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co227_a2

# co227_a2 - x 4 = 2 4 x 1 x 2-3 x 3 4 x 4 = 1 2 x 1 3 x 2-2...

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Assignment 2 Due: Wednesday October 12 at the BEGINNING of class 1. Convert the following linear programs into standard equality form: (a) minimize x 1 - 6 x 2 + 4 x 3 subject to 8 x 2 + x 3 5 x 1 - 8 x 2 0 x 2 0 (b) maximize 5 x 1 + 2 x 2 - x 3 subject to - 5 x 1 + 3 x 2 + x 3 = - 2 x 1 + x 2 - 3 x 3 7 2 x 1 + x 3 2 x 1 0 2. Write a linear program in standard equality form that satisﬁes the following: (a) A linear program with one variable that is unbounded. (b) A linear program with one variable that is infeasible. (c) A linear program with one variable that has an optimal solution. 3. Given the following linear program, prove whether the linear program is infeasible, unbounded, or has an optimal solution with an appropriate certiﬁcate and show the validity of the certiﬁcate. You may use any of the given vectors in your analysis. (a) maximize x 1 - x 2 + x 3 subject to 2 x 1 - x 2 + x 3 = 2 - x 1 + 2 x 2 - 5 x 3 = 2 4 x 1 - x 2 - x 3 = 6 x 1 ,x 2 ,x 3 0 r = (1 , 3 , 1) T , x = (2 , 2 , 0) T , y = ( - 2 , - 1 , 1) T (b) maximize x 1 + x 2 + 2 x 3 - x 4 subject to 3 x 1 + 5 x 2 + 6 x 3

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Unformatted text preview: + x 4 = 2 4 x 1 + x 2-3 x 3 + 4 x 4 = 1 2 x 1 + 3 x 2-2 x 3 = 5 x 1 ,x 2 ,x 3 ,x 4 ≥ r = (1 , 2 , 3 , 4) T , x = (2 , 1 , ,-1) T , y = (1 , 2 ,-1) T (c) maximize 2 x 1 + x 2 + 4 x 3 subject to 2 x 1-x 2 + x 3 = 2-x 1 + 2 x 2-5 x 3 = 2 4 x 1-x 2-x 3 = 6 x 1 ,x 2 ,x 3 ≥ r = (1 , 3 , 1) T , x = (2 , 2 , 0) T , y = (-2 ,-1 , 1) T 4. For the following, choose a basis, ﬁnd the corresponding basic solution and improve the solution. (a) maximize x 1-x 3 + 4 x 5 subject to 3 x 1 + 5 x 3 + x 4-x 5 = 3-x 1 + x 3-2 x 5 + x 6 = 2 x 1 + x 2 + 2 x 3 + 3 x 5 = 3 x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ≥ (b) maximize-x 1 + 2 x 3 + 4 x 6 subject to x 1 + 4 x 3 + x 5 + x 6 = 4 x 2-x 3 + 3 x 6 = 2 2 x 3 + x 4-2 x 6 = 1 x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 ≥...
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co227_a2 - x 4 = 2 4 x 1 x 2-3 x 3 4 x 4 = 1 2 x 1 3 x 2-2...

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