Explanation Given: Minimize w = 4 y 1 + 2 y 2 Subject to: y 1 + 3 y 2 ≥ 6 2 y 1 + 8 y 2 ≤ 21 With y 1 ≥ 0 , y 2 ≥ 0 Explanation: The problem has two constraints for the objective function. We need one surplus variable and one slack variable to make the linear equation. Convert objective function to maximize: z = − w = − 4 x 1 − 2 x 2 The problem can be restated in maximum as Maximize z = − 4 x 1 − 2 x 2 Subject to: x 1 + 3 x 2 ≥ 6 2 x 1 + 8 x 2 ≤ 21 With x 1 ≥ 0 , x 2 ≥ 0 Subtract surplus variable s 1 : x 1 + 3 x 2 ≥ 6 x 1 + 3 x 2 − s 1 = 6 Add slack variable s 2 : 2 x 1 + 8 x 2 ≤ 21 2 x 1 + 8 x 2 + s 2 = 21 Before building tableau, all the variables should be on left side of the equation and constants on right. Rewrite the objective function: 4 x 1 + 2 x 2 + z = 0 Initial simplex tableau is as follows: [ x 1 x 2 s 1 s 2 z 1 3 − 1 0 0 6 2 8 0 1 0 21 4 2 0 0 1 0 ] To get maximum of the function, slack variable s 1 should be zero. The farthest left of − 1 is 1, is pivot column. The smallest quotient formed by dividing each number in the pivot column from the right side of the tableau of the corresponding row becomes pivot row. Corresponding quotients are: 6 1 = 6 21 2 = 10.5 Quotient 6 is smallest, specifying 1 is the pivot in the 1st row, 1st column. [ x 1 x 2 s 1 s 2 z 1 3 − 1 0 0 6 2 8 0 1 0 21 4 2 0 0 1 0 ] Perform row operations to make all zeros except pivot in x 1 column

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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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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Solution Summary: The author explains that the problem has two constraints for the objective function: one surplus variable and one slack variable to make the linear equation.

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