 Lecture 19 2 The Assignment Problem • The assignment problem can be interpreted as that of assigning n items to n people so as to maximize the total “value” of the assigned items. • An LP formulation is as follows: max n X i =1 n X j =1 c ij f ij s.t. n X i =1 f ij = 1 , j = 1 , . . . , n n X j =1 f ij = 1 , i = 1 , . . . , n f ij ≥ , ∀ i, j • Here, c ij can be interpreted as the value of item i to person j . • Note that this can be interpreted as a network flow problem , so there always exists an optimal solution for which f ij ∈ { , 1 } . • This allows us to interpret the solution as an assignment. IE406 Lecture 19 3 The Dual of the Assignment Problem • The dual problem has the following form: min n X i =1 p j + n X j =1 r i s.t. r i + p j ≥ c ij , ∀ i, j. • Here, we will interpret r i as the price of item i and p j as the person profit of person j . • In order to minimize ∑ n i =1 r i , we must have r i = max j =1 ,...,n { c ij- p j } • Hence, we can rewrite the dual as min n X j =1 p j + n X i =1 max j { c ij- p j } IE406 Lecture 19 4 • This is an unconstrained optimization problem with a piecewise concave objective function. IE406 Lecture 19 5 The Complementary Slackness Conditions • The complementary slackness conditions tell us that f ij > ⇒ r i + p j = c ij • Substituting the previous form for r i , we get f ij > ⇒ c ij- p j = max k { c ik- p k } • In other words, this says that each person should be assigned the item that maximizes their personal profit....
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Lecture19 - Introduction to Mathematical Programming IE406...

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