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# hw5 - 3 Prove that Simplex will never cycle if an LP has...

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AMS 540 / MBA 540 (Fall, 2010) Estie Arkin Homework Set # 5 Due in class on Tuesday, October 19, 2010. 1). Consider the following LP: min z = - 2 x 1 - 3 x 2 + x 3 + 12 x 4 - 2 x 1 - 9 x 2 + x 3 + 9 x 4 + x 5 = 0 1 / 3 x 1 + x 2 - 1 / 3 x 3 - 2 x 4 + x 6 = 0 x 1 , x 2 , x 3 , x 4 0 (a). The following anti cycling rule was suggested: Choose an entering variable that maximizes z j - c j , and break ties in the min ratio rule by favoring the row with least index (row 1 over row 2). Apply this rule to the LP and show that cycling occurs. (This should take 6 pivots.) (b). Apply one of the anti cycling rules discussed in class to solve this LP. (You should get that the LP is unbounded.) 2). The following is the ±rst tableau of phase I for an LP, the arti±cial variables a 1 , a 2 , a 3 were added to constraints 1,2,3 respectively. Let a 1 = x 7 , a 2 = x 8 , a 3 = x 9 . w x 1 x 2 x 3 x 4 x 5 x 6 a 1 a 2 a 3 RHS 1 0 0 1 0 2 -1 0 0 0 3 0 1 -1 1 0 2 0 1 0 0 0 0 -2 1 0 0 -2 0 0 1 0 3 0 1 0 1 0 1 -1 0 0 1 0 0 0 2 1 1 2 1 0 0 0 4 (a). Using Bland’s rule, which variable enters the basis, and which variable leaves the basis? (b). Using the lexicographic rule, which variable enters the basis, and which variable leaves the basis?
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Unformatted text preview: 3). Prove that Simplex will never cycle if an LP has the property that for every degenerate BFS the winner of the min ratio test is unique. 4). Consider the following LP: min z = 2 x 1 + 15 x 2 + 5 x 3 + 6 x 4 s . t . x 1 + 6 x 2 + 3 x 3 + x 4 ≥ 2-2 x 1 + 5 x 2-x 3 + 3 x 4 ≤ -3 x 1 , x 2 , x 3 , x 4 ≥ (a). Give the dual of the LP. (b). Solve the dual geometrically. (c). From the solution of the dual problem, ±nd the solution of the original problem. Reminder: The midterm is Thursday October 21 in class, covering material from homeworks 1-5. The exam is closed notes. However, you may bring 1 page of notes, written by you (not typed or xeroxed). This page will be turned in with the exam....
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