HW_4_solution

# HW_4_solution - Solution problem 1(questions comments...

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Solution problem 1: (questions, comments, suggestions please to Steffen Rebennack, [email protected] ) As in the problem stated, we first transform the linear program into standard form. For this, we have to have equality constraints; hence we introduce the (positive) slack variables s 1 , s 2 and s 3 . They will be the initial basis giving us the following tableau: z x 1 x 2 x 3 s 1 s 2 s 3 RHS z 1 -3 -1 -1 0 0 0 0 s 1 0 1 1 0 1 0 0 6 s 2 0 0 1 1 0 1 0 12 s 3 0 1 1 2 0 0 1 20 We are maximizing, so we want a variable to enter the basis which has a neg- ative entry in row 0 of our tableau. Here, we choose variable x 1 to enter the basis. The minimum ratio test gives us that the variable s 1 leaves the basis. The updated tableau reads then as follows z x 1 x 2 x 3 s 1 s 2 s 3 RHS z 1 0 2 -1 3 0 0 18 x 1 0 1 1 0 1 0 0 6 s 2 0 0 1 1 0 1 0 12 s 3 0 0 0 2 -1 0 1 14 Again, we choose a variable with negative coefficient in row 0. This time, it is the unique choice x 3 . The minimum ratio test give us the leaving variable which is in this case s 3 . After the pivoting step, we get z x 1 x 2 x 3 s 1 s 2 s 3 RHS z 1 0 2 0 2.5 0 0.5 25 x 1 0 1 1 0 1 0 0 6 s 2 0 0 1 0 0.5 1 -0.5 5 x 3 0 0 0 1 -0.5 0 0.5 7 As all the coefficients in row 0 are positive, we have found an optimal solution. Finally, the optimal solution is x 1 = 6 x 2 = 0 x 3 = 7 with the objective function value 25. 1

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Solution problem 2: (questions, comments, suggestions please to Zeki Caner Taskin, [email protected] ) We first write the problem in tableu form. z x 1 x 2 x 3 x 4 RHS z 1 -1 3 -1 1 0 x 1 0 1 -1 1 2 15 x 3 0 -2 1 1 -1 0 We first apply Gauss-Jordan elimination to convert the x 1 column to identity.
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