To determine To calculate: The maximization problem. Explanation Given: The provided table, Grape A (bushels) Grape B (bushels) Sugar (Pounds) Profit (Dollars) Fruity 2 2 2 12 Crystal 1 3 1 15 The winery has available 110 bushels of grape A, 125 bushels of grape B , and 90 pounds of sugar. Formula used: Calculation: Consider the maximization problem, first formulate the problem in the system of linear equations. Let x 1 be the number of gallons of fruity wine then, x 2 be the number of gallons of crystal wine. Consider the provided table, Grape A (bushels) Grape B (bushels) Sugar (Pounds) Profit (Dollars) Fruity 2 2 2 12 Crystal 1 3 1 15 Let z be the objective function to maximize as the profit is given fruity and crystal wine are $ 12 , $ 15 respectively. So, the objective function is: z = 12 x 1 + 15 x 2 From two types of wines, grape A bushels is 110 and the fruity and crystal wines are 2 , 1 respectively, so the constraints is: 2 x 1 + x 2 ≤ 110 From two types of wines, grape B bushels is 125 and the fruity and crystal wines are 2 , 3 respectively, so the constraints is: 2 x 1 + 3 x 2 ≤ 125 From two types of wines, sugar is 90 and the fruity and crystal wines are 2 , 1 respectively, so the constraints are: 2 x 1 + x 2 ≤ 90 Therefore, formulate the problem which is: Maximize, z = 12 x 1 + 15 x 2 Subject to, 2 x 1 + x 2 ≤ 110 , 2 x 1 + 3 x 2 ≤ 125 , 2 x 1 + x 2 ≤ 90 x 1 ≥ 0 , x 2 ≥ 0 Use two stages method to solve the maximization problem as there are various steps which must be followed: Step (1): Modify each equation constraint by an equivalent pair of inequality constraint Step (2): Write each constraint with a positive constant. Step (3): If minimization problem then convert into maximization by replacing z = − w . Step (4): Add or subtract surplus variables according to the need of the constraints into equations. Step (5): Write initial simplex table. Step (6): Find a basic feasible solution if a solution exists, which is stage I Step (7): If basic feasible solution is found, then do the simplex method further which is a stage and get the optimal solution. Further, to convert linear inequality into the linear equation, take a slack variable and surplus variable which is done by adding and subtracting a non-negative variable to each constraint. If linear equality has less than or equal to sign, then use slack variable by adding as the equation is less than or equal to the value, if it is equal then the slack variable is zero. If linear equality has greater than or equal to sign then, use the surplus variable by subtracting as the equation is greater than or equal to the value, if it is equal then the surplus variable is zero which is: 2 x 1 + x 2 + s 1 = 110 , 2 x 1 + 3 x 2 + s 2 = 125 , 2 x 1 + x 2 + s 3 = 90 where s 1 , s 2 , s 3 are greater than or equal to zero. Then, rewrite constraints by adding or subtracting the non-negative slack and surplus variables which are: Maximize, z = 12 x 1 + 15 x 2 Subject to, 2 x 1 + x 2 + s 1 = 110 , 2 x 1 + 3 x 2 + s 2 = 125 , 2 x 1 + x 2 + s 3 = 90 , x 1 ≥ 0 , x 2 ≥ 0 , s 1 ≥ 0 , s 2 ≥ 0 , s 3 ≥ 0 Then, convert the linear programming into a system of linear equations in which all the coefficient of variables is written on the left of the equal sign and the constants are on the right. Then, convert the objective function into linear equations which are: − 12 x 1 − 15 x 2 + z = 0 Then, write the constraint and objective function into the augmented matrix which is in the form of a simplex table: x 1 x 2 s 1 s 2 s 3 [ 2 1 1 0 0 110 2 3 0 1 0 125 2 1 0 0 1 90 − 12 − 15 0 0 0 0 ] Here, the last row shows the coefficients of the objective function and last columns shows the constants. The basic variables correspond to columns which have an entry of 1 and rest is zero its value corresponds to the last column which is constant, and the non-basic variables correspond to the other columns and its value is zero. In the above, nonbasic variables are x 1 , x 2 and the basic variables are s 1 , s 2 , s 3 . Here, x 1 = 0 , x 2 = 0 , , s 1 = 110 , s 2 = 125 , s 3 = 90 In the above simplex, the table has a basic feasible solution as the basic variable has no negative value, so it is said to be basic feasible solution and stage I is completed. In the last row, all the entries except the last and second last entries are called indicators. The simplex table represents a system of four linear equations in six variables which is x 1 , x 2 , , s 1 , s 2 , s 3 , z as there are more variables than equations then the system is dependent, and it has infinitely many solutions. Apply the simplex method for stage II: x 1 x 2 s 1 s 2 s 3 [ 2 1 1 0 0 110 2 3 0 1 0 125 2 1 0 0 1 90 − 12 − 15 0 0 0 0 ] In the simplex, choose the pivot entry as the pivot is the most negative indicator, which is − 15 . Then, the column containing the most negative indicator is called pivot column, which is x 2 Now all the positive entry of pivot column is divided by the last column which is constant. First, 110 ÷ 1 = 110 The second one, 125 ÷ 3 = 125 3 ≈ 41.66 Then last one, 90 ÷ 1 = 90 After the row with the smallest quotient is called the pivot row, which is 125 3 and it occurs in the second row. The entry in the pivot row and pivot column is called pivot, which is 3 . So, the simplex table with pivot element is: x 1 x 2 s 1 s 2 s 3 [ 2 1 1 0 0 110 2 3 0 1 0 125 2 1 0 0 1 90 − 12 − 15 0 0 0 0 ] In this, the pivot is 3 , the pivot is selected, then do the row operations in the initial simplex table to get the new simplex table by eliminating pivot column all value to zero except the pivot, which is changed by row transform into 1

