Explanation Given: A political interview takes 45 minutes and a market interview takes 55 minutes. At least eight interviews needed in week. She has to earn at least $60, one political interview pays $8 and one market interview pays$10. One political interview earns six bonus points and one market interview earns five bonus a minimum of 40 bonus points is needed. Explanation: Let her conduct x 1 number of political interviews and x 2 number of market interviews. Then, the provided statements would convert into this system of linear inequalities: x 1 + x 2 ≥ 8 8 x 1 + 10 x 2 ≥ 60 6 x 1 + 5 x 2 ≥ 40 also, x 1 , x 2 ≥ 0 where the objective function to be minimized is z = 45 ( x 1 ) + 55 ( x 2 ) . Use the above information to form this matrix: [ 1 1 8 8 10 60 6 5 40 45 55 0 ] . Transpose this to obtain a dual matrix. [ 1 8 6 45 1 10 5 55 8 60 40 0 ] _ The dual problem thus is y 1 + 8 y 2 + 6 y 3 ≤ 45 y 1 + 10 y 2 + 5 y 3 ≤ 55 w = 8 y 1 + 60 y 2 + 40 y 3 with, x 1 , x 2 ≥ 0 . Where the maximum value of w is the minimum value of z . Introduce slack variables s 1 and s 2 to solve the dual problem y 1 + 8 y 2 + 6 y 3 + s 1 = 45 y 1 + 10 y 2 + 5 y 3 + s 2 = 55 w − 8 y 1 − 60 y 2 − 40 y 3 = 0 This system of linear equation gives the following tableau: [ y 1 y 2 y 3 s 1 s 2 w 1 8 6 1 0 0 45 1 10 5 0 1 0 55 − 8 − 60 − 40 0 0 1 0 ] _ This tableau gives a negative value for the last row; hence, choose the most negative number in the last row and find pivot in this column. Since, -60 is the most negative number use ratio test to find the pivot element in this column. 45 8 = 5.625 , 55 10 = 5.5 Since 10 gives the least ratio use this element as the pivot element. Use row operation 1 10 R 2 → R 2 to change the pivot element to 1 and use R 1 − 8 R 2 → R 1 , R 3 + 60 R 2 → R 3 to change the remaining elements in the pivot column to 0. [ y 1 y 2 y 3 s 1 s 2 w 1 8 6 1 0 0 45 1 10 1 1 2 0 1 10 0 11 2 − 8 − 60 − 40 0 0 1 0 ] [ y 1 y 2 y 3 s 1 s 2 w 1 5 0 2 1 − 4 5 0 1 1 10 1 1 2 0 1 10 0 11 2 − 2 0 − 10 0 6 1 330 ] _ This tableau gives a negative value for the last row; hence, choose the most negative number in the last row and find pivot in this column. Since, -10 is the most negative number use ratio test to find the pivot element in this column. 1 2 = 0.5 , 11 2 1 2 = 11 Since 2 gives the least ratio use this element as the pivot element. Use row operation 1 2 R 1 → R 1 to change the pivot element to 1 and use R 2 − 1 2 R 1 → R 2 , R 3 + 10 R 1 → R 3 to change the remaining elements in the pivot column to 0

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4.3, Problem 23E

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The amount of each interview to minimize time.

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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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The amount of each interview to minimize time.

Solution Summary: The author calculates the minimum hours to be worked in a week when eight political interviews and 0 market interviews are taken. The provided statements would convert into this system of linear inequalities.

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The amount of each interview to minimize time.

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Chapter 4 Solutions