Homework 4

# Homework 4 - evidence for your answer 1 3 Simplex Method...

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CO350 Homework 4 – Winter 2007 Due: Friday, Feb. 2, 2007. Hand in at the start of class. 1. Simplex Method Consider the following LP (P). max 4 x 1 + 2 x 2 + 6 x 3 3 x 1 + 2 x 2 10 x 2 + 2 x 3 8 2 x 1 + x 2 + x 3 8 x 1 , x 2 , x 3 0 (a) Find an optimal solution for (P) using the Simplex Method. Clearly show the details of each iteration. (b) What is the dual (D) of (P)? (c) Find a solution y to (D) and use y to argue, using the Weak Duality Theorem, that the solution found in (a) is indeed optimal for (P). 2. Simplex Method and Unboundedness The following LP (P), from Section 2.3 of your notes, has already been shown to be unbounded. max 2 x 1 - 3 x 2 - x 1 + x 2 1 x 1 - 2 x 2 1 x 1 , x 2 0 (a) Apply the Simplex Method to (P). Clearly show the details of each iteration. (b) Suppose M is a large positive real number. Using (a), ﬁnd a feasible solution for (P) whose objective function value is M . (c) What is the dual (D) of (P)? (d) Does (D) have an optimal solution, or is (D) unbounded or is (D) infeasible? Provide

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Unformatted text preview: evidence for your answer. 1 3. Simplex Method, Does this ever end? (from Chvatal) Consider the following LP (P). max 10 x 1-57 x 2-9 x 3-24 x 4 . 5 x 1-5 . 5 x 2-2 . 5 x 3 + 9 x 4 ≤ . 5 x 1-1 . 5 x 2-. 5 x 3 + x 4 ≤ x 1 ≤ 1 x 1 , x 2 , x 3 x 4 ≥ (a) Perform two iterations of the Simplex Method on (P) using the two rules below. Clearly show the details of each iteration. Also, for each tableau, clearly identify the basis B , the corresponding basic feasible solution, and the current objective function value. i. Choose as entering variable the non-basic variable with the largest coeﬃcient in the z-row. ii. If two or more basic variables compete for leaving the basis, choose the candidate with the smallest subscript. (b) In the three tableaux, what has changed? (c) In the three tableaux, what has not changed? 2...
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