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LP - Linear Programming Sample Problems Problem 1 Complete...

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Linear Programming Sample Problems Problem 1. Complete the following tableau, where the objective function is z = 3 x 1 - 2 x 2 + 2 x 4 - x 5 . c * x 1 x 2 x 3 x 4 x 5 b 3 1 1 0 0 1 - 1 2 0 1 0 5 4 5 0 0 1 9 Next, maximize the objective function using the simplex method. Solution. First read off the basic solution: x 3 = 1 , x 4 = 5 , x 5 = 9 , x 1 = x 2 = 0. Then compute the “truncated” vector c * by picking the components of c corresponding to the basic variables x 3 , x 4 , x 5 : This gives c * = (0 , 2 , - 1). Next, find the value of z for this solution by computing z = c * b = 0 · 1 + 2 · 5 + ( - 1) · 9 = 1. (You can also find this by plugging in the values for x 1 , . . . , x 5 into the formula for z . Finally we compute the simplex indicators using the formula c * A - c . The first simplex indicator s 1 is computed as s 1 = 0 2 - 1 3 - 1 4 - 3 = (0 · 2 + 1 · ( - 1) + ( - 1) · 4) - 3 = - 9 . The other simplex indicators are found similarly. Putting all of this together gives the following tableau: c * x 1 x 2 x 3 x 4 x 5 b 0 3 1 1 0 0 1 2 - 1 2 0 1 0 5 - 1 4 5 0 0 1 9 - 9 1 0 0 0 1 Next, we note that the first simplex indicator is negative. Because not all entries of the corresponding column are negative, we can improve the solution. This is done by using row
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reduction, pivoting on one of the elements in the first column. We pivot on the first element
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