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

See similar textbooks

Videos

Question

Chapter 4.3, Problem 15E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.3, Problem 14E

Chapter 4.3, Problem 16E

Students have asked these similar questions

Write out and explain your steps to each of these problems.

Calculate the area and perimeter

65% of all violent felons in the prison system are repeat offenders. If 43 violent felons are randomly selected, find the probability that a. Exactly 28 of them are repeat offenders. b. At most 28 of them are repeat offenders. c. At least 28 of them are repeat offenders. d. Between 22 and 26 (including 22 and 26) of them are repeat offenders.

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...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

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.

Videos

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

Want to see the full answer?

Chapter 4 Solutions