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ornotes3-2007

# ornotes3-2007 - Matrix Notation Max CX s.t(X is a vector...

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Matrix Notation Max CX (X is a vector with n elements) s.t. AX b (there are m constraints) and X 0 Convert by adding slacks so that S = b – AX So the constraints are now AX + IS = b where I is an identity matrix of dimension m x m Redefine the X vector to contain both the original X's and the slacks. Redefine the C vector to contain the original C along with the zeros for the slacks, and the new A matrix will contain the original A matrix along with the identity matrix for the slacks. Max CX (X is a vector with n + m elements) s.t. AX b (A has dimensions m, n+m) The A matrix is not square, so it can't be inverted. Basic and nonbasic variables z The solution to the LP problem will have a set of potentially nonzero variables equal in number to the number of constraints. z Such a solution is called a Basic Solution and the associated variables are commonly called Basic Variables . z The other variables are set to zero and are called the nonbasic variables . z Note: if there are 3 constraints, no more than 3 X variables can be non-zero and so on.

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Partition the problem MAX C B X B B B + C NB X NB s.t. A B X B B B + A NB X NB = b X B , X B NB 0. Subscript B represents basic variables and NB, non-basic (0 valued) variables and corresponding coefficients. AB will be square because only as many variables as there are constraints can be in the solution. Your text calls this matrix "B" for basis matrix.
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ornotes3-2007 - Matrix Notation Max CX s.t(X is a vector...

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