By Michael R. Greenberg
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Extra info for Applied Linear Programming. For the Socioeconomic and Environmental Sciences
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. The element a22 i n ^ι is converted into 1 by multiplication of row 2 by 3. Step 3 row 2 3[0 i -i][-f 1 0] equals new row 2 [0 1 - l ] [ - 2 3 0] Step 4 Multiply new row 2 by — ^ and add it to row 1.
The inverse is 30 / 2: ALGEBRAIC METHODS and X Assume that the total resource picture is going to change from B = to Bl = One could once again transform the system AB into IX through synthetic elimination and find that X = Or Ä~l can be used to obtain the second set of X values. Specifically, given the original equation, ÄX = B, we premultiply both sides by A'1 and get Ä~ lÄX = Ä~1B. Since Ä~1Ä = I, the expression reduces to X = Ä~ 1B. In short, if we wish to test alternative values of B, we can perform these tests with matrix multiplications by Ά~ *.
4')] to illustrate the development and use of this information. Maximize: Z = 2XX + X2 + 0X 3 + 0X 4 + 0X5 k Subject tO : X1 + 2X2 + Ί ^ 2XX + 3X2 3X, + X2 (1) variabless l a c ' + X± N = 10 (2) =12 (3) + X5 = 15 (4) Xl >0 (5) X2 >0 (6) Optimal solution: Z = lOf ; Xx =ψ;Χ2= f ; X3 =^;X4 = 0; X5 = 0. Inspection of the graphical solution (Fig. 1) indicates that Eqs. (3) and (4) are the restrictive constraints. If we change the availability of the resource in constraint Eq. (2) from Xx + 2X2 + X3 = 10 to Xt + 2X2 + X3 = 9, the optimal value of the solution will remain at lOf.
Applied Linear Programming. For the Socioeconomic and Environmental Sciences by Michael R. Greenberg