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Unformatted text preview: 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 1 1 2 1 5 4 5 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 = 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 3 1 1 1 2 1 2 1 5 1 4 5 1 9 9 1 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 reduction, pivoting on one of the elements in the first column. We pivot on the first elementreduction, pivoting on one of the elements in the first column....
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This note was uploaded on 01/06/2011 for the course MAT 2342 taught by Professor Hofstra during the Spring '08 term at University of Ottawa.
 Spring '08
 Hofstra
 Linear Algebra, Algebra, Linear Programming

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