HW7Sol - Fall 2009 IEOR 160 Industrial Engineering...

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Unformatted text preview: Fall 2009 IEOR 160 Industrial Engineering & Operations Research October 22, 2009 Page 1 of 2 HOMEWORK 7 SOLUTIONS Section 12.9 2. KT conditions yield (1) 1- 2 x 1 = 0 (2)- 1- 2 x 2 = 0 (3) (1- x 2 1- x 2 2 ) = 0 , 0. Trying > 0, we find from (1) and (2) that x 1 = 1 2 and x 2 =- 1 2 . Then (3) yields = 2 2 . Finally we find that x 1 =- x 2 = 2 2 . Since the constraint is convex and the objective function is linear, we know that we have found an optimal solution to the problem. 5. Let L 1 be the labor purchased at $15 per hour and L 2 the aabor purchased at $25 per hour. We wish to solve max 270( L 1 + L 2 ) 1 / 2 K 1 / 3- 15 L 1- 25 L 2- 5 K subject to L 1 160 and all variables nonnegative. Ignoring the non-negativity restrictions, the KT conditions yield (1) 135( L 1 + L 2 ) 1 / 2 K 1 / 3- 15- = 0 (2) 135( L 1 + L 2 ) 1 / 2 K 1 / 3- 25 = 0 (3) 90( L 1 + L 2 ) 1 / 2 K 2 / 3- 5 = 0 (4) (160 L 1 ) = 0, 0....
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HW7Sol - Fall 2009 IEOR 160 Industrial Engineering...

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