Explanation Procedure used: “ Formation of the Dual Problem: Given a minimization problem with ≥ problem constraints, Step 1: Use the coefficients and constants in the problem constraints and the objective function to form a matrix A with the coefficients of the objective function in the last row. Step 2: Interchange the rows and columns of matrix A to form the matrix A T , the transpose of A. Step 3: Use the rows of A T to form a maximization problem with ≤ problem Constraints”. Calculation: Summarize the relevant data in a table. L M N Minimum requirements Calcium 20 10 10 300 Iron 10 10 10 200 Vitamin 10 15 10 240 Cholesterol 20 24 18 To decide how many ounces of each food should be used to meet the minimum requirements and simultaneously minimize the cholesterol intake. This will involve three decision variables: x 1 = Number of foods L contains calcium x 2 = Number of foods M contains iron x 3 = Number of foods N contains vitamin The minimum requirements 300 units of calcium. 20 x 1 + 10 x 2 + 10 x 3 ≥ 300 The minimum requirements 200 units of iron. 10 x 1 + 10 x 2 + 10 x 3 ≥ 200 The minimum requirements 240 units of vitamin. 10 x 1 + 15 x 2 + 10 x 3 ≥ 240 Using the cholesterol intake in the table, the total cholesterol intakes are: C = 20 x 1 + 24 x 2 + 18 x 3 To solve the following linear programing problem: Minimize C = 20 x 1 + 24 x 2 + 18 x 3 subject to 20 x 1 + 10 x 2 + 10 x 3 ≥ 300 10 x 1 + 10 x 2 + 10 x 3 ≥ 200 10 x 1 + 15 x 2 + 10 x 3 ≥ 240 x 1 , x 2 , x 3 ≥ 0 Apply the steps which are given in the procedure used: A = [ 20 10 10 300 10 10 10 200 10 15 10 240 20 24 18 1 ] In given matrix, Interchanging the the first row with first column, second row with second column, and so on. A T = [ 20 10 10 20 10 10 15 24 10 10 10 18 300 200 240 1 ] To avoid the confusion, use different variables. Maximize P = 300 y 1 + 200 y 2 + 240 y 3 subject to 20 y 1 + 10 y 2 + 10 y 3 ≤ 20 10 y 1 + 10 y 2 + 15 y 3 ≤ 24 10 y 1 + 10 y 2 + 10 y 3 ≤ 18 y 1 , y 2 , y 3 ≥ 0 Hence, the simplex dual maximization problem is Maximize P = 300 y 1 + 200 y 2 + 240 y 3 subject to 20 y 1 + 10 y 2 + 10 y 3 ≤ 20 10 y 1 + 10 y 2 + 15 y 3 ≤ 24 10 y 1 + 10 y 2 + 10 y 3 ≤ 18 y 1 , y 2 , y 3 ≥ 0 . Using the variables x 1 and x 2 from the original problem as the slack variables in the dual problem and obtain the initial system for the dual problem: 20 y 1 + 10 y 2 + 10 y 3 + x 1 = 20 10 y 1 + 10 y 2 + 15 y 3 + x 2 = 24 10 y 1 + 10 y 2 + 10 y 3 + x 3 = 18 − 300 y 1 − 200 y 2 − 240 y 3 + P = 0 Now to form a simplex tableau and solve the dual problem: y 1 y 2 y 3 x 1 x 2 x 3 P x 1 x 2 x 3 P [ 20 10 10 1 0 0 0 20 10 10 15 0 1 0 0 24 10 10 10 0 0 1 0 18 − 300 − 200 − 240 0 0 0 1 0 ] To select the pivot column, select the most negative number in the bottom row of the tableau and to the left of the P column. The column containing this element is the pivot column . If there is a tie for the most negative indicator, choose either column. Hence, first column is the pivot column. To select the pivot row, check for the smallest quotient for each row above the dashed line. If there is a tie for the smallest quotient, choose either row. 20 20 = 1 ( smallest ) 24 10 = 2.4 18 10 = 1.8 So, the first row is selected as the pivot row. The element at the intersection of the pivot column and the pivot row is called the pivot element. So, the element enclosed in rectangle 20 is the pivot element. y 1 y 2 y 3 x 1 x 2 x 3 P x 1 x 2 x 3 P [ 20 10 10 1 0 0 0 20 10 10 15 0 1 0 0 24 10 10 10 0 0 1 0 18 − 300 − 200 − 240 0 0 0 1 0 ] 0.05 R 1 → R 1 y 1 y 2 y 3 x 1 x 2 x 3 P ∼ x 1 x 2 x 3 P [ 1 0.5 0.5 0.05 0 0 0 1 10 10 15 0 1 0 0 24 10 10 10 0 0 1 0 18 − 300 − 200 − 240 0 0 0 1 0 ] R 2 − 10 R 1 → R 2 R 3 − 10 R 1 → R 3 R 4 + 300 R 1 → R 4 y 1 y 2 y 3 x 1 x 2 x 3 P ∼ y 1 x 2 x 3 P [ 1 0

16th Edition

ISBN: 9781323174869

Author: Barnett

Publisher: PEARSON EDUCATION (COLLEGE)

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Chapter 6.3, Problem 55E

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The minimum cholesterol intake and minimize the cholesterol intake by using the simplex dual problem method.

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Chapter 6.3, Problem 54E

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Chapter 6 Solutions

COLLEGE MATH FOR MATH2040 RMU PKG

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

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