Explanation Procedure used: Table method (Two Decision Variables) : Assume that a standard maximization problem in standard form has two decision variables x 1 and x 2 and m problem constraints. Step 1: Use slack variables s 1 , s 2 , … s m to convert the i -system to an e- system. Step 2: From a table with ( m + 2 ) ( m + 1 ) 2 rows and m + 2 columns labeled x 1 , x 2 , s 1 , s 2 , … s m . In the first row, assign 0 to x 1 and x 2 . In second row, assign 0 to x 1 and s 1 . Continue until the rows contain all possible combinations of assigning two 0’s to the variables. Step 3: Complete each row to a solution of e - system, if possible. Because two of the variables have the value 0, this involves solving a system of m linear equations in m variables. Use the Gauss-Jordan method, or another method if you find it easier. If the system has no solution or infinite many solutions, do not complete the row. Step 4: Find the maximum value of P over those completed rows that have no negative values. It is the optimal value of the objective function, provided the optimal value exists. Definition used: The basic solution ( x 1 , x 2 , s 1 , s 2 , s 3 ) is called a feasible solution, if all variables in the solution is non negative. Calculation: The given linear programming problem is, Maximize P = 36 x 1 + 24 x 2 Subject to { x 1 + 2 x 2 ≤ 8 x 1 + x 2 ≤ 5 2 x 1 + x 2 ≤ 8 x 1 , x 2 ≥ 0 Convert the above system inequality into the system equation as; x 1 + 2 x 2 = 8 x 1 + x 2 = 5 2 x 1 + x 2 = 8 Let the slack variable for the above system equation be s 1 , s 2 , and s 3 . Add the slack variables s 1 , s 2 , and s 3 to the above equation and obtain the e - system as; x 1 + 2 x 2 + s 1 = 8 x 1 + x 2 + s 2 = 5 2 x 1 + x 2 + s 3 = 8 Here, the number of problem constrains is m = 3 , so there will be ( 3 + 2 ) ( 3 + 1 ) 2 = 10 rows in the table of all basic solutions. Since the constraints contain two decision variables, x 1 and x 2 , assign two zeros to each row of the table in all possible combinations. Construct a table of all basic solution by assigning zero to the decision variables and slack variable as follows. Substitute the value x 1 = 0 and x 2 = 0 in the e -system and solve for s 1 and s 2 . ( 0 ) + 2 ( 0 ) + s 1 = 8 s 1 = 8 0 + 0 + s 2 = 5 s 2 = 5 Substitute x 1 = 0 and x 2 = 0 in 2 x 1 + x 2 + s 3 = 8 and simplify for s 3 ; 2 ( 0 ) + 0 + s 3 = 8 s 3 = 8 Hence, the basic solutions to the e -system is ( 0 , 0 , 8 , 5 , 8 ) . Substitute the value x 1 = 0 and s 1 = 0 in the e -system and solve for x 2 and s 2 . 