X 2 4 4 marks hence from rst principles derive the

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Unformatted text preview: x 2 4 (4 marks) Hence, from rst principles , derive the dual of the linear programming problem minimise v = 4 y 1 y 2 subject to y 1 + 2 y 2 = 4 2 y 1 + 2 y 2 3 y 1 + y 2 8 y 1 unrestricted y 2 (11 marks) AMA405 4 Continued AMA405 5 The optimal tableau for the problem maximise z = x 1 + x 2 + 3 x 3 subject to 2 x 1 + x 2 + x 3 12 (1) x 1 x 2 + x 3 8 (2) x 1 + 2 x 2 x 3 6 (3) x 1 ; x 2 ; x 3 is x 1 x 2 x 3 x 4 x 5 x 6 Solution z 4 2 1 32 x 2 1 = 2 1 1 = 2 1 = 2 2 x 3 3 = 2 1 1 = 2 1 = 2 10 x 6 3 = 2 1 = 2 3 = 2 1 12 where x 4 , x 5 and x 6 are the slack variables associated with constraints (1), (2) and (3) respectively. (i) Write down the values of the dual variables and, using the inverse of the basis matrix, verify their values. (5 marks) (ii) Verify, by adding them to the optimal tableau separately , whether or not the addition of the constraints (a) 2 x 1 + x 2 4 (b) x 1 + x 2 + x 3 10 changes the optimal solution and, if so, determine the new solution. (14 marks) (iii) A new variable is added to the problem which has cost coe cient 4 and coe cients 1, 1 and 2 in constraints (1), (2) and (3), respectively. Indicate whether the addition of this new variable alters the solution of the problem, giving reasons for your answer. If the answer is YES, then indicate brie y what changes to the optimal tableau would be necessary before the Simplex algorithm could be applied to determine the new solution. DO NOT SOLVE THE RESULTING PROBLEM. (6 marks) End of Question Paper AMA405 5...
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