Explanation Given: Find x 1 ≥ 0 and x 2 ≥ 0 such that x 1 + 2 x 2 ≤ 18 x 1 + 3 x 2 ≥ 12 2 x 1 + 2 x 2 ≤ 24 and z = 5 x 1 − 10 x 2 is maximized. Explanation: The following information is provided: Maximize z = 5 x 1 − 10 x 2 Subject to: x 1 + 2 x 2 ≤ 18 (I) x 1 + 3 x 2 ≥ 12 (II) 2 x 1 + 2 x 2 ≤ 24 (III) Step 1: First, add slack variable and subtract surplus variable in order to get equality constraints. Add slack variable (say s 1 ) to equation (I): x 1 + 2 x 2 + s 1 = 18 Subtract surplus variable (say s 2 ) from equation (II): x 1 + 3 x 2 − s 2 = 12 Add slack variable to equation (III): 2 x 1 + 2 x 2 + s 3 = 24 Step 2: Represent the system of linear equations obtained above in the simplex tableau. This is the initial tableau: [ x 1 x 2 s 1 s 2 s 3 z s 1 1 2 1 0 0 0 18 s 2 1 3 0 − 1 0 0 12 s 3 2 2 0 0 1 0 24 z − 5 10 0 0 0 1 0 ] _ The solution from the initial tableau is ( x 1 , x 2 , s 1 , s 2 , s 3 ) = ( 0 , 0 , 0 , 18 , − 12 , 24 ) . This is not a feasible solution because the value of surplus variable ( s 2 ) is negative. Step 3: If any basic variable has a negative value, identify the nonzero number in that variable’s column, and note that row. In the initial tableau, the basic variable s 2 is negative and the nonzero number in this variable column is − 1 , which is in row 2, that is, in R 2 . Step 4: Now, in the row located in step 3, find the most positive entry in the leftmost column. The most positive entry in column 1 is 2 . Step 5: From the column found in step 4, pick a pivot by the test ratio (or by quotients). Column 1 is the pivot column. Apply the test ratio: 18 1 = 18 , 12 1 = 12 , 24 2 = 12 Same ratio is obtained corresponding to row 1 and row 2. Hence, give priority to the element present at the intersection of 2nd row and 1st column. Therefore, 1 is the pivot element, which follows that s 2 is the depart variable and x 1 is the enter variable. Therefore, [ x 1 x 2 s 1 s 2 s 3 z s 1 1 2 1 0 0 0 18 x 1 1 3 0 − 1 0 0 12 s 3 2 2 0 0 1 0 24 z − 5 10 0 0 0 1 0 ] _ Step 6:

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 4.4, Problem 14E

To determine

x1≥0,x2≥0, and value of z by using the simplex method.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.4, Problem 13E

Chapter 4.4, Problem 15E

Students have asked these similar questions

Strick handwritten solutions need

Only human experts solved it. No ai solutions need okk

Only 100% sure experts solve it correct complete solutions need to get full marks it's my quiz okkkk.take your time but solve full accurate okkk Geometry expert solve it

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.

x 1 ≥ 0 , x 2 ≥ 0 , and value of z by using the simplex method.

Solution Summary: The author explains how to determine the value of z by using the simplex method.

Concept explainers

Videos

x1≥0,x2≥0, and value of z by using the simplex method.

Want to see the full answer?

Chapter 4 Solutions