Explanation Given: Minimize w = 2 y 1 + y 2 + 3 y 3 Subject to y 1 + y 2 + y 3 ≥ 100 2 y 1 + y 2 ≥ 50 with y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 Explanation: The equation provided is Minimize w = 2 y 1 + y 2 + 3 y 3 Subject to y 1 + y 2 + y 3 ≥ 100 2 y 1 + y 2 ≥ 50 with y 1 ≥ 0 , y 2 ≥ 0 , y 3 ≥ 0 To change the problem into dual, first write the augmented matrix for the provided problem: [ 1 1 1 100 2 1 0 50 2 1 3 0 ] _ Now, find the transpose of the above matrix: [ 1 2 2 1 1 1 1 0 3 100 50 0 ] _ In this matrix, the last row represents the objective function and the last column represents the constants. When dual problem is derived from original problem and there are m variables and n constraints in original problem, then in dual of the problem there are n variables and m constraints. Hence, the dual problem is derived from the above transpose matrix as follows (use x 1 , x 2 instead of y 1 , y 2 , y 3 and z instead of w ): Maximize z = 100 x 1 + 50 x 2 Therefore, the objective function is z = 100 x 1 + 50 x 2 The constraints were as follows: x 1 + 2 x 2 ≤ 3 x 1 + x 2 ≤ 2 x 1 ≤ 3 with x 1 ≥ 0 , x 2 ≥ 0 , and x 3 ≥ 0 Now, apply the simplex method for maximization. Step 1: Add slack variables or basic variable to each constraint to convert it into linear equations. As a result, x 1 + 2 x 2 + s 1 = 3 x 1 + x 2 + s 2 = 2 x 1 + s 3 = 3 with x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 0 , s 1 ≥ 0 , s 2 ≥ 0 , and s 3 ≥ 0 Step 2: Represent the system of linear equations obtained above in simplex tableau. This is initial tableau. [ x 1 x 2 s 1 s 2 s 3 z s 1 1 2 1 0 0 0 2 s 2 1 1 0 1 0 0 1 s 3 1 0 0 0 1 0 3 z − 100 − 50 0 0 0 1 0 ] _ This tableau leads to the solution x 1 = 0 , x 2 = 0 , s 1 = 2 , s 2 = 1 , a n d s 3 = 3 with z = 0

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4.3, Problem 15E

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To solve: The provided linear programming problem using the simplex method.

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Chapter 4.3, Problem 14E

Chapter 4.3, Problem 16E

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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 provided linear programming problem using the simplex method.

Solution Summary: The author explains the linear programming problem using the simplex method. The required minimum value is underset_w=100.

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To solve: The provided linear programming problem using the simplex method.

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Chapter 4 Solutions