{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

simplex

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

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

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 0 } 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 0 1 3 1 0 - 1 3 - 3 1 2 3 and ¯ A = 0 1 - 1 2 0 1 3 1 0 1 0 - 1 3 0 0 - 3 1 2 3 ¯ b = 30 10 20

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

View Full Document
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 3 × 5 example there are 5 basic feasible solutions and five other basic solutions that are not feasible. For exam- ple, if we take B = (1 , 2 , 3), then the corresponding basic solution has x 3 < 0. It would
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online