Explanation Given: Maximize z = 10 x 1 + 12 x 2 Subject to: 4 x 1 + x 2 ≤ 20 5 x 1 + x 2 ≤ 50 2 x 1 + + 2 x 2 ≤ 24 with x 1 ≥ 0 , x 2 ≥ 0 Explanation: The objective function for the provided problem is z = 10 x 1 + 12 x 2 . The constraints were as follows: 4 x 1 + x 2 ≤ 20 5 x 1 + x 2 ≤ 50 2 x 1 + + 2 x 2 ≤ 24 with x 1 ≥ 0 , x 2 ≥ 0 Here, we apply simplex method for maximization. Step 1: Add slack variables (or basic variables) to each constraint to convert it into linear equation. Basic variables are the variables that are nonzero. As a result 4 x 1 + 2 x 2 + s 1 = 20 5 x 1 + x 2 + s 2 = 50 2 x 1 + 2 x 2 + s 3 = 24 with x 1 ≥ 0 , x 2 ≥ 0 , s 1 ≥ 0 , s 2 ≥ 0 and s 3 ≥ 0 Here, s 1 , s 2 , and s 3 are basic variables and x 1 , x 2 are nonbasic variables. Step 2: Represent the system of linear equations obtained above in simplex tableau. This is the initial tableau: [ x 1 x 2 s 1 s 2 s 3 z s 1 4 2 1 0 0 0 20 s 2 5 1 0 1 0 0 50 s 3 2 2 0 0 1 0 24 z − 10 − 12 0 0 0 1 0 ] _ This tableau leads to the solution x 1 = 0 , x 2 = 0 , s 1 = 20 , and s 2 = 50 , s 3 = 24 with z = 0 . Step 3: Select a pivot column, that is, the negative entry in the bottom row with the largest numerical value. In the initial tableau, the negative entry in the bottom row with the largest numerical value is − 12 .Hence, the column that corresponds to − 12 is the pivot column, which follows that x 2 is the entry. Step 4: Select the pivot. The pivot must be a positive number. Compute the ratio a b for each positive entry b in the pivot column, where a is the number in the right most column in that row. This is known as the test ratio. Now, apply the test ratio for the pivot: 20 2 = 10 , 50 1 = 50 , 24 2 = 12 Therefore, 2 is the pivot because the smallest positive ratio is obtained by the element of R 1 , which follows that s 1 is the depart variable. Step 5: Use the pivot to make other entries in the pivot column zero. Convert the pivot entry to 1 (if it is not): [ x 1 x 2 s 1 s 2 s 3 z s 1 4 2 1 0 0 0 20 s 2 5 1 0 1 0 0 50 s 3 2 2 0 0 1 0 24 z − 10 − 12 0 0 0 1 0 ] _ To make pivot element 1 , apply R 1 → R 1 2

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4.2, Problem 9E

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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 solution of the linear programming problem by simplex method.

Solution Summary: The author explains the simplex method for solving the linear programming problem. Add slack variables to each constraint to convert it into a linear equation.

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