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midtermsolutions - IEOR 162 Linear Programming Spring 2007...

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IEOR 162 Linear Programming Spring 2007 Midterm Solutions 1. a. False. Consider the LP max { x 1 + x 2 : x 1 + x 2 2 , x 1 , x 2 0 } , for which one optimal solution is x 1 = x 2 = 1, which is clearly not a basic feasible solution (since only 1 of the 3 constraints is active at that point). b. False. The problem could be unbounded, in which case x is not an optimal solution. c. False. The correct statement is x is a basic feasible solution if and only if x is an extreme point. d. False. Consider the LP max { x 1 + x 2 : x 1 + x 2 2 , x 1 , x 2 0 } , for which the following two points are feasible: x 1 = (2 , 0) and x 2 = (0 , 2). Now let α = 2, then αx 1 + (1 - α ) x 2 = (4 , 0) - (0 , 2) = (4 , - 2), which is clearly not a feasible solution. e. False. At each iteration of the simplex method, it is possible for all of the basic variables’ values to change as well as one non-basic variable. 2. a. Standard Form: - max - 5 x 1 s.t. 2 x 1 - x + 3 + x - 3 + s 1 = 0 x 2 + 3 x + 3 - 3 x - 3 - e 2 = 0 x 1 + x 2 + s 3 = 0 6 x 1 - 7 x 2 = 0 x 1 , x 2 , x + 3 , x - 3 , s 1 , e 2 , s 3 0 b.
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