t2key - MAT419 Linear Optimization Mar 6 2008 TEST 2 NAME...

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MAT419 Linear Optimization Mar. 6, 2008 TEST 2 NAME KEY 1. Suppose the following LOP P has the final tableau, as shown. Max . z = 215 x 1 +68 x 2 209 x 3 s . t . 2 x 1 x 3 17 x 1 +4 x 2 3 x 3 12 +5 x 2 +6 x 3 26 & x 1 , x 2 , x 3 0 7 4 0 3 1 0 0 63 0 8 7 1 2 0 0 7 0 83 0 6 12 7 0 140 0 336 0 854 203 0 7 12082 (a) Use only the final tableau to write the optimal solution x and corresponding dual optimal solution y . x = (63 , 0 , 7) / 7 = (9 , 0 , 1) y = (854 , 203 , 0) / 7 = (122 , 29 , 0) . (b) Prove (without solving P ) that your results from part (a) are correct. 2(9) (1) 17 , (9) + 4(0) 3(1) ≤ − 12 , 5(0) + 6(1) 26 , and x 1 ,x 2 ,x 3 0 , so x is primal-feasible. 2(122) (29) 215 , 4(29) + 5(0) 68 , (122) 3(29) + 6(0) ≥ − 209 , and y 1 ,y 2 ,y 3 0 , so y is dual-feasible. Finally, 215(9) + 68(0) 209(1) = 17(122) 12(29) + 26(0) , so x and y are primal-dual optimal. 2. Consider the following LOP. Max . z = 34 x 1 +33 x 2 2 x 3 s . t . 2 x 1 +6 x 2 +3 x 3 18 x 1 +3 x 2 +5 x 3 11 3 x 1 2 x 3 15 5 x 1 +4 x 2 x 3 17 & x 1 , x 2 , x 3
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