HW7Sol

# 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 λ = 0 (2)- 1- 2 x 2 λ = 0 (3) λ (1- x 2 1- x 2 2 ) = 0 , λ ≥ 0. Trying λ > 0, we find from (1) and (2) that x 1 = 1 2 λ and x 2 =- 1 2 λ . Then (3) yields λ = √ 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- λ = 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) λ (160 L 1 ) = 0, λ ≥ 0....
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HW7Sol - Fall 2009 IEOR 160 Industrial Engineering&...

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