Mathematics with Applications In the Management, Natural and Social Sciences (11th Edition)

11th Edition

ISBN: 9780321931078

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 7, Problem 59RE

To determine

To calculate: The maximization problem.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7, Problem 58RE

Chapter 7, Problem 60RE

Chapter 7 Solutions

Mathematics with Applications In the Management, Natural and Social Sciences (11th Edition)

Ch. 7.1 - Checkpoint 1 Graph the given...Ch. 7.1 - Prob. 2CP Ch. 7.1 - Prob. 3CP Ch. 7.1 - Prob. 4CP Ch. 7.1 - Prob. 5CP Ch. 7.1 - Checkpoint 6 Graph the feasible region of the...Ch. 7.1 - Prob. 7CP Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...

Ch. 7.1 - Prob. 4E Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Match the inequality with its graph, which is one...Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Prob. 9E Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Prob. 11E Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Prob. 17E Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Prob. 19E Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Prob. 22E Ch. 7.1 - Prob. 23E Ch. 7.1 - Graph each of the given linear inequalities. (See...Ch. 7.1 - Prob. 25E Ch. 7.1 - Prob. 26E Ch. 7.1 - Prob. 27E Ch. 7.1 - Prob. 28E Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Prob. 32E Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Prob. 34E Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Prob. 37E Ch. 7.1 - Prob. 38E Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Prob. 40E Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Prob. 43E Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Graph the feasible region for the given systems of...Ch. 7.1 - Prob. 47E Ch. 7.1 - Prob. 48E Ch. 7.1 - Prob. 49E Ch. 7.1 - Prob. 50E Ch. 7.1 - Prob. 51E Ch. 7.1 - 52. Business A manufacturer of electric shavers...Ch. 7.1 - Prob. 53E Ch. 7.1 - Prob. 54E Ch. 7.1 - Prob. 55E Ch. 7.1 - Prob. 56E Ch. 7.2 - Checkpoint 1 Suppose the objective function in...Ch. 7.2 - Prob. 2CP Ch. 7.2 - Prob. 3CP Ch. 7.2 - Checkpoint 4 Use the region of feasible solutions...Ch. 7.2 - Prob. 5CP Ch. 7.2 - Prob. 6CP Ch. 7.2 - Exercises 1–6 show regions of feasible solutions....Ch. 7.2 - Exercises 1–6 show regions of feasible solutions....Ch. 7.2 - Exercises 1–6 show regions