Lecture20 - Advanced Operations Research Techniques IE316...

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Unformatted text preview: Advanced Operations Research Techniques IE316 Lecture 20 Dr. Ted Ralphs IE316 Lecture 20 1 Reading for This Lecture • Bertsimas 7.8-7.10 IE316 Lecture 20 2 The Assignment Problem • The assignment problem is that of assigning n people to n projects so as to minimize cost. • An LP formulation is as follows: min 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 • 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. IE316 Lecture 20 3 The Dual of the Assignment Problem • The dual problem has the following form: max n X i =1 + n X j =1 s.t. r i + p j ≤ c ij , ∀ i,j. • In order to maximize ∑ n i =1 r i , we must have r i = min j =1 ,...,n { c ij- p j } • Hence, we can rewrite the dual as max n X j =1 p j + n X i =1 min j { c ij- p j } • This is an unconstrained optimization problem with a piecewise concave objective function. IE316 Lecture 20 4 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 > ⇒ p j- c ij = max k { p k- c ik } • If we view p k as the profit associated with project k , then this says that each person should be assigned to the most profitable project....
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This note was uploaded on 08/06/2008 for the course IE 316 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture20 - Advanced Operations Research Techniques IE316...

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