0 + 2 x 2 + 0 = 8 x 2 = 4 0 + x 2 + s 2 = 5 4 + s 2 = 5 ( Substitute x 2 = 4 ) s 2 = 1 Substitute x 1 = 0 and x 2 = 4 in 2 x 1 + x 2 + s 3 = 8 and simplify for s 3 ; 2 ( 0 ) + 4 + s 3 = 8 4 + s 3 = 8 s 3 = 4 Hence, the basic solution to the e -system is ( 0 , 4 , 0 , 1 , 4 ) . Substitute the value x 1 = 0 and s 2 = 0 in the e -system and solve for x 2 and s 1 . 0 + x 2 + 0 = 5 x 2 = 5 0 + 2 x 2 + s 1 = 8 2 ( 5 ) + s 1 = 8 ( Substitute x 2 = 5 ) s 1 = − 2 Substitute x 1 = 0 and x 2 = 5 in 2 x 1 + x 2 + s 3 = 8 and simplify for s 3 ; 2 ( 0 ) + 5 + s 3 = 8 5 + s 3 = 8 s 3 = 3 Hence, the basic solution to the e -system is ( 0 , 5 , − 2 , 0 , 3 ) . Substitute the value x 1 = 0 and s 3 = 0 in the e -system and solve for x 2 and s 1 . 2 ( 0 ) + x 2 + 0 = 8 x 2 = 8 0 + 2 x 2 + s 1 = 8 2 ( 8 ) + s 1 = 24 ( Substitute x 2 = 8 ) s 1 = 8 Substitute x 1 = 8 and x 2 = 8 in x 1 + x 2 + s 2 = 5 and simplify for s 2 ; 0 + 8 + s 2 = 5 8 + s 2 = 5 s 2 = − 3 Hence, the basic solution to the e -system is ( 0 , 8 , 8 , − 3 , 0 ) . Substitute the value x 2 = 0 and s 1 = 0 in the e -system and solve for x 1 and s 2 . x 1 + 2 ( 0 ) + 0 = 8 x 1 = 8 x 1 + 0 + s 2 = 5 8 + s 2 = 5 ( Substitute x 1 = 8 ) s 2 = − 3 Substitute x 1 = 8 and x 2 = 0 in 2 x 1 + x 2 + s 3 = 8 and simplify for s 3 ; 2 ( 8 ) + 0 + s 3 = 8 16 + s 3 = 8 s 3 = − 8 Hence, the basic solution to the e -system is ( 8 , 0 , 0 , 9 , − 8 ) . Substitute the value x 2 = 0 and s 2 = 0 in the e -system and solve for x 1 and s 1 . x 1 + 0 + 0 = 5 x 1 = 5 x 1 + 2 x 2 + s 1 = 8 5 + 2 ( 0 ) + s 1 = 8 ( Substitute x 1 = 5 ) s 1 = 3 Substitute x 1 = 5 and x 2 = 0 in 2 x 1 + x 2 + s 3 = 8 and simplify for s 3 ; 2 ( 5 ) + 0 + s 3 = 8 10 + s 3 = 8 s 3 = − 2 Hence, the basic solution to the e -system is ( 5 , 0 , 3 , 0 , − 2 ) . Substitute the value x 2 = 0 and s 3 = 0 in the e -system and solve for x 1 and s 2 . 2 x 1 + 0 + 0 = 8 x 1 = 4 x 1 + 0 + s 2 = 5 4 + s 2 = 5 ( Substitute x 1 = 4 ) s 2 = 1 Substitute x 1 = 1 and x 2 = 0 in x 1 + 2 x 2 + s 1 = 8 and simplify for s 1 ; 1 + 2 ( 0 ) + s 1 = 8 1 + s 1 = 8 s 1 = 7 Hence, the basic solution to the e -system is ( 4 , 0 , 7 , 1 , 0 )