of feasible solutions....Ch. 7.2 - Exercises 1–6 show regions of feasible solutions....Ch. 7.2 - Exercises 1–6 show regions of feasible solutions....Ch. 7.2 - Prob. 6E Ch. 7.2 - Use graphical methods to solve Exercises 7–12....Ch. 7.2 - Use graphical methods to solve Exercises 7–12....Ch. 7.2 - Prob. 9E Ch. 7.2 - Use graphical methods to solve Exercises 7–12....Ch. 7.2 - Use graphical methods to solve Exercises 7–12....Ch. 7.2 - Use graphical methods to solve Exercises 7–12....Ch. 7.2 - Find the minimum and maximum values of (if...Ch. 7.2 - Find the minimum and maximum values of (if...Ch. 7.2 - Find the minimum and maximum values of (if...Ch. 7.2 - Find the minimum and maximum values of (if...Ch. 7.2 - Prob. 17E Ch. 7.2 - 18. Find values and that maximize subject to...Ch. 7.2 - Prob. 19E Ch. 7.2 - Prob. 20E Ch. 7.3 - Prob. 1CP Ch. 7.3 - Prob. 2CP Ch. 7.3 - Prob. 3CP Ch. 7.3 - Prob. 1E Ch. 7.3 - Prob. 2E Ch. 7.3 - Prob. 3E Ch. 7.3 - Write the constraints in Exercises 1–4 as linear...Ch. 7.3 - Prob. 5E Ch. 7.3 - Solve these linear programming problems, which are...Ch. 7.3 - Prob. 7E Ch. 7.3 - Solve these linear programming problems, which are...Ch. 7.3 - Prob. 9E Ch. 7.3 - Prob. 10E Ch. 7.3 - Prob. 11E Ch. 7.3 - Solve these linear programming problems, which are...Ch. 7.3 - Prob. 13E Ch. 7.3 - Solve the following linear programming problems....Ch. 7.3 - Prob. 15E Ch. 7.3 - Prob. 16E Ch. 7.3 - Prob. 17E Ch. 7.3 - Prob. 18E Ch. 7.3 - Prob. 19E Ch. 7.3 - Prob. 20E Ch. 7.3 - Prob. 21E Ch. 7.3 - Solve the following linear programming problems....Ch. 7.3 - Prob. 23E Ch. 7.3 - Prob. 24E Ch. 7.3 - Prob. 25E Ch. 7.3 - Prob. 26E Ch. 7.3 - Prob. 27E Ch. 7.3 - Prob. 28E Ch. 7.3 - Prob. 29E Ch. 7.4 - Prob. 1CP Ch. 7.4 - Prob. 2CP Ch. 7.4 - Prob. 3CP Ch. 7.4 - Prob. 4CP Ch. 7.4 - Prob. 5CP Ch. 7.4 - Prob. 6CP Ch. 7.4 - Prob. 1E Ch. 7.4 - Prob. 2E Ch. 7.4 - In Exercises 1–4, (a) determine the number of...Ch. 7.4 - Prob. 4E Ch. 7.4 - Prob. 5E Ch. 7.4 - Prob. 6E Ch. 7.4 - Prob. 7E Ch. 7.4 - Prob. 8E Ch. 7.4 - Prob. 9E Ch. 7.4 - Prob. 10E Ch. 7.4 - Prob. 11E Ch. 7.4 - Prob. 12E Ch. 7.4 - Prob. 13E Ch. 7.4 - Prob. 14E Ch. 7.4 - Prob. 15E Ch. 7.4 - Prob. 16E Ch. 7.4 - Prob. 17E Ch. 7.4 - Prob. 18E Ch. 7.4 - Prob. 19E Ch. 7.4 - Prob. 20E Ch. 7.4 - Prob. 21E Ch. 7.4 - Use the simplex method to solve Exercises...Ch. 7.4 - Prob. 