An Introduction to Numerical Mathematics by Eduard L. Stiefel (Auth.)

By Eduard L. Stiefel (Auth.)

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12). Let us now develop the table of the linear program (58) in the normal notation, where we take as objective function exactly that function ζ = χι + x2 which has to be made minimal: 36 2. 5 1 = ζ = (61) 0 Since the last row is positive, the table can be reduced by the simplex algorithm. We will take a look at the results of this process with no specific aim in mind at this time, other than general interest. 2, a row with a negative element in the last column must be selected as pivot row; we chose the third row.

In other words, Max must always use that strategy which is given by the saddle column; all other strategies are inactive. In the same way, Min always has to play the saddle row. Incidentally, the value of the game is equal to the saddle element. 7 Chebyshev Approximation A surveyor wants to measure the length of two adjoining segments A Β and BC of a straight line somewhere in a tract of land. Measuring each segment separately with a tape, he finds that A Β = 15 yards, BC = 6 yards. As a control, he also measures the entire line and finds that AC = 20 yards.

2. S O L U T I O N OF A PROGRAM W I T H T H E E X C H A N G E M E T H O D 25 condition that all variables y ι and xjc have to be positive: Vi > 0, xk> 0. Frequently it will also be required to minimize the objective function. This is nothing new because the problem ζ = min is equivalent to 11 (—z) - max. The use of equations as constraints can also be reduced to the above case. In fact, an equation y ι = 0 can be replaced by the two inequalities yi > 0 and (—yi) > 0. The formal scheme of this linear program has the form ...

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