Explanation Given: The cost per pound of bluegrass, rye, and Bermuda is 16, 14, and 12 cents, respectively. In each batch, there must be at least 25% of bluegrass seeds. Amount of Bermuda seeds must be no more than 2 3 the amount of rye seeds. The company must make exactly 6000lb of this mixture. Explanation: Let the mixture contain x 1 lb of bluegrass seeds, x 2 lb of rye seeds, and x 3 lb of Bermuda seeds. Then, the provided statements would convert into the following system of linear inequalities. 0.25 ( x 1 + x 2 + x 3 ) ≤ x 1 x 3 ≤ 2 3 x 2 x 1 + x 2 + x 3 = 6000 Also, x 1 , x 2 , x 3 ≥ 0 where the objective function to be minimized is z = 16 x 1 + 14 x 2 + 12 x 3 ; hence, assume an objective function w such that w = − z then the maximum value of w would be the minimum value of z . Simplify the system of linear inequalities and introduce surplus variable s 1 , slack variable s 2 , and artificial variable a 1 to convert it into a system of linear equations. − 0.75 x 1 + 0.25 x 2 + 0.25 x 3 + s 1 = 0 2 3 x 2 − x 3 − s 2 = 0 x 1 + x 2 + x 3 + a 1 = 6000 w + 16 x 1 + 14 x 2 + 12 x 3 = 0 This system of linear equation gives the following tableau: [ x 1 x 2 x 3 s 1 s 2 a 1 w − 0.75 0.25 0.25 1 0 0 0 0 0 2 3 − 1 0 − 1 0 0 0 1 1 1 0 0 1 0 6000 16 14 12 0 0 0 1 0 ] _ The solution of this tableau gives a negative value for the basic variable s 2 ; hence, choose a positive value farthest to the left in the second row. Now use ratio test to find the pivot element in this column. 0 0.25 = 0 , 0 2 3 = 0 , 6000 1 = 0 Since both the elements 0.25 and 2 3 give the least ratio choose the element 2 3 as choosing this element would eliminate the negative basic variable. Use row operation 3 2 R 2 → R 2 to change the pivot element to 1 and use R 1 − 0.25 R 2 → R 1 , R 3 − R 2 → R 3 , R 4 − 14 R 2 → R 4 To change the remaining elements in the pivot column to 0. [ x 1 x 2 x 3 s 1 s 2 a 1 w − 0.75 0.25 0.25 1 0 0 0 0 0 1 − 1.5 0 − 1.5 0 0 0 1 1 1 0 0 1 0 6000 16 14 12 0 0 0 1 0 ] [ x 1 x 2 x 3 s 1 s 2 a 1 w − 0

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4.4, Problem 29E

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The amount of each kind of seed needed to minimize the cost.

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Chapter 4 Solutions

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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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The amount of each kind of seed needed to minimize the cost.

Solution Summary: The author calculates the cost per pound of bluegrass, rye, and Bermuda. Then, the provided statements would convert into the following system of linear inequalities.

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