# HW5 - problem P1 to determine a basic feasible solution of...

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5th Homework for IEOR162 Linear Programming Semester: Fall 2010 Instructor: Dr. Sarah Drewes Due to: Thursday, October 28, 2010 (at the beginning of class) P1 Consider the following LP, that was ﬁrst introduced by Klee and Minty : max 10 x 1 + 10 x 2 s.t. x 1 1 , 2 · x 1 + x 2 100 . x 0 . a) Graph the feasible region. b) Solve the problem using the simplex algorithm. Start in the point ( x 1 ,x 2 ) = (0 , 0) and use the following pivoting rule of Bland : Choose as entering variable the variable among those with negative row 0 coeﬃcient with the smallest index, hence in the order x 1 ,x 2 ,s 1 ,s 2 . Ratio test: if more than one variable wins the ratio test, choose the pivot variable with the smallest row index in the tableau. c) Change the objective function in such a way, that – independent of the pivot rule – the optimal solution is found in one simplex iteration. Explain why it is independent of the pivot rule! P2 Solve Phase I of the two-phase simplex algorithm with the pivoting rule of Bland (compare

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Unformatted text preview: problem P1) to determine a basic feasible solution of the following LP: min 2 x 1 + x 2-5 x 3 s.t. x 1 + 2 x 2-4 x 3 + x 4 = 3 2 x 1-3 x 2 + x 3 + 5 x 4 =-4 7 x 1-10 x 3 + 13 x 4 = x 1 ,...,x 4 ≥ , 1 P3 Consider the following LP: max 2 x 1 + 3 x 2-x 3-12 x 4 s.t.-2 x 1-9 x 2 + x 3 + 9 x 4 ≤ 1 3 x 1 + x 2-1 3 x 3-2 x 4 ≤ x 1 ,...,x 4 ≥ Transform into standard form and start the simplex algorithm with basic variables BV := { s 1 ,s 2 } and use the following rules: – Choose the entering variable with the ’most negative’ coeﬃcient in row 0. – Ratio test: if more than one variable wins the ratio test, choose the pivot variable with the smallest row index in the tableau. Show that the simplex algorithm cycles. 2...
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HW5 - problem P1 to determine a basic feasible solution of...

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