Explanation Given: Maximize z = 20 x 1 + 30 x 2 subject to: 5 x 1 + 10 x 2 ≤ 120 10 x 1 + 15 x 2 ≥ 200 with x 1 ≥ 0 , x 2 ≥ 0 Explanation: The problem has two constraints for the objective function. We need two slack variables to make the linear equation. Adding slack variable s 1 5 x 1 + 10 x 2 ≤ 120 5 x 1 + 10 x 2 + s 1 = 120 Adding surplus variable s 2 10 x 1 + 15 x 2 ≥ 200 10 x 1 + 15 x 2 − s 2 = 200 Before building tableau, all the variables should be on left side of the equation and constants on right. Rewrite the objective function: − 20 x 1 − 30 x 2 + z = 0 Initial simplex tableau is as follows: [ x 1 x 2 s 1 s 2 z 5 10 1 0 0 120 10 15 0 − 1 0 200 − 20 − 30 0 0 1 0 ] To get maximum of the function, pivot procedure is applied till all the indictors are non-negative and solutions are read from the tableau. Step 1: Find the largest negative indicator (number in the last row), because it indicates the variable with the largest coefficient in the objective function. It becomes the pivot column. From the above tableau − 30 is largest negative number. Step 2: 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: 120 10 = 12 200 15 = 13.33 Quotient 12 is smallest, specifying 10 is the pivot in the 1st row, 2nd column. [ x 1 x 2 s 1 s 2 z 5 10 1 0 0 120 10 15 0 − 1 0 200 − 20 − 30 0 0 1 0 ] Perform row operations to make all zeros except pivot in x 2 column. Row 2: 4 R 2 − 6 R 1 Row 3: 3 R 1 + R 3 [ x 1 x 2 s 1 s 2 z 5 10 1 0 0 120 10 0 − 6 − 4 0 80 − 5 0 3 0 1 360 ] From the above tableau − 5 is largest negative number. Repeat step 2

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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Solve the problem using simplex method.

Solution Summary: The author explains that the problem has two constraints for the objective function: we need two slack variables to make the linear equation.

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