Explanation Given: Minimize w = 4 y 1 + 7 y 2 + 9 y 3 Subject to 2 y 1 + 3 y 2 + 4 y 3 ≥ 45 y 1 + 5 y 2 + 2 y 3 ≥ 40 With y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 Explanation: Use the given information to write the matrix as follows. [ 2 3 4 45 1 5 2 40 4 7 9 0 ] _ Transpose this matrix for the dual problem by interchanging rows and columns of the above matrix. [ 2 1 4 3 5 7 4 2 9 45 40 0 ] _ Dual problem for the transpose matrix is as follows: Maximize z = 45 x 1 + 40 x 2 subject to 2 x 1 + x 2 ≤ 4 3 x 1 + 5 x 2 ≤ 7 4 x 1 + 2 x 2 ≤ 9 with x 1 ≥ 0 , x 2 ≥ 0 Solve the standard maximization problem using simplex method. Add slack variables to all the equations. There are three equations, so there would be three slack variables s 1 , s 2 , s 3 . 2 x 1 + x 2 + s 1 = 4 3 x 1 + 5 x 2 + s 2 = 7 4 x 1 + 2 x 2 + s 3 = 9 − 45 x 1 − 40 x 2 + z = 0 with x 1 ≥ 0 , x 2 ≥ 0 , s 1 ≥ 0 , s 2 ≥ 0 , s 3 ≥ 0 The first tableau for these equations is given below. Apply R 2 → R 2 − 5 R 1 , R 3 → R 3 − 2 R 1 , and R 4 → R 4 + 15 R 2 to get x 1 x 2 s 1 s 2 s 3 z [ 2 1 1 0 0 0 4 3 5 0 1 0 0 7 4 2 0 0 1 0 9 − 45 − 40 0 0 0 1 0 ] Now, apply R 2 → − 1 7 R 2 and R 4 → 1 35 R 4 to get x 1 x 2 s 1 s 2 s 3 z [ 2 1 1 0 0 0 4 − 7 0 − 5 1 0 0 − 13 0 0 − 2 0 1 0 1 0 35 0 15 0 1 105 ] Next, apply R 1 → R 1 − 2 R 2 . x 1 x 2 s 1 s 2 s 3 z [ 2 1 1 0 0 0 4 1 0 5 7 − 1 7 0 0 13 7 0 0 − 2 0 1 0 1 0 1 0 15 35 0 1 35 105 35 ] Now, apply R 4 → R 4 − R 1

11th Edition

ISBN: 9780321979438

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 4.3, Problem 16E

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The solution of the given minimization problem using simplex method.

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Chapter 4 Solutions

Finite Mathematics (11th Edition)

Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 2E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 4E Ch. 4.1 - For Exercises 5-8. (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - Introduce slack variables as necessary, then...Ch. 4.1 - Introduce slack variables as necessary, then write...

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