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sec5 - 5 The tableau Solving simple linear programming...

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5 The tableau Solving simple linear programming problems graphically suggests the impor- tance of extreme points of the feasible region. In the last section, we related the geometric idea of an extreme point to the algebraic idea of a basic feasi- ble solution. Our aim now is to capitalize on this algebraic idea to design a solution algorithm for linear programming problems. Consider a linear programming problem in standard equality form: maximize c T x subject to Ax = b x 0 . By definition, a basis B for the matrix A is a list of indices such that the matrix of corresponding columns A B is invertible. We know from linear alge- bra that, since A B is invertible, some sequence of elementary row operations (adding multiples of rows to other rows and interchanging rows) reduces A B to the identity matrix. Suppose we apply exactly that sequence of row oper- ations to the system Ax = b . As we saw before, we can rewrite the original system (5.1) A B x B + A N x N = b, where the list N consists of the nonbasic indices (those indices not in B ). The vectors x B and x N have components respectively x i for i B (the basic variables) and x j for j N (the nonbasic variables). Applying our sequence of row operations, we must arrive at an equivalent system, having the form (5.2) Ix B + ¯ Ax N = ¯ b, or in other words x i + j N ¯ a ij x j = ¯ b i ( i B ) . This form of the system of constraints is particularly well-suited for un- derstanding the basis B . Given any values of the nonbasic variables x j (for j N ), we can easily read off the corresponding values of the basic variables x i (for i B ). In particular, if we set the nonbasic variables to zero, we obtain the corresponding basic solution x B = ¯ b and x N = 0. This basic solu- tion is feasible exactly when ¯ b 0. Notice that each basic variable appears exactly once in the system.
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