23E Ch. 7.4 - Prob. 24E Ch. 7.4 - Prob. 25E Ch. 7.4 - Prob. 26E Ch. 7.4 - Prob. 27E Ch. 7.4 - Prob. 28E Ch. 7.4 - Prob. 29E Ch. 7.4 - Prob. 30E Ch. 7.4 - Prob. 31E Ch. 7.4 - Prob. 32E Ch. 7.4 - Use the simplex method to solve Exercises...Ch. 7.4 - Prob. 34E Ch. 7.4 - Use the simplex method to solve Exercises...Ch. 7.4 - Use the simplex method to solve Exercises...Ch. 7.4 - Prob. 37E Ch. 7.4 - Prob. 38E Ch. 7.5 - Prob. 1CP Ch. 7.5 - Prob. 1E Ch. 7.5 - Prob. 2E Ch. 7.5 - Prob. 3E Ch. 7.5 - Set up the initial simplex tableau for each of the...Ch. 7.5 - In each of the given exercises, (a) use the...Ch. 7.5 - In each of the given exercises, (a) use the...Ch. 7.5 - In each of the given exercises, (a) use the...Ch. 7.5 - Prob. 8E Ch. 7.5 - Prob. 9E Ch. 7.5 - Prob. 10E Ch. 7.5 - Prob. 11E Ch. 7.5 - Prob. 12E Ch. 7.5 - Prob. 13E Ch. 7.5 - Prob. 14E Ch. 7.5 - Prob. 15E Ch. 7.5 - Prob. 16E Ch. 7.5 - Prob. 17E Ch. 7.5 - Prob. 18E Ch. 7.5 - Prob. 19E Ch. 7.5 - Prob. 20E Ch. 7.5 - Prob. 21E Ch. 7.5 - Prob. 22E Ch. 7.6 - Checkpoint 1 Give the transpose of each...Ch. 7.6 - Prob. 2CP Ch. 7.6 - Prob. 3CP Ch. 7.6 - Prob. 4CP Ch. 7.6 - Prob. 5CP Ch. 7.6 - Prob. 6CP Ch. 7.6 - Prob. 1E Ch. 7.6 - Prob. 2E Ch. 7.6 - Prob. 3E Ch. 7.6 - Prob. 4E Ch. 7.6 - Prob. 5E Ch. 7.6 - Prob. 6E Ch. 7.6 - Prob. 7E Ch. 7.6 - Prob. 8E Ch. 7.6 - Prob. 9E Ch. 7.6 - Prob. 10E Ch. 7.6 - Prob. 11E Ch. 7.6 - Prob. 12E Ch. 7.6 - Prob. 13E Ch. 7.6 - Prob. 14E Ch. 7.6 - Prob. 15E Ch. 7.6 - Prob. 16E Ch. 7.6 - Prob. 17E Ch. 7.6 - Prob. 18E Ch. 7.6 - Prob. 19E Ch. 7.6 - Prob. 20E Ch. 7.6 - Prob. 21E Ch. 7.6 - Prob. 22E Ch. 7.6 - Prob. 23E Ch. 7.6 - Prob. 24E Ch. 7.6 - Prob. 25E Ch. 7.6 - Prob. 26E Ch. 7.6 - Prob. 27E Ch. 7.6 - 28. Business An animal food must provide at least...Ch. 7.6 - Prob. 29E Ch. 7.6 - 30. Business Joan McKee has a part-time job...Ch. 7.6 - Prob. 31E Ch. 7.6 - Prob. 32E Ch. 7.6 - Prob. 33E Ch. 7.6 - Prob. 34E Ch. 7.7 - Prob. 1CP Ch. 7.7 - Prob. 2CP Ch. 7.7 - Prob. 3CP Ch. 7.7 - Prob. 4CP Ch. 7.7 - Prob. 5CP Ch. 7.7 - Prob. 1E Ch. 7.7 - Prob. 2E Ch. 7.7 - Prob. 3E Ch. 7.7 - Prob. 4E Ch. 7.7 - Prob. 5E Ch. 7.7 - Prob. 6E Ch. 7.7 - Prob. 7E Ch. 7.7 - Prob. 8E Ch. 7.7 - Prob. 9E Ch. 7.7 - Prob. 10E Ch. 7.7 - Prob. 11E Ch. 7.7 - Prob. 12E Ch. 7.7 - Prob. 13E Ch. 7.7 - Prob. 14E Ch. 7.7 - Prob. 15E Ch. 7.7 - Prob. 16E Ch. 7.7 - Prob. 17E Ch. 