Explanation Given: At least 0.01 of the content of Sugarlo must come from ingredients 3 and 4, 0.01 must come from ingredients 2 and 5, and 0.01 from ingredients 1 and 4. Sugarlo has a sugar content of no more than 10%; the sugar content and the cost per gallon of the ingredients are given below. 1 2 3 4 5 Water Sugar content 0.28 0.19 0.43 0.57 0.22 0 Cost ($) 0.48 0.32 0.53 0.28 0.43 0.04 Explanation: Let the mixture contain x 1 gal, x 2 gal, x 3 gal, x 4 gal, x 5 gal, and x 6 gal of ingredients I, II, III, IV, V, and water, respectively. Then the provided statements would convert into the following system of linear inequalities. x 3 + x 4 ≥ 150 x 2 + x 5 ≥ 150 x 1 + x 4 ≥ 150 0.28 x 1 + 0.19 x 2 + 0.43 x 3 + 0.57 x 4 + 0.22 x 5 + 0 x 6 ≤ 150 x 1 + x 2 + x 3 + x 4 + x 5 + x 6 = 15 , 000 Also, x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ≥ 0 where the objective function to be minimized is z = 0.48 x 1 + 0.32 x 2 + 0.53 x 3 + 0.28 x 4 + 0.43 x 5 + 0.04 x 6 ; 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 slack variables s 1 , s 2 , and s 3 . Introduce surplus variable s 4 and artificial variable a 1 to convert it into a system of linear equations. x 3 + x 4 − s 1 = 150 x 2 + x 5 − s 2 = 150 x 1 + x 4 − s 3 = 150 0.28 x 1 + 0.19 x 2 + 0.43 x 3 + 0.57 x 4 + 0.22 x 5 + 0 x 6 + s 4 = 15 , 000 x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + a 1 = 15 , 000 w + 0.48 x 1 + 0.32 x 2 + 0.53 x 3 + 0.28 x 4 + 0.43 x 5 + 0.04 x 6 = 0 This system of the linear equation gives the following tableau: [ x 1 x 2 x 3 x 4 x 5 x 6 s 1 s 2 s 3 s 4 a 1 w 0 0 1 1 0 0 − 1 0 0 0 0 0 150 0 1 0 0 1 0 0 − 1 0 0 0 0 150 1 0 0 1 0 0 0 0 − 1 0 0 0 150 0.28 0.19 0.43 0.57 0.22 0 0 0 0 1 0 0 15 , 000 1 1 1 1 1 1 0 0 0 0 1 0 15 , 000 0.48 0.32 0.53 0.28 0.43 0.04 0 0 0 0 0 1 0 ] _ The solution of this tableau gives a negative value for the basic variable s 1 ; hence, choose a positive value farthest from the left in the fourth row. Now use ratio test to find the pivot element in this column: 150 1 = 150 , 15 , 000 0.43 = 34883.7 , 15 , 000 1 = 15 , 000 Choose 1 as it gives the least ratio. Use the third element in the first row as the pivot. Use row operation R 4 − 0.43 R 1 → R 4 , R 5 − R 1 → R 5 , R 6 − 0.53 R 1 → R 6 to change the remaining elements in the pivot column to 0. [ x 1 x 2 x 3 x 4 x 5 x 6 s 1 s 2 s 3 s 4 a 1 w 0 0 1 1 0 0 − 1 0 0 0 0 0 150 0 1 0 0 1 0 0 − 1 0 0 0 0 150 1 0 0 1 0 0 0 0 − 1 0 0 0 150 0.28 0.19 0 0.14 0.22 0 0.43 0 0 1 0 0 14 , 935.5 1 1 0 0 1 1 1 0 0 0 1 0 14 , 850 0.48 0.32 0 − 0.25 0.43 0.04 0.53 0 0 0 0 1 − 79.5 ] _ The solution of this tableau gives a negative value for the basic variable s 2 ; hence, choose a positive value farthest from the left in the second row. Now use ratio test to find the pivot element in this column: 150 1 = 150 , 14 , 935.5 0.19 = 78 , 607.8 , 14 , 850 1 = 14 , 850 Choose 1 as the pivot element as the least ratio is obtained for this element. Use the second element in the second row as the pivot. Use row operation R 4 − 0.19 R 2 → R 4 , R 5 − R 2 → R 5 , R 6 − 0.32 R 2 → R 6 to change the remaining elements in the pivot column to 0. [ x 1 x 2 x 3 x 4 x 5 x 6 s 1 s 2 s 3 s 4 a 1 w 0 0 1 1 0 0 − 1 0 0 0 0 0 150 0 1 0 0 1 0 0 − 1 0 0 0 0 150 1 0 0 1 0 0 0 0 − 1 0 0 0 150 0.28 0 0 0.14 0.03 0 0.43 0.19 0 1 0 0 14 , 907 1 0 0 0 0 1 1 1 0 0 1 0 14 , 700 0.48 0 0 − 0.25 0.11 0.04 0.53 0.32 0 0 0 1 − 127.5 ] _ The solution of this tableau gives a negative value for the basic variable s 3 ; hence, choose a positive value farthest from the left in the second row. Now use ratio test to find the pivot element in this column. 150 1 = 150 , 14 , 907 0

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

11th Edition

ISBN: 9780133886818

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

Publisher: PEARSON

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Chapter 4.4, Problem 30E

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Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (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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The amount of each ingredient to be used in preparing 15,000 gals of Sugarlo to minimize the cost.

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