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137553341

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

See similar textbooks

Videos

Question

Chapter 6.1, Problem 3MP

To determine

The solution of the given linear programming problem using table method.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.1, Problem 2MP

Chapter 6.1, Problem 4MP

Students have asked these similar questions

Is the area to the right of the left-tail critical value Chi-squared(1-alpha) of the chi-squared distribution always (1-alpha), and is the area to the left of this critical value always alpha? Does this apply to all chi-squared distributions?Please see image attached

Find the Laplace transform of f(t). f(t) = sin(2t)e-4t

Find the Laplace transform of f(t). f(t) = 8t6 + 3t4 - 2t + 5

Chapter 6 Solutions

Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

Ch. 6.1 - Refer to Example 1. Find the basic solution for...Ch. 6.1 - Construct the table of basic solutions and use it...Ch. 6.1 - Prob. 3MP Ch. 6.1 - Prob. 4MP Ch. 6.1 - Prob. 1ED Ch. 6.1 - The following linear programming problem has only...Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E

Ch. 6.1 - In how many ways can two variables be chosen from...Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - Prob. 8E Ch. 6.1 - Problems 9–12 refer to the system 9. Find the...Ch. 6.1 - Prob. 10E Ch. 6.1 - Prob. 11E Ch. 6.1 - Prob. 12E Ch. 6.1 - In Problems 13–20, write the e-system obtained via...Ch. 6.1 - In Problems 13–20, write the e-system obtained via...Ch. 6.1 - In Problems 13–20, write the e-system obtained via...Ch. 6.1 - Prob. 16E Ch. 6.1 - Prob. 17E Ch. 6.1 - Prob. 18E Ch. 6.1 - Prob. 19E Ch. 6.1 - Prob. 20E Ch. 6.1 - Prob. 21E Ch. 6.1 - Prob. 22E Ch. 6.1 - Prob. 23E Ch. 6.1 - Prob. 24E Ch. 6.1 - Prob. 25E Ch. 6.1 - Prob. 26E Ch. 6.1 - Prob. 27E Ch. 6.1 - Prob. 28E Ch. 6.1 - Prob. 29E Ch. 6.1 - Prob. 30E Ch. 6.1 - Prob. 31E Ch. 6.1 - Prob. 32E Ch. 6.1 - Prob. 33E Ch. 6.1 - Prob. 34E Ch. 6.1 - Prob. 35E Ch. 6.1 - Prob. 36E Ch. 6.1 - Prob. 37E Ch. 6.1 - Prob. 38E Ch. 6.1 - Problems 31–40 refer to the partially completed...Ch. 6.1 - Prob. 40E Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - Prob. 42E Ch. 6.1 - Prob. 43E Ch. 6.1 - Prob. 44E Ch. 6.1 - Prob. 45E Ch. 6.1 - Prob. 46E Ch. 6.1 - In Problems 41-48, convert the given i-system to...Ch. 6.1 - Prob. 48E Ch. 6.1 - Prob. 49E Ch. 6.1 - Prob. 50E Ch. 6.1 - Prob. 51E Ch. 6.1 - Prob. 52E Ch. 6.1 - Prob. 53E Ch. 6.1 - Prob. 54E Ch. 6.1 - Prob. 55E Ch. 6.1 - Prob. 56E Ch. 6.1 - Prob. 57E Ch. 6.1 - Prob. 58E Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - Prob. 60E Ch. 6.1 - Prob. 61E Ch. 6.1 - Prob. 62E Ch. 6.1 - Prob. 63E Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - In Problems 59-66, solve the given linear...Ch. 6.1 - Prob. 66E Ch. 6.1 - Prob. 67E Ch. 6.1 - Prob. 