simplex - OR3300/5300 Fall 2011 Prof. Bland Review of the...

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Unformatted text preview: OR3300/5300 Fall 2011 Prof. Bland Review of the Simplex Method Part I: Standard Form Systems and Ordered Bases Let A be an m n matrix and let b be an m 1 vector. We will be interested in linear programming problems in which the constraints take the form Ax = b,x 0. Note that the set S = { x R I n : Ax = b,x } of feasible solutions to such l.p. problems is a polyhedron. This polyhedron may not be bounded (for example if m = 0), but if it is nonempty, it has at least one extreme point. (We will give an intuitive argument in class.) For a little while we are going to ignore the nonnegativity constraints and focus on the linear system Ax = b . For each index 1 j n let A j denote the jth column of A . Now suppose that for the list B = ( B 1 ,...,B m ) of indices in { 1 ,...,n } the m m matrix A B = [ A B 1 ,...,A B m ] is nonsingular. Then we say that B is basic for A , and that B is an ordered basis correspond- ing to the linear system Ax = b , where ( A, b ) := A- 1 B ( A,b ) The variables x B 1 , ,x B m are basic variables , and the other variables x j , for j not in B , are called nonbasic . It is convenient to denote by N the set of indices of the nonbasic variables. For example let A = 2 2 1 0 0 4 2 0 1 0 3 6 0 0 1 b = 80 120 210 Then B = (3 , 4 , 5) is basic for A , and, since A B = I 3 , A = A, b = b , so this choice of B is an ordered basis for the system Ax = b . Another B that is basic for A is B = (2 , 1 , 4). For this choice of B we have: A B = [ A 2 ,A 1 ,A 4 ] = 2 2 0 2 4 1 6 3 0 A- 1 B = - 1 2 1 3 1 0- 1 3- 3 1 2 3 and A = 0 1- 1 2 1 3 1 0 1 0- 1 3 0 0- 3 1 2 3 b = 30 10 20 For any m n matrix A , m 1 vector b and B that is basic for A , the matrix A = A- 1 B A has a very special form: specifically A B = I m . We say that such a matrix A is in standard form , and we also say that the linear system Ax = b is in standard form. In each row i = 1 , ,m the basic variable x B i has a coefficient of 1 but each of the other basic variables have coefficient 0. We can think of x B i as the basic or dependent variable in row i of A . The nonbasic variables x j play the role of independent variables in this system. For any arbitrary assignment of values to { x j : j N } setting x B i = b i- j N a ij x j ( i = 1 ,...,m ) assures that Ax = b and, therefore, that Ax = b . Of special interest is the unique solution of Ax = b in which every nonbasic variable x j = 0 ; it has x B = b . We call this a basic solution of Ax = b . The time has come to bring the nonnegativity constraints back into the picture. If b is nonnegative, then we say that the solution given by x B = b,x N = 0 is a basic feasible solution of our linear programming problem. In our 3of our linear programming problem....
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This note was uploaded on 03/08/2012 for the course ORIE 3300 at Cornell University (Engineering School).

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simplex - OR3300/5300 Fall 2011 Prof. Bland Review of the...

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