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

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Namely, when changing x0 Z ' CB B & 1b & (CB B & 1 a0 & c0) x0 . ¯ ), then Since the first term of the equation is equal to the value of the current objective function, ( Z 1 A degenerate solution is defined to be one where at least one basic variable equals zero. copyright Bruce A. McCarl and Thomas H. Spreen 3-6 it can be rewritten as Z ' Z & (CB B & 1 a0 & c0)X0 ,. For maximization problems, the objective value will increase for any entering nonbasic variable if its term, CBB-1 a0 - c0 , is negative.

Ranging analysis deals with the question: what is the range of values for a particular parameter for which the current solution remains optimal? Ranging analysis of right-hand-side (bi) and objective function coefficients (cj) is common; many computer programs available to solve LP problems have options to conduct ranging analyses although GAMS does not easily support such features (See chapter 19 for details). 1 Ranging Right-Hand-Sides copyright Bruce A. McCarl and Thomas H. Spreen 3-14 Let us study what happens if we alter the right hand side (RHS).

One slack variable is added for each constraint equation. Rewriting the constraints gives AX + IS = b, where I is an M x M identity matrix and S is a Mx1 vector. Also the slack variables appear in the objective function with zero coefficients. 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.