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lp computation

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

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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

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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
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