7.7 - Prob. 18E Ch. 7.7 - Prob. 19E Ch. 7.7 - Prob. 20E Ch. 7.7 - Prob. 21E Ch. 7.7 - Prob. 22E Ch. 7.7 - Prob. 23E Ch. 7.7 - Prob. 24E Ch. 7.7 - Prob. 25E Ch. 7.7 - Prob. 26E Ch. 7.7 - Prob. 27E Ch. 7.7 - Prob. 28E Ch. 7.7 - Prob. 29E Ch. 7.7 - Prob. 30E Ch. 7.7 - Prob. 31E Ch. 7.7 - Prob. 32E Ch. 7.7 - Prob. 33E Ch. 7.7 - Prob. 34E Ch. 7.7 - Prob. 35E Ch. 7.7 - Use the two-stage method to solve Exercises 33–40....Ch. 7.7 - Prob. 37E Ch. 7.7 - Prob. 38E Ch. 7.7 - Prob. 39E Ch. 7.7 - Prob. 40E Ch. 7.7 - Prob. 41E Ch. 7.7 - Prob. 42E Ch. 7.7 - Prob. 43E Ch. 7 - Prob. 1CE Ch. 7 - 2. Consider preparing a stir-fry using beef, oil,...Ch. 7 - Prob. EP Ch. 7 - Prob. 1RE Ch. 7 - Prob. 2RE Ch. 7 - Graph each of the given linear inequalities. 3. Ch. 7 - Prob. 4RE Ch. 7 - Prob. 5RE Ch. 7 - Prob. 6RE Ch. 7 - Prob. 7RE Ch. 7 - Prob. 8RE Ch. 7 - Prob. 9RE Ch. 7 - Prob. 10RE Ch. 7 - Prob. 11RE Ch. 7 - Prob. 12RE Ch. 7 - Prob. 13RE Ch. 7 - Use the graphical method to solve Exercises...Ch. 7 - Use the graphical method to solve Exercises...Ch. 7 - Prob. 16RE Ch. 7 - Prob. 17RE Ch. 7 - Prob. 18RE Ch. 7 - 19. Finance The BlackRock Equity Dividend Fund...Ch. 7 - Prob. 20RE Ch. 7 - Prob. 21RE Ch. 7 - Prob. 22RE Ch. 7 - Prob. 23RE Ch. 7 - Prob. 24RE Ch. 7 - Prob. 25RE Ch. 7 - Prob. 26RE Ch. 7 - Prob. 27RE Ch. 7 - Prob. 28RE Ch. 7 - Prob. 29RE Ch. 7 - Prob. 30RE Ch. 7 - Prob. 31RE Ch. 7 - Prob. 32RE Ch. 7 - Prob. 33RE Ch. 7 - Prob. 34RE Ch. 7 - Prob. 35RE Ch. 7 - Prob. 36RE Ch. 7 - 37. When is it necessary to use the simplex method...Ch. 7 - Prob. 38RE Ch. 7 - 39. What kind of problem can be solved with the...Ch. 7 - 40. In solving a linear programming problem, you...Ch. 7 - Prob. 41RE Ch. 7 - Prob. 42RE Ch. 7 - Prob. 43RE Ch. 7 - Prob. 44RE Ch. 7 - Prob. 45RE Ch. 7 - Use the method of duals to solve these...Ch. 7 - Prob. 47RE Ch. 7 - Prob. 48RE Ch. 7 - Prob. 49RE Ch. 7 - Prob. 50RE Ch. 7 - Prob. 51RE Ch. 7 - Prob. 52RE Ch. 7 - Use the two-stage method to solve these...Ch. 7 - Prob. 54RE Ch. 7 - Prob. 55RE Ch. 7 - Prob. 56RE Ch. 7 - Business Solve the following maximization...Ch. 7 - Prob. 58RE Ch. 7 - Business Solve the following maximization...Ch. 7 - Prob. 60RE Ch. 7 - Prob. 61RE Ch. 7 - Business Solve the following minimization...Ch. 7 - Business Solve these mixed-constraint...Ch. 7 - Business Solve these mixed-constraint...

