By Michael R. Greenberg
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Maximize : Z = 2Xl + X2 + 0X3 + 0X4 + 0X5 (1) subject to Xx + 2X2 + X3 2XX + 3X2 + XA =10 (2) =12 (3) + X5 = 18 3*! + X2 X1 >0 (5) X2 >0 (6) Solution: Z = 12; Xx =6;X2 = 0; X3 = 4; X5 = 0; X6 = 0 (see Fig. 3). 18 '' 15 - 12 - 9 ~ \ 6 (40 /Optimum solution: Z = 1 2 ; Xy = 6; X2 = 0 /x2) m>K^\^^il 3 3 6 9 X, Fig.
O o] 0] = [ - 2 -2 — 2 3 4-row 2 2 1 new row 2 0 1 3 — 4 3 1 — 1 3 2 3 0 0 0 1 0 2 ~~ 3 1 0 — Since the element a3l is already zero, additional transformations of column 1 are not required. 1 Faddeeva (1959) provides an excellent summary of these methods. ESSENTIAL MATRIX METHODS / 25 At the conclusion of the first column operation, A has been transformed into Äx and the identity matrix into a matrix B. i o o -I i o B = At = 0 0 1 The third, fourth, and fifth steps are to convert a22 in Αλ into 1, and ai2 and a32 into zeros.
Indeed, this procedure is the typical method used in input-output analysis. In linear programming the process of estimating the impact of a change in one or more elements of B on X is called parametric programming. Parametric programming will be reviewed in Chapter 3. 5. BASIS VECTORS IN LINEAR PROGRAMMING The simultaneous equation systems that have been considered in this chapter are of the simplest form: all the matrices are square and the constraints are equalities. As you saw in the previous chapter, linear programming constraints may be equalities or inequalities.