Question 1

Solve the following problem by the simplex method. Max 100x₁ + 120x₂ + 85x₃ s.t. 3x₁ + 1x₂ + 6x₃ $\le$ 120 5x₁ + 8x₂ + 2x₃ $\le$ 160 x₁ , x₂ , x₃ $\ge$ 0

Accepted Answer

The answer of Solve the following problem by the simplex...

Question 2

A solution is optimal when all values in the c_j -z_j row of the simplex tableau are either zero or positive.

Accepted Answer

The answer of A solution is optimal when all values...

Question 3

The variable to remove from the current basis is the variable with the smallest positive c_j - z_j value.

Accepted Answer

In the simplex method, the variable to remove from the current basis is determined by the smallest non-positive ratio of the right-hand side to the pivot column, not by the smallest positive c_j - z_j value.

Question 4

Artificial variables are added for the purpose of obtaining an initial basic feasible solution.

Accepted Answer

Artificial variables are used in the two-phase method to obtain an initial basic feasible solution for the problem. Once the initial