This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
 Fall '08
 Ralphs

Click to edit the document details