Explanation Given: Consider the linear programming problem as follows. Minimize z = x 1 + 2 x 2 subject to − 2 x 1 + x 2 ≥ 1 x 1 − 2 x 2 ≥ 1 with x 1 ≥ 0 , x 2 ≥ 0 Calculations: Obtain the augmented matrix, [ − 2 1 1 1 − 2 1 1 2 0 ] . Observe that the last row is the coefficients of the objective function z = x 1 + 2 x 2 and the upper two rows are the coefficients of the two linear constraints - 2 x 1 + x 2 ≥ 1 and x 1 − 2 x 2 ≥ 1 with their constants. Obtain the transpose of the augmented matrix, [ − 2 1 1 1 − 2 2 1 1 0 ] . Observe that the last row becomes the coefficients of the objective function that is w = y 1 + y 2 and the upper three rows becomes the coefficients as well as constant of the three linear constraints that are − 2 y 1 + y 2 ≤ 1 and y 1 − 2 y 2 ≤ 2 . Hence, the dual of the linear programming problem is obtained as follows. Maximize w = y 1 + y 2 subject to − 2 y 1 + y 2 ≤ 1 y 1 − 2 y 2 ≤ 2 with y 1 ≥ 0 , y 2 ≥ 0 Add slack variable to the above linear programming problem. Maximize w = y 1 + y 2 subject to − 2 y 1 + y 2 + s 1 = 1 y 1 − 2 y 2 + s 2 = 2 with y 1 ≥ 0 , y 2 ≥ 0 , s 1 ≥ 0 , s 2 ≥ 0 Rewrite the objective function, z − y 1 − y 2 = 0 . Now, obtain the simplex tableau, y 1 y 2 s 1 s 2 z [ − 2 1 1 0 0 1 1 − 2 0 1 0 2 − 1 − 1 0 0 1 0 ]

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 4.3, Problem 17E

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To choose: The true statement from the given statements.

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Chapter 4.3, Problem 16E

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Chapter 4 Solutions

Finite Mathematics and Calculus with Applications

Ch. 4.1 - Prob. 1YT Ch. 4.1 - Prob. 2YT Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - For Exercises 58, (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E

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Chapter 4 Solutions