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# Tutorial 7 - s.t x 1 5 x 2 2 x 3 = 30 x 1-5 x 2-6 x 3 ≤...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA3252 Linear and Network Optimization Tutorial 7 1. Consider the following linear programming problem: min 3 x 1 + 4 x 3 s.t. 2 x 1 + x 2 - x 3 - 2 x 1 + 3 x 2 - 5 x 3 7 x 1 0 , x 2 0 (a) Write the standard form of the above LP problem. (b) Verify that the dual of the above LP problem obtained directly from the table of primal-dual relation given in the lecture and the dual of the standard form LP problem in part (a) are equivalent. 2. Consider the following linear programming problem (P) ( P ) min c 0 x s.t. Ax b x 0 (a) Write down the dual problem ( D ). (b) State and prove the Weak Duality Theorem for the linear programming prob- lem ( P ) and its dual problem ( D ). 3. Consider the following linear programming problem: max 2 x 1 + 4 x 2 + 4 x 3 - 3 x 4 s.t. x 1 + x 2 + x 3 = 4 x 1 + 4 x 2 + x 4 = 8 x 1 , x 2 , x 3 , x 4 0 . (a) Write the associated dual problem. (b) Given that x 2 and x 3 yield optimal basic variables to the primal problem, determine the associated optimal dual solution. 1

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4. Consider the following linear programming problem: min - 5 x 1 - 2 x 2 - 3 x 3 s.t.
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Unformatted text preview: s.t. x 1 + 5 x 2 + 2 x 3 = 30 x 1-5 x 2-6 x 3 ≤ 40 x 1 ,x 2 ,x 3 ≥ Let x 4 be the artiﬁcial variable of the ﬁrst constraint and x 5 be the slack variable of the second constraint. Both variables are the starting basic variables for the implementation of the big-M method. The partial optimal simplex tableau is given as follows: basic x 1 x 2 x 3 x 4 x 5 Solution ¯ c 23 7 (5 + M ) 150 x 1 1 x 5 1 (a) Write the associated dual problem and determine its optimal solution from the above tableau. (b) Complete the optimal tableau. 5. Consider the following primal LP problem: min 6 x 1-5 x 3 s.t. 6 x 1-3 x 2 + x 3 = 2 3 x 1 + 4 x 2 + x 3 ≤ 5 x 1-7 x 2 ≤ 5 x 1 ≥ ,x 2 ≤ . (a) Write the dual of the above linear programming problem. (b) Use the Complementary Slackness Theorem to check whether the solution ( x 1 ,x 2 ,x 3 ) = (0 , , 2) is an optimal solution. 2...
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