# ExerciseSet3.pdf - Linear Programming 2013-2014 Exercise...

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Linear Programming 2013-2014 Exercise Set 3 Duality and Complementary Slackness Exercise 1 Consider the linear programming problem max 9 x 1 + 12 x 2 s.t. x 1 + 2 x 2 6 2 x 1 + x 2 6 x 1 , x 2 0 . a. Find the best (= smallest) upper bound for this problem by only using the first restriction. Hint: see pages 2–4 of the slides “Duality and Complementary Slackness”. b. Find the best upper bound for this problem by only using the second restriction. c. Formulate the dual problem and find an optimal solution for this problem. d. Which positive linear combination of both restrictions gives the best upper bound? Exercise 2 Consider the linear programming problem max 3 x 1 + 5 x 2 + 4 x 3 s.t. 3 x 1 + 2 x 2 89 x 2 + x 3 3 x 1 + 2 x 2 + x 3 10 x 1 , x 2 , x 3 0 . a. Use the simplex method with the smallest index rule to solve this problem. b. Write down the dual problem and find the optimal solution of the dual problem. 1

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Exercise 3 Consider the linear programming problem max 2 x 1 + 3 x 2 s.t. - 2 x 1 + x 2 2 x 1 - x 2

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