68E Ch. 6.1 - Prob. 69E Ch. 6.1 - Prob. 70E Ch. 6.1 - In Problems 67-70, explain why the linear...Ch. 6.1 - Prob. 72E Ch. 6.1 - Prob. 73E Ch. 6.1 - Prob. 74E Ch. 6.1 - Prob. 75E Ch. 6.1 - Prob. 76E Ch. 6.2 - Solve the following linear programming problem...Ch. 6.2 - Solve using the simplex method: Ch. 6.2 - Prob. 3MP Ch. 6.2 - Prob. 1ED Ch. 6.2 - Prob. 1E Ch. 6.2 - Prob. 2E Ch. 6.2 - Prob. 3E Ch. 6.2 - Prob. 4E Ch. 6.2 - Prob. 5E Ch. 6.2 - Prob. 6E Ch. 6.2 - Prob. 7E Ch. 6.2 - Prob. 8E Ch. 6.2 - In Problems 9–12, Using slack variables, write the...Ch. 6.2 - Prob. 10E Ch. 6.2 - Prob. 11E Ch. 6.2 - Prob. 12E Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Prob. 14E Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Prob. 16E Ch. 6.2 - Prob. 17E Ch. 6.2 - Repeat Problem 17 with P = x1 + 2x2. 17. Ch. 6.2 - Prob. 19E Ch. 6.2 - Prob. 20E Ch. 6.2 - Prob. 21E Ch. 6.2 - Prob. 22E Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Prob. 24E Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Solve the linear programming problems in Problems...Ch. 6.2 - Prob. 28E Ch. 6.2 - Prob. 29E Ch. 6.2 - Prob. 30E Ch. 6.2 - Prob. 31E Ch. 6.2 - Prob. 32E Ch. 6.2 - Prob. 33E Ch. 6.2 - Prob. 34E Ch. 6.2 - Prob. 35E Ch. 6.2 - Prob. 36E Ch. 6.2 - Prob. 37E Ch. 6.2 - Prob. 38E Ch. 6.2 - In Problems 37–40, there is a tie for the choice...Ch. 6.2 - Prob. 40E Ch. 6.2 - Prob. 41E Ch. 6.2 - Prob. 42E Ch. 6.2 - Prob. 43E Ch. 6.2 - Prob. 44E Ch. 6.2 - Prob. 45E Ch. 6.2 - Prob. 46E Ch. 6.2 - Prob. 47E Ch. 6.2 - Prob. 48E Ch. 6.2 - Prob. 49E Ch. 6.2 - Prob. 50E Ch. 6.2 - Prob. 51E Ch. 6.2 - Prob. 52E Ch. 6.2 - Prob. 53E Ch. 6.2 - Prob. 54E Ch. 6.2 - Prob. 55E Ch. 6.2 - Prob. 56E Ch. 6.3 - Form the dual problem: Ch. 6.3 - Prob. 2MP Ch. 6.3 - Prob. 3MP Ch. 6.3 - Repeat Example 4 if the shipping charge from plant...Ch. 6.3 - Prob. 1ED Ch. 6.3 - Prob. 2ED Ch. 6.3 - Prob. 1E Ch. 6.3 - Prob. 2E Ch. 6.3 - Prob. 3E Ch. 6.3 - Prob. 4E Ch. 6.3 - Prob. 5E Ch. 6.3 - Prob. 6E Ch. 6.3 - Prob. 7E Ch. 6.3 - In Problems 1–8, find the transpose of each...Ch. 6.3 - In Problems 9 and 10, Form the dual problem. Write...Ch. 6.3 - Prob. 10E Ch. 6.3 - In Problems 11 and 12, a minimization problem, the...Ch. 6.3 - Prob. 12E Ch. 6.3 - Prob. 13E Ch. 6.3 - In Problems 13–20, Form the dual problem. Find the...Ch. 6.3 - In Problems 13–20, Form the dual problem. Find the...Ch. 6.3 - Prob. 16E Ch. 6.3 - In Problems 13–20, Form the dual problem. Find the...Ch. 6.3 - Prob. 18E Ch. 6.3 - Prob. 19E Ch. 6.3 - Prob. 20E Ch. 6.3 - Prob. 21E Ch. 6.3 - Prob. 22E Ch. 6.3 - Solve the linear programming problems in Problems...Ch. 6.3 - Prob. 24E Ch. 6.3 - Prob. 25E Ch. 6.3 - Prob. 26E Ch. 6.3 - Prob. 27E Ch. 6.3 - Prob. 28E Ch. 6.3 - Prob. 29E Ch. 6.3 - Prob. 30E Ch. 6.3 - Prob. 31E Ch. 6.3 - Prob. 32E Ch. 6.3 - Prob. 33E Ch. 6.3 - Prob. 34E Ch. 6.3 - Prob. 35E Ch. 6.3 - Prob. 36E Ch. 6.3 - Prob. 37E Ch. 6.3 - Prob. 38E Ch. 6.3 - Prob. 39E Ch. 6.3 - Prob. 40E Ch. 6.3 - Prob. 41E Ch. 6.3 - Prob. 42E Ch. 6.3 - Prob. 43E Ch. 6.3 - Prob. 44E Ch. 6.3 - Solve the linear programming problems in Problems...Ch. 6.3 - Prob. 46E Ch. 6.3 - Prob. 47E Ch. 6.3 - Prob. 48E Ch. 6.3 - Prob. 49E Ch. 6.3 - Mining. A mining company operates two mines, each...Ch. 6.3 - Prob. 51E Ch. 6.3 - Prob. 52E Ch. 6.3 - Prob. 53E Ch. 6.3 - Prob. 54E Ch. 6.3 - Prob. 55E Ch. 6.3 - Prob. 56E Ch. 6.3 - Prob. 57E Ch. 6.3 - Prob. 58E Ch. 6.4 - Repeat Example 1 for EXAMPLE 1 Finding the...Ch. 6.4 - Prob. 2MP Ch. 6.4 - Prob. 3MP Ch. 6.4 - Prob. 4MP Ch. 6.4 - Prob. 5MP Ch. 6.4 - Interpret the values of the slack and surplus...Ch. 6.4 - Prob. 1E Ch. 6.4 - Prob. 2E Ch. 6.4 - Prob. 3E Ch. 6.4 - Prob. 4E Ch. 6.4 - Prob. 5E Ch. 6.4 - Prob. 6E Ch. 6.4 - Prob. 7E Ch. 6.4 - Prob. 8E Ch. 6.4 - Use the big M method to solve Problems 9–22. 9. Ch. 6.4 - Prob. 10E Ch. 6.4 - Prob. 11E Ch. 6.4 - Prob. 12E Ch. 6.4 - Use the big M method to solve Problems 9–22. 13. Ch. 6.4 - Prob. 14E Ch. 6.4 - Use the big M method to solve Problems 9–22. 15. Ch. 6.4 - Prob. 16E Ch. 6.4 - Prob. 17E Ch. 6.4 - Prob. 18E Ch. 6.4 - Prob. 19E Ch. 6.4 - Prob. 20E Ch. 6.4 - Use the big M method to solve Problems 9-22. 21....Ch. 6.4 - Prob. 22E Ch. 6.4 - Prob. 23E Ch. 6.4 - Prob. 24E Ch. 6.4 - Problems 25–32 are mixed. Some can be solved by...Ch. 6.4 - Prob. 26E Ch. 6.4 - Prob. 27E Ch. 6.4 - Prob. 28E Ch. 6.4 - Prob. 29E Ch. 6.4 - Prob. 30E Ch. 6.4 - Prob. 31E Ch. 6.4 - Prob. 32E Ch. 6.4 - In Problems 33–38, construct a mathematical model...Ch. 6.4 - In Problems 33–38, construct a mathematical model...Ch. 6.4 - In Problems 33-38, construct a mathematical model...Ch. 6.4 - Prob. 36E Ch. 6.4 - Prob. 37E Ch. 6.4 - Prob. 38E Ch. 6.4 - Prob. 39E Ch. 6.4 - Prob. 40E Ch. 6.4 - Prob. 41E Ch. 6.4 - Prob. 42E Ch. 6.4 - Prob. 43E Ch. 6.4 - Prob. 44E Ch. 6.4 - Prob. 45E Ch. 6.4 - Prob. 46E Ch. 6.4 - Prob. 47E Ch. 6 - Prob. 1RE Ch. 6 - Prob. 2RE Ch. 6 - Prob. 3RE Ch. 6 - Prob. 4RE Ch. 6 - Prob. 5RE Ch. 6 - Prob. 6RE Ch. 6 - Prob. 7RE Ch. 6 - Prob. 8RE Ch. 6 - Prob. 9RE Ch. 6 - Prob. 10RE Ch. 6 - Prob. 11RE Ch. 6 - Prob. 12RE Ch. 6 - Prob. 13RE Ch. 6 - Prob. 14RE Ch. 6 - Prob. 15RE Ch. 6 - Prob. 16RE Ch. 6 - Prob. 17RE Ch. 6 - Prob. 18RE Ch. 6 - Prob. 19RE Ch. 6 - Prob. 20RE Ch. 6 - Prob. 21RE Ch. 6 - Prob. 22RE Ch. 6 - Prob. 23RE Ch. 6 - Prob. 24RE Ch. 6 - Prob. 25RE Ch. 6 - Prob. 26RE Ch. 6 - Prob. 27RE Ch. 6 - Prob. 28RE Ch. 6 - In Problems 28 and 29, Introduce slack, surplus,...Ch. 6 - Prob. 30RE Ch. 6 - Prob. 31RE Ch. 6 - Prob. 32RE Ch. 6 - Prob. 33RE Ch. 6 - Prob. 34RE Ch. 6 - Prob. 35RE Ch. 6 - Prob. 36RE Ch. 6 - Prob. 37RE Ch. 6 - Prob. 38RE Ch. 6 - Prob. 39RE Ch. 6 - Prob. 40RE Ch. 6 - Prob. 41RE

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.