Explanation “Procedure used: Selecting Basic and Non-basic Variables for the Simplex Method. Given a simplex tableau. Step 1: Numbers of variables- Determine the number of basic variables and the number of non-basic variables. These numbers do not change during the simplex process. Step 2: Selecting basic variables - A variable can be selected as a basic variable only if it corresponds to a column in the tableau that has exactly one nonzero element (usually 1) and the nonzero element in the column is not in the same row as the nonzero element in the column of another basic variable. This procedure always selects P as a basic variable, since the P column never changes during the simplex process. Step 3: Selecting non-basic variables - After the basic variables are selected in step 2, the remaining variables are selected as the non-basic variables. The tableau columns under the non-basic variables usually contain more than one nonzero element. Selecting the Pivot Element: Step 1: Locate the most negative indicator in the bottom row of the tableau to the left of the P column (the negative number with the largest absolute value). The column containing this element is the pivot column. If there is a tie for the most negative indicator, choose either column. Step 2: Divide each positive element in the pivot column above the dashed line into the corresponding element in the last column. The pivot row is the row corresponding to the smallest quotient obtained. If there is a tie for the smallest quotient, choose either row. If the pivot column above the dashed line has no positive elements, there is no solution, and stop. Step 3: The pivot (or pivot element) is the element at the intersection of the pivot column and pivot row. Performing a Pivot Operation: A pivot operation, or pivoting, consists of performing row operations as follows: Step 1: Multiply the pivot row by the reciprocal of the pivot element to transform the pivot element into a 1. (If the pivot element is already a 1, omit this step.) Step 2: Add multiples of the pivot row to other rows in the tableau to transform all other nonzero elements in the pivot column into 0's”. Calculation: The given linear programming problem is: Maximize P = 5 x 1 + 3 x 2 subject to 5 x 1 + 4 x 2 ≤ 100 2 x 1 + x 2 ≤ 28 4 x 1 + x 2 ≤ 42 x 1 ≤ 10 x 1 , x 2 ≥ 0 Convert problem constraints into a system of equations using slack variables: There are m = 4 problem constraints, so it needs four slack variables s 1 , s 2 , s 3 , s 4 to convert the i-system to the e-system. 5 x 1 + 4 x 2 + s 1 = 100 2 x 1 + x 2 + s 2 = 28 4 x 1 + x 2 + s 3 = 42 x 1 + s 4 = 10 − 5 x 1 − 3 x 2 + P = 0 Form the simplex tableau: x 1 x 2 s 1 s 2 s 3 s 4 P s 1 s 2 s 3 s 4 P [ 5 4 1 0 0 0 0 100 2 1 0 1 0 0 0 28 4 1 0 0 1 0 0 42 1 0 0 0 0 1 0 10 − 5 − 3 0 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, the 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. 100 5 = 20 28 2 = 14 42 4 = 10.5 10 1 = 10 So, the fourth 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. x 1 x 2 s 1 s 2 s 3 s 4 P s 1 s 2 s 3 s 4 P [ 5 4 1 0 0 0 0 100 2 1 0 1 0 0 0 28 4 1 0 0 1 0 0 42 1 0 0 0 0 1 0 10 − 5 − 3 0 0 0 0 1 0 ] So, the element enclosed in rectangle 1 is the pivot element. The variable on the top of the pivot column is called as the entering variable. the variable to the left of the pivot row is the exiting variable So, x 1 is entering variable and s 4 is the exiting variable. x 1 x 2 s 1 s 2 s 3 s 4 P s 1 s 2 s 3 s 4 P [ 5 4 1 0 0 0 0 100 2 1 0 1 0 0 0 28 4 1 0 0 1 0 0 42 1 0 0 0 0 1 0 10 − 5 − 3 0 0 0 0 1 0 ] ( R 1 − 5 R 4 → R 1 R 2 − 2 R 4 → R 2 R 3 − 4 R 4 → R 3 R 5 + 5 R 4 → R 5 ) x 1 x 2 s 1 s 2 s 3 s 4 P ∼ s 1 s 2 s 3 x 1 P [ 0 4 1 0 0 − 5 0 50 0 1 0 1 0 − 2 0 8 0 1 0 0 1 − 4 0 2 1 0 0 0 0 1 0 10 0 − 3 0 0 0 5 1 50 ] basic solution ( x 1 , x 2 , s 1 , s 2 , s 3 , s 4 ) = ( 10 , 0 , 50 , 8 , 2 , 0 ) Repeat pivot operation and obtain 1 as the pivot element. x 1 x 2 s 1 s 2 s 3 s 4 P s 1 s 2 s 3 x 1 P [ 0 4 1 0 0 − 5 0 50 0 1 0 1 0 − 2 0 8 0 1 0 0 1 − 4 0 2 1 0 0 0 0 1 0 10 0 − 3 0 0 0 5 1 50 ] ( R 1 − 4 R 3 → R 1 R 2 − R 3 → R 2 R 5 + 3 R 3 → R 5 ) x 1 x 2 s 1 s 2 s 3 s 4 P ∼ s 1 s 2 x 2 x 1 P [ 0 0 1 0 − 4 11 0 42 0 0 0 1 − 1 2 0 6 0 1 0 0 1 − 4 0 2 1 0 0 0 0 1 0 10 0 0 0 0 3 − 7 1 56 ] basic solution ( x 1 , x 2 , s 1 , s 2 , s 3 , s 4 ) = ( 10 , 2 , 42 , 6 , 0 , 0 ) Repeat pivot operation and obtain 2 as the pivot element

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+

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Chapter 6.2, Problem 34E

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To solve: The linear programming problem by the simplex method, write the basic solution at each step and graph the feasible region.

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Chapter 6.2, Problem 33E

Chapter 6.2, Problem 35E

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= 2.035 765 03 -9 ws 64 7 sin &3-9sin 04 = 1.134

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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

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