Lecture20

Lecture20 - Advanced Operations Research Techniques IE316...

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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 .

Page1 / 13

Lecture20 - Advanced Operations Research Techniques IE316...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online