homework-11

homework-11 - Problem 2 Consider the LP problem min 2 x 1 x...

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MthSc810 – Mathematical Programming Fall 2011, Homework #11. Due Thursday, November 17, 2011, 6PM EDT. From the syllabus: homework will be penalized 50% for each day they are late. After two days, they will not be accepted. No exception. Problem 1. Consider the LP problem min x 1 + 2 x 2 s.t. 3 x 1 + x 2 9 x 1 + x 2 5 x 1 , x 2 0 . 1. Put it in standard form. 2. Verify that the optimal basis is B = { 1 , 2 } by computing B 1 b and the reduced costs of the nonbasic variables. 3. Suppose a variable x 5 , unrestricted in sign, is added to the original problem, with objective coe±cient α and column A 5 = (1 , 1) . How does the standard form change? 4. Determine the values of α such that the optimal solution of the new problem has the same entries for x 1 and x 2 . 5. Determine the values of α such that the problem is bounded.
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Unformatted text preview: Problem 2. Consider the LP problem: min 2 x 1 + x 2 s.t. 3 x 1 + x 2 ≥ 6 x 1 + x 2 ≥ 4 x 1 , x 2 ≥ . 1. After putting it in standard form, verify that the basis B = { 1 , 2 } is optimal. 2. Add the constraint x 1 +3 x 2 = α . For what value of α is the problem feasible? 3. Choose an α such that the problem is feasible and perform the necessary operations to ²nd the new optimal solution. Problem 3. Consider the LP problem: min (1-α ) x 1 + (2 + α ) x 2 s.t. 3 x 1 + x 2 ≤ 9 x 1 + x 2 ≥ 5 x 1 , x 2 ≥ . For α = 0, this is equivalent to the LP in Problem 1. Compute the values of α such that the problem is infeasible, unbounded, or has a ²nite optimal solution....
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