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.

Recommended textbooks for you

Discrete Mathematics and Its Applications ( 8th I...
Math
ISBN:9781259676512
Author:Kenneth H Rosen
Publisher:McGraw-Hill Education
Mathematics for Elementary Teachers with Activiti...
Math
ISBN:9780134392790
Author:Beckmann, Sybilla
Publisher:PEARSON
Calculus Volume 1
Math
ISBN:9781938168024
Author:Strang, Gilbert
Publisher:OpenStax College
Thinking Mathematically (7th Edition)
Math
ISBN:9780134683713
Author:Robert F. Blitzer
Publisher:PEARSON
Discrete Mathematics With Applications
Math
ISBN:9781337694193
Author:EPP, Susanna S.
Publisher:Cengage Learning,
Pathways To Math Literacy (looseleaf)
Math
ISBN:9781259985607
Author:David Sobecki Professor, Brian A. Mercer
Publisher:McGraw-Hill Education

Discrete Mathematics and Its Applications ( 8th I...

Math

ISBN:9781259676512

Author:Kenneth H Rosen

Publisher:McGraw-Hill Education

Mathematics for Elementary Teachers with Activiti...

Math

ISBN:9780134392790

Author:Beckmann, Sybilla

Publisher:PEARSON

Calculus Volume 1

Math

ISBN:9781938168024

Author:Strang, Gilbert

Publisher:OpenStax College

Thinking Mathematically (7th Edition)

Math

ISBN:9780134683713

Author:Robert F. Blitzer

Publisher:PEARSON

Discrete Mathematics With Applications

Math

ISBN:9781337694193

Author:EPP, Susanna S.

Publisher:Cengage Learning,

Pathways To Math Literacy (looseleaf)

Math

ISBN:9781259985607

Author:David Sobecki Professor, Brian A. Mercer

Publisher:McGraw-Hill Education

SEE MORE TEXTBOOKS

Solve ANY Optimization Problem in 5 Steps w/ Examples. What are they and How do you solve them?; Author: Ace Tutors;https://www.youtube.com/watch?v=BfOSKc_sncg;License: Standard YouTube License, CC-BY

Optimization Problems in Calculus; Author: Professor Dave Explains;https://www.youtube.com/watch?v=q1U6AmIa_uQ;License: Standard YouTube License, CC-BY

Introduction to Optimization; Author: Math with Dr. Claire;https://www.youtube.com/watch?v=YLzgYm2tN8E;License: Standard YouTube License, CC-BY