{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lp computation

lp computation - Economics 146 Fall 2007 Linear Programming...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Economics 146 – Fall, 2007 Linear Programming – Part II Last revision: 10/8/07. Computational Form for Linear Programming. Although the standard LP form and the computational LP form both use the notation A , b , c to represent the data of the problem, the actual matrices and vectors are different for the two forms. See the numerical example below the definition of computational form. IMPORTANT NOTE! The terms standard form and computational form are not the same terminology used in the textbook. There are several different versions of the terminology in use in textbooks about LP. The book’s use of standard form is what we call computational form . Other terms that are found in the literature (albeit not in a consistent fashion) in addition to standard form and computational form are canonical form and symmetric form . For purposes of computation – especially for using the Simplex Method – we pose the LP problem as a special form using linear equalities. An LP problem is in computational form if it is written as: mimimize b T y (1) subject to Ay = c (2) and y 0 (3) where the data of the problem is A is an m × n matrix b R n c R m and the variables for the problem are y R n and the assumptions of the form are m < n c 0 We define: A feasible solution for the LP problem in computational form is y R n which satisfies (2) and (3). 1
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
A basic feasible solution is a feasible solution y which has no more than m positive components. A non-degenerate basic feasible solution is a feasible solution y which has exactly m positive components. It will be important to note that the constraint (2) states that a feasible solution y (if one exists) provides a way to write the vector c as a linear combination of the columns of A . If we denote the columns of A as a 1 , a 2 , . . . , a n , then the constraint (2) can be written as y 1 a 1 + y 2 a 2 + · · · + y n a n = c (4) We will illustrate the computational form by converting an LP in standard form to an equivalent LP in computational form. Start with the standard form problem which seeks to find x 1 , x 2 , x 3 so as to: maximize - x 1 +3 x 2 - 2 x 3 subject to +3 x 1 - x 2 +2 x 3 7 - 2 x 1 +4 x 2 12 - 4 x 1 +3 x 2 +8 x 3 10 and x 1 , x 2 , x 3 0 (5) To change this into computational form we will add a slack variable to each inequality to recast it as an equality. We will also change the maximization problem into a minimization problem by multiplying the objective function by - 1. Thus the problem in computational form is to find x 1 , x 2 , x 3 , x 4 , x 5 , x 6 so as to minimize x 1 - 3 x 2 +2 x 3 subject to 3 x 1 - x 2 +2 x 3 + x 4 = 7 - 2 x 1 +4 x 2 + x 5 = 12 - 4 x 1 +3 x 2 +8 x 3 + x 6 = 10 and x 1 , . . . , x
Image of page 2
Image of page 3
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