Explanation Given: Minimize w = 7 y 1 + 2 y 2 + 3 y 3 And subject to: y 1 + y 2 + 2 y 3 ≥ 48 y 1 + y 2 ≥ 12 y 3 ≥ 10 3 y 1 + y 3 ≥ 30 With y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 Explanation: Consider the details: Minimize w = 7 y 1 + 2 y 2 + 3 y 3 And subject to: y 1 + y 2 + 2 y 3 ≥ 48 y 1 + y 2 ≥ 12 y 3 ≥ 10 3 y 1 + y 3 ≥ 30 With y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 Use the dual method. The augmented matrix is: [ 1 1 2 48 1 1 0 12 0 3 0 0 1 1 10 30 7 2 3 0 ] And transpose of the matrix is: [ 1 1 0 3 7 1 1 0 0 2 2 0 1 1 3 48 12 10 30 0 ] Further, write the dual problem. Maximize z = 48 x 1 + 12 x 2 + 10 x 3 + 30 x 4 And subject to: x 1 + x 2 + 3 x 4 ≤ 7 x 1 + x 2 ≤ 2 2 x 1 + x 3 + x 4 ≤ 3 With x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 , x 4 ≥ 0 Initial simplex tableau is: [ x 1 x 2 x 3 x 4 s 1 s 2 s 3 z 1 1 0 3 1 0 0 0 7 1 1 0 0 0 1 0 0 2 2 0 1 1 0 0 1 0 3 − 48 − 12 − 10 − 30 0 0 0 1 0 ] Pivot 2 in column 2, row 3. R 1 → − R 3 + 2 R 1 R 2 → − R 3 + 2 R 2 R 4 → 24 R 3 + R 4 [ x 1 x 2 x 3 x 4 s 1 s 2 s 3 z 0 2 − 1 5 2 0 − 1 0 10 0 2 − 1 − 1 0 2 − 1 0 1 2 0 1 1 0 0 1 0 3 0 − 12 − 14 − 6 0 0 24 1 72 ] Pivot 2 in column 2, row 2. R 1 → R 1 − R 2 R 4 → 6 R 2 + R 4 [ x 1 x 2 x 3 x 4 s 1 s 2 s 3 z 0 0 0 6 2 − 2 0 0 10 0 2 − 1 − 1 0 2 − 1 0 1 2 0 1 1 0 0 1 0 3 0 0 8 − 12 0 12 18 1 78 ] Pivot 6 in column 4, row 1. And R 1 → 1 6 R 1 R 2 → 1 12 R 2 R 3 → 1 12 R 3 [ x 1 x 2 x 3 x 4 s 1 s 2 s 3 z 1 0 0 1 1 3 − 1 3 0 0 5 3 0 1 1 2 0 1 6 5 6 1 2 0 4 3 0 1 1 2 0 − 1 6 1 6 1 2 0 2 3 0 0 8 0 4 18 18 1 98 ] Therefore, the minimum value is 98 and y 1 = 4 , y 2 = 8 , y 3 = 18 _ . Now, use the section 4.4 method. First, change the objective. Maximize z = − w = − 7 y 1 − 2 y 2 − 3 y 3 The constraints are not changed. Initial simple tableau is: [ y 1 y 2 y 3 s 1 s 2 s 3 s 4 z 1 1 2 − 1 0 0 0 0 48 1 1 0 0 − 1 0 0 0 12 0 0 1 0 0 − 1 0 0 10 3 0 1 0 0 0 − 1 0 30 7 2 3 0 0 0 0 1 0 ] Pivot 3 in column 1, row 4

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4, Problem 27RE

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The minimization of the problem and then determine the solution of the problem by using the dual method and simplex method.

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Chapter 4, Problem 26RE

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Chapter 4 Solutions

Finite Mathematics (11th Edition)

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The minimization of the problem and then determine the solution of the problem by using the dual method and simplex method.

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The minimization of the problem and then determine the solution of the problem by using the dual method and simplex method.

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