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By Eduard L. Stiefel (Auth.)

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METHOD OF D U A L S O L U T I O N 37 These values are > 0 because, after the last EX-step, the last column has become positive. Hence, all five constraints *i>0; x2 > 0; yx > 0; y2 > 0; y3 > 0 are satisfied. 5; # 2 = 3 , and the minimal value of the objective function is ζ = χχ -f χ2 = 4,5. Since the values in the lower right-hand corners of the tables are constantly increasing during the simplex algorithm, they con­ stitute lower bounds for the minimum. The objective function is thus not systematically decreased by the algorithm until the minimal value has been attained, but, on the contrary, it is constantly increased.

The objective function is thus not systematically decreased by the algorithm until the minimal value has been attained, but, on the contrary, it is constantly increased. The procedure described here is called the dual-solution method. It can always be used when a linear objective function of the independent variables xjc has co­ efficients > 0 and is to be made minimal. , the variables on the upper border of the last table—equal to zero. For a geometric interpretation of the dual solution method, the corresponding variables on the upper border of the previous tables shall also be set equal to zero.

2#i + x2 — 6, (59) = max. 2/3 = ζ— Xl -1 -2 -2 1 X2 -4 -3 -1 1 1 -8 -12 -6 0 Now the last row of the c\ is no longer positive. To proceed in this case, we first Χ look for special values of the independent variables #i, x 2 which satisfy 1 I f this last element is equal to zero, we may arbitrarily regard it as either positive or negative. 34 2. L I N E A R PROGRAMMING the five constraints in (58). " Geometrically (Fig. 3), this means finding a feasible point or, in other words, a point which lies inside the shaded convex polygon.

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