IEOR162_hw09

# IEOR162_hw09 - IEOR 162 Fall 2011 Homework 09 1(Modified...

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Unformatted text preview: IEOR 162, Fall 2011 Homework 09 1. (Modified from Problem 6.10.1; 10 points) Glassco manufactures glasses: wine, beer, champagne, and whiskey. Each type of glass requires time in the molding shop, time in the packaging shop, and a certain amount of glass. The resources required to make each type of glass are given below. Wine Beer Champagne Whiskey Molding time (minutes) 4 9 7 10 Packaging time (minutes) 1 1 3 40 Glass (oz) 3 4 2 1 Selling price \$6 \$10 \$9 \$20 Currently, 600 minutes of molding time, 400 minutes of packaging time, and 500 oz of glass are available. Assuming that Glassco wants to maximize revenue by solving the following LP max 6 x 1 + 10 x 2 + 9 x 3 + 20 x 4 s.t. 4 x 1 + 9 x 2 + 7 x 3 + 10 x 4 ≤ 600 (Molding constraint) x 1 + x 2 + 3 x 3 + 40 x 4 ≤ 400 (Packaging constraint) 3 x 1 + 4 x 2 + 2 x 3 + x 4 ≤ 500 (Glass constraint) x i ≥ ∀ i = 1 ,..., 4 . (1) It can be shown that the optimal solution to this LP is ( x * 1 ,x * 2 ,x * 3 ,x * 4 ) = ( 400 3 , , , 20 3 ) and the corre- sponding objective value is z * = 2800 3 . (a) (2 points) Find the dual LP for the Glassco problem in (1). (b) (4 points) Using the given optimal primal solution and the Theorem of Complementary Slackness, find the optimal solution to the dual of the Glassco problem. (c) (4 points; 2 point each) Find an example of each of the complementary slackness conditions. Interpret each example in terms of shadow prices. i. i th primal slack > 0 implies i th dual variable = 0 (This is equation (40) in Section 6.10 of the textbook). ii. i th dual variable > 0 implies i th primal slack = 0 (This is equation (41) in Section 6.10 of the textbook)....
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IEOR162_hw09 - IEOR 162 Fall 2011 Homework 09 1(Modified...

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