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M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY 15.053 – Optimization Methods in Management Science (Spring 2007) Problem Set 5 Due March 22 nd , 2007 at 4:30 pm. You will need 119 points out of 140 to receive a grade of 5. Problem 1: Weak and Strong Duality (36 Points; 4 Each) The idea behind this problem is to explore what types of relationships the optimal solutions of the primal and dual can have. Look at the following grid, for each empty box determine if the situation could occur. If it could occur, give an example. If it could not occur, explain why using weak and strong duality. PRIMAL/ DUAL Finite Optimal Solution Unbounded Solution Infeasible Solution Finite Optimal Solution Part A: Part B: Part C: Unbounded Solution Part D: Part E: Part F: Infeasible Solution Part G: Part H: Part I: For example in Part F you need to determine if it is possible for the primal to be unbounded and the dual to be infeasible. Problem 2: Dual Simplex (28 Points; 4 Points Each) The reason we wrote this problem is to have you practice the basics of dual simplex. Also we wanted to point out the dangers of drinking and driving. Things aren’t so simple to Paris and Nicole these days, both having DUI’s. They are too out of it and forgot how to use the simplex method. Since they are in a duel, they decide to use the dual simplex method to solve their problem. However they are too drunk on sugar to do so and ask you, a super talented 053 student for help. Consider the following primal problem Page 0 of 7
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max. .. z = − x 1 x 2 st : x + 2 x x = 2 1 2 3 1 4 1 = x x , , , 4 3 2 1 x x x x 0 Part A: Take the dual. Part B: Graph the feasible region of the dual Part C: Assume x 3 and x 4 are slack variables, write the original problem and graph its feasible region. Label the corner points of the feasible set. Part D: Letting x3 and x4 be the initial basic variables write out the initial tableau for the simplex method. Is this current solution feasible? Part E: What corner point on your graph does this solution correspond to in the primal problem? Part F:
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