Explanation Given Information: The linear programing problem with mixed constraint is given as: Maximize f = 3 x + 2 y subject to − x + 2 y ≤ 20 − 3 x + 2 y ≤ − 36 x + y ≤ 22 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: Provided the LPP is, Maximize f = 3 x + 2 y subject to the constraints − x + 2 y ≤ 20 − 3 x + 2 y ≤ − 36 x + y ≤ 22 Since, above maximization problem has mixed constraint so first convert all the constraints into standard form. Express all constraints as ≤ constraints. To form the simplex form convert all the inequalities into equations, add slack variable into the inequalities. − x + 2 y + s 1 = 20 − 3 x + 2 y + s 2 = − 36 x + y + s 3 = 22 x ≥ 0 , y ≥ 0 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 s 1 s 2 s 3 f [ − 1 2 1 0 0 0 20 − 3 2 0 1 0 0 − 36 1 1 0 0 1 0 22 − 3 − 2 0 0 0 1 0 ] Since ( − 3 ) is the most negative entry therefore 2 nd column is pivot column and ratio is minimum in first row corresponding to element ( − 3 ) , hence ( − 3 ) is pivot element for this table. Solve the given problem by simplex method. Apply the row operations to solve this matrix: − 1 3 R 2 → R 2 R 2 + R 1 → R 1 − R 2 + R 3 → R 3 3 R 2 + R 4 → R 4 The matrix will become: x y s 1 s 2 s 3 f [ 0 4 3 1 − 1 3 0 0 32 1 − 2 3 − 1 − 1 3 0 0 12 0 5 3 1 1 3 1 0 10 0 − 4 0 − 1 0 1 36 ] As − 4 is the most negative entry, the first column is the pivot column and 5 3 in the row 3 is pivot entry

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 4.5, Problem 18E

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

Maximize f=3x+2y subject to

−x+2y≤20−3x+2y≤−36x+y≤22

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Chapter 4.5, Problem 17E

Chapter 4.5, Problem 19E

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The optimal solution of the LPP with the help of simplex method. Maximize f = 3 x + 2 y subject to − x + 2 y ≤ 20 − 3 x + 2 y ≤ − 36 x + y ≤ 22

Solution Summary: The author calculates the optimal solution of the linear programming problem with the help of simplex method.

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

Maximize f=3x+2y subject to

−x+2y≤20−3x+2y≤−36x+y≤22

Want to see the full answer?

Chapter 4 Solutions

The optimal solution of the LPP with the help of simplex method. Maximize f = 3 x + 2 y subject to − x + 2 y ≤ 20 − 3 x + 2 y ≤ − 36 x + y ≤ 22

Solution Summary: The author calculates the optimal solution of the linear programming problem with the help of simplex method.

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To calculate: The optimal solution of the LPP with the help of simplex method. Maximize f=3x+2y subject to −x+2y≤20−3x+2y≤−36x+y≤22

Want to see the full answer?

Chapter 4 Solutions

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

Maximize f=3x+2y subject to

−x+2y≤20−3x+2y≤−36x+y≤22