class_8_linear_programming(2).pdf - Linear programming...

Info icon This preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming August 12, 2017
Image of page 1

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

View Full Document Right Arrow Icon
Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Overview 1 Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs)
Image of page 2
Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming Many problems can be naturally described as maximization/minimization of a linear function under linear constrains. For example: Alice and Bob are flying to Paris for a vacation. They want to take with them as many books as possible , however they can’t carry more then 12 pounds of books. Alice will take x 1 books and Bob will take x 2 books. Alice’s books weigh 3 pounds each, and Bob’s books weigh 1 pound each. After hours of discussions, they decided that given the length of their vacation, and how fast they read, they should make sure that: x 2 2 - 3 4 x 1 2 3 x 2 + 10 3
Image of page 3

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

View Full Document Right Arrow Icon
Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming Many problems can be naturally described as maximization/minimization of a linear function under linear constrains. For example: Alice and Bob are flying to Paris for a vacation. They want to take with them as many books as possible , however they can’t carry more then 12 pounds of books. Alice will take x 1 books and Bob will take x 2 books. Alice’s books weigh 3 pounds each, and Bob’s books weigh 1 pound each. After hours of discussions, they decided that given the length of their vacation, and how fast they read, they should make sure that: x 2 2 - 3 4 x 1 2 3 x 2 + 10 3
Image of page 4
Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming We can express the problem as: Maximize: x 1 + x 2 subject to the linear constrains: 3 x 1 + x 2 12 3 x 1 - 2 x 2 10 4 x 1 - 2 x 2 ≥ - 3 x 1 , x 2 0
Image of page 5

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

View Full Document Right Arrow Icon
Linear programming Linear programming Linear problems Standard form and Slack form The simplex algorithm A few facts regarding the simplex algorithm (without proofs) Linear programming Another example: We can express the problem of finding maximal flow in a flow network as follows: Maximize: X v V f ( s , v ) - X v V f ( v , s ) subject to: f ( u , v ) c ( u , v ) for any ( u , v ) V × V X v V f ( u , v ) = X v V f ( v , u ) for any u V - { s , t } f ( u , v ) 0 for any ( u , v ) V × V (Here we assume all edges exist and the capacities of edges that don’t actually exist are 0.)
Image of page 6
Linear programming Linear programming Linear problems Standard form and Slack form
Image of page 7

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

View Full Document Right Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern