homework-11-solutions

homework-11-solutions - MthSc810 Mathematical Programming...

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MthSc810 – Mathematical Programming Fall 2011, solutions to Homework #11. 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 coefficient 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. Solution. 1. min x 1 + 2 x 2 s.t. 3 x 1 + x 2 + x 3 = 9 x 1 + x 2 x 4 = 5 x 1 , x 2 , x 3 , x 4 0 . 2. B = parenleftbigg 3 1 1 1 parenrightbigg , and B 1 = 1 2 parenleftbigg 1 1 1 3 parenrightbigg . Hence B 1 b = (2 , 3) (0 , 0) . The reduced cost vector is ¯ c N = c N c B B 1 N = (0 , 0) (1 , 2) parenleftBigg 1 2 1 2 1 2 3 2 parenrightBigg parenleftbigg 1 0 0 1 parenrightbigg = (1 , 2) parenleftBigg 1 2 1 2 1 2 3 2 parenrightBigg = ( 1 2 , 5 2 ) (0 , 0), implying dual feasibility and optimality. 3. Introduce two variables x 5 , x 6 both nonnegative so that the new unrestricted variable is described as x 5 x 6 .
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