# Applied mathematical programming using algebraic systems by McCarl B.A., Spreen T.H.

By McCarl B.A., Spreen T.H.

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Furthermore, since the basic variables must remain nonnegative the solution must satisfy XB ' B & 1 b & B & 1 a0 x0 \$ 0. This equation permits the derivation of a bound on the maximum amount the nonbasic variable x0 can be changed while the basic variables remain non-negative. Namely, x0 may increase until one of the basic variables becomes zero. Suppose that the first element of XB to become zero is xBi*. Solving for xBi* gives xBi ( ' (B & 1 b)i ( & (B & 1 a0)i ( x0 ' 0 where ( )I denotes the ith element of the vector.

Thus, we add an 1xM vector of zero's to the objective function and conditions constraining the slack variables to be nonnegative. t. AX % X, copyright Bruce A. McCarl and Thomas H. Spreen 3-2 IS ' b S \$ 0. Throughout the rest of this section we redefine the X vector to contain both the original X's and the slacks. Similarly, the new C vector will 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.

In fact, they are usually estimated by statistical techniques. Thus, after developing a LP model, it is often useful to conduct sensitivity analysis by varying one of the exogenous parameters and observing the sensitivity of the optimal solution to that variation. For example, in the van shop problem the net return per fancy van is \$2,000, but this value depends upon the van cost, the cost of materials and the sale price all of which could be random variables. Considerable research has been directed toward incorporating uncertainty into programming models.