Explanation Given Information: The provided constraints are; 5 x − y + w ≥ 1 , 000 z + w ≤ 2 , 000 x + y ≤ 500 x ≥ 0 , y ≥ 0 , z ≥ 0 , w ≥ 0 Formula used: To convert the minimization problem to the maximization problem, multiply the objective function by − 1 . To compute solution of general linear programming problems, use slack and surplus variables to covert inequalities into equations and star the rows for which the active variables have negative value. If there are starred rows, use Phase I to proceed. Phase I: To remove the stars that is to make active variables have positive values. Start from the first starred row and locate the largest positive number and pivot the column, use test ratios and the smallest test ratio corresponds to a pivot and clear the pivot column. Check again and star the rows for which the active variables have negative value and repeat the process until there are no starred rows and move to Phase II. Phase II: Use the simplex method to solve the obtained standard maximization tableau. Choose the smallest negative number on the left side of the bottom row and pivot the column. Select the pivot which is the smallest of the test ratios a b , where a is the entry in the right most column and b is the corresponding entry in the pivot column. Clear the pivot column using the pivot and label the pivot row with the label of the pivot column. Repeat till all the entries in the bottom row are non negative. If there are no negative entries in the bottom row of the standard maximization tableau, then the obtained tableau represents the solution to the problem. Use row operations to clear the column of the augmented matrix. Calculation: Consider the objective function c = x + y + z + w of the minimization problem. Recall that to convert the minimization problem to the maximization problem, multiply the objective function by − 1 as; − c = − ( x + y + z + w ) p = − x − y − z − w Thus, the corresponding maximization problem is Maximize p = − x − y − z − w subject to constraints, 5 x − y + w ≥ 1 , 000 z + w ≤ 2 , 000 x + y ≤ 500 x ≥ 0 , y ≥ 0 , z ≥ 0 , w ≥ 0 Recall that to compute solution of general linear programming problems, use slack and surplus variables to covert inequalities into equations and star the rows for which the active variables have negative value. Subtract the surplus variable s , t and add the slack variable u as; 5 x − y + w − s = 1 , 000 z + w + t = 2 , 000 x + y + u = 500 Thus, the equations are 5 x − y + w − s = 1 , 000 z + w + t = 2 , 000 x + y + u = 500 x + y + z + w + p = 0 Write the equations in augmented matrix as; x y z w s t u p s 5 − 1 0 1 − 1 0 0 0 1000 t 0 0 1 1 0 1 0 0 2000 u 1 1 0 0 0 0 1 0 500 p 1 1 1 1 0 0 0 1 0 Check the values of the active variables s and t as; s = 1000 − 1 = − 1000 t = 2000 1 = 2000 And, u = 500 1 = 500 Since, the value of s is negative, star the s row. Thus, the tableau is; x y z w s t u p * s 5 − 1 0 1 − 1 0 0 0 1000 t 0 0 1 1 0 1 0 0 2000 u 1 1 0 0 0 0 1 0 500 p 1 1 1 1 0 0 0 1 0 Since, there are starred rows, use Phase I to proceed

7th Edition

ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 5.4, Problem 17E

To determine

To calculate: The minimum value of c=x+y+z+w subject to constraints,

5x−y+w≥1,000z+w≤2,000x+y≤500x≥0,y≥0,z≥0,w≥0

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.4, Problem 16E

Chapter 5.4, Problem 18E

Students have asked these similar questions

question 8

Two construction companies are bidding against one another for the right to construct a new community center building. The first construction company, Fine Line Homes, believes that its competitor, Buffalo Valley Construction, will place a bid for this project according to the distribution shown in this table: Buffalo Valley's Bid Bid Probability $160,000 0.2 $165,000 0.5 $170,000 0.2 $175,000 0.1 Furthermore, Fine Line Homes estimates that it will cost $160,000 for its own company to construct this building. Given its fine reputation and long-standing service within the local community, Fine Line Homes believes that it will likely be awarded the project in the event that it and Buffalo Valley Construction submit exactly the same bids. Find the bid that maximizes Fine Line’s expected profit. Max expected profit $ ________ . Bid that maximizes profit $ ________ .

Find the area of the surface obtained by rotating the circle x² + y² = r² about the line y = r.

Chapter 5 Solutions

Finite Mathematics

Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - Prob. 6E Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - Prob. 9E Ch. 5.1 - In Exercises 1-26, sketch the region that...

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 minimum value of c = x + y + z + w subject to constraints, 5 x − y + w ≥ 1 , 000 z + w ≤ 2 , 000 x + y ≤ 500 x ≥ 0 , y ≥ 0 , z ≥ 0 , w ≥ 0

Solution Summary: The author explains how to calculate the minimum value of c=x+y+z+w subject to constraints.

Videos

To calculate: The minimum value of c=x+y+z+w subject to constraints,

5x−y+w≥1,000z+w≤2,000x+y≤500x≥0,y≥0,z≥0,w≥0

Want to see the full answer?

Chapter 5 Solutions

The minimum value of c = x + y + z + w subject to constraints, 5 x − y + w ≥ 1 , 000 z + w ≤ 2 , 000 x + y ≤ 500 x ≥ 0 , y ≥ 0 , z ≥ 0 , w ≥ 0

Solution Summary: The author explains how to calculate the minimum value of c=x+y+z+w subject to constraints.

Videos

To calculate: The minimum value of c=x+y+z+w subject to constraints, 5x−y+w≥1,000z+w≤2,000x+y≤500x≥0,y≥0,z≥0,w≥0

Want to see the full answer?

Chapter 5 Solutions

To calculate: The minimum value of c=x+y+z+w subject to constraints,

5x−y+w≥1,000z+w≤2,000x+y≤500x≥0,y≥0,z≥0,w≥0