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sol5 - CO 350 Linear Optimization(Spring 2009 Solutions to...

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CO 350 Linear Optimization (Spring 2009) Solutions to Assignment 5 [10] 1. (a) The variables that may enter are x 4 and x 5 . For x 4 all coefficients in A 4 are positive. Since in each case b i = 0 any basis element may be leaving. Thus we have ( x 4 , x 2 ),( x 4 , x 3 ) and ( x 4 , x 6 ). If x 5 is entering then x 2 is leaving, so we have ( x 5 , x 2 ). (b) The current matrix is 2 1 0 2 - 4 0 β - 1 1 x 4 enters so we compute (after substituting β = 1) the lexicographic minimum of (0 , 1 , - 2 , 0) = min (0 , 2 , 1 , 0) 2 , (0 , 2 , - 4 , 0) 2 , (0 , 1 , - 1 , 1) 1 therefore x 3 is leaving. (c) We have the same matrix as in part (b). Note that (0 , 1 , 0 . 5 , 0) > L (0 , 1 , - 2 , 0), so we need to ensure that (0 , 1 , - 2 , 0) > L 0 , 1 , - 1 β , 1 β ⇒ - 2 > - 1 β 0 . 5 > β. To have x 6 leaving the basis, we need 0 . 5 > β > 0. [20] 2. A = 1 1 2 0 2 2 3 3 1 2 , b = 3 7 , c = [5 , 8 , 4 , 2 , 3] T . We use x 4 as basic variable. Phase 1 is the LP A = 1 1 2 0 2 1 2 3 3 1 2 0 , b = 3 7 , c = [0 , 0 , 0 , 0 , 0 , - 1] T . Iteration 1, B = { 6 , 4 } and x * B = [3 , 7] T 1. Solve A T B y = c B 1 0 0 1 y 1 y 2 = - 1 0 y = - 1 0 2. Choose entering variable c 1 = c 1 - A T 1 y = 0 - [1 , 2][ - 1 , 0] T = 1 > 0 So x 1 enters the basis. 1
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3. Choose leaving variable - solve A B d = A 1 1 0 0 1 d 6 d 4 = 1 2 d 6 d 4 = 2 1 t = min x * 6 d 6 , x * 4 d 4 = min 3 1 , 7 2 = 3 r = 6 so x 6 is leaving. 4. x * 1 := t = 3 and x * 4 := x * 4 - td 4 = 7 - 3 × 2 = 1 5. B = { 1 , 4 } and all artificial variables are zero, hence the first phase is over. For Phase II we have B = { 1 , 4 } and corresponding x * = [3 , 1] T together with A = 1 1 2 0 2 2 3 3 1 2 , b = 3 7 , c = [5 , 8 , 4 , 2 , 3] T . Iteration 1, 1. Solve A T B y = c B 1 2 0 1 y 1 y 2 = 5 2 y = 1 2 2. Choose entering variable c 2 = c 2 - A T 2 y = 8 - [1 , 3][1 , 2] T = 1 > 0 So x 2 enters the basis. 3. Choose leaving variable - solve A B d = A 2 1 0 2 1 d 1 d 4 = 1 3 d = 1 1 t = min x * 1 d 1 , x * 4 d 4 = min 3 1 , 1 1 = 1 hence x 4 is leaving. 4. x * 2 := t = 1 and x * 1 := x * 1 - td 1 = 3 - 1 × 1 = 2 5. B = { 1 , 2 } Iteration 2, 1. Solve A T B y = c B 1 2 1 3 y 1 y 2 = 5 8 y = - 1 3 2
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2. Choose entering variable c 3 = c 3 - A T 3 y = 4 - [2 , 3][ - 1 , 3] T = - 3 0 c 4 = c 4 - A T 4 y = 2 - [0 , 1][ - 1 , 3] T = - 1 0 c 5 = c 5 - A T 5 y = 3 - [2 , 2][ - 1 , 3] T = - 1 0 No entering variable can be selected. We have optimality, the optimal solution is x * = [2 , 1 , 0 , 0 , 0] T and the optimal value is 18. [20] 3. Iteration 1: B = { 6 , 7 , 8 , 9 } , x * B = 18 3 28 39 , A B = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 . Solve A T B y = c B = 0 0 0 0 to get y = 0 0 0 0 .
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