Explanation Given Information: The provided optimization problem is given as Maximize f = 20 x + 12 y + 12 z subject to x + z ≤ 40 x + y ≤ 30 y + z ≤ 40 Formula used: To solve the linear programming problem by simplex method, follow the steps below: Step 1: Use slack variables and write the constraint inequalities in equation form. Step 2: Write the equations in a simplex matrix. Step 3: Choose the most negative number on the left side of the bottom row and pivot the column. Step 4: Select the pivot entry which is the smallest of the test ratios a b , where, a is entry in the right most column and b is the corresponding entry in the pivot column. Step 5: Make the pivot entry as 1 and other entries of pivot column as 0 by the use of row operations. Step 6: Repeat the above steps till all the entries in the bottom row are non-negative. Calculation: Consider the provided function, Maximize f = 20 x + 12 y + 12 z subject to x + z ≤ 40 x + y ≤ 30 y + z ≤ 40 To form the simplex table convert all the inequalities into equations by adding slack variable into the inequalities. x + z + s 1 = 40 x + y + s 2 = 30 y + z + s 3 = 40 Where s 1 , s 2 and s 3 are two slack variables with s 1 , s 2 , s 3 ≥ 0 . Simplex matrix for the LPP is written below: x y z s 1 s 2 s 3 f [ 1 0 1 1 0 0 0 40 1 1 0 0 1 0 0 30 0 1 1 0 0 1 0 40 − 20 − 12 − 12 0 0 0 1 0 ] Now, to find the pivot element, first find the pivot row and pivot column. Most negative value in last row is ( − 20 ) in first column therefore variable x will enter in the simplex table. To finding the leaving variable, calculate minimum ratio. Minimum ratio = min { 40 1 , 30 1 } = 30 Second row is pivot row and leave the table. Pivot element is 1. Apply the row operations to solve this matrix: − R 2 + R 1 → R 1 R 4 + 20 R 2 → R 4 The simplex matrix will become: x y z s 1 s 2 s 3 f [ 0 − 1 1 1 − 1 0 0 10 1 1 0 0 1 0 0 30 0 1 1 0 0 1 0 40 0

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 4.3, Problem 26E

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To calculate: The optimal solution of the maximization problem with the help of simplex method.

Maximize f=20x+12y+12z subject to

x+z≤40x+y≤30y+z≤40

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Chapter 4.3, Problem 25E

Chapter 4.3, Problem 27E

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The optimal solution of the maximization problem with the help of simplex method. Maximize f = 20 x + 12 y + 12 z subject to x + z ≤ 40 x + y ≤ 30 y + z ≤ 40

Solution Summary: The author explains how to calculate the optimal solution of the linear programming problem by simplex method.

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To calculate: The optimal solution of the maximization problem with the help of simplex method.

Maximize f=20x+12y+12z subject to

x+z≤40x+y≤30y+z≤40

Want to see the full answer?

Chapter 4 Solutions

The optimal solution of the maximization problem with the help of simplex method. Maximize f = 20 x + 12 y + 12 z subject to x + z ≤ 40 x + y ≤ 30 y + z ≤ 40

Solution Summary: The author explains how to calculate the optimal solution of the linear programming problem by simplex method.

Videos

To calculate: The optimal solution of the maximization problem with the help of simplex method. Maximize f=20x+12y+12z subject to x+z≤40x+y≤30y+z≤40

Want to see the full answer?

Chapter 4 Solutions

To calculate: The optimal solution of the maximization problem with the help of simplex method.

Maximize f=20x+12y+12z subject to

x+z≤40x+y≤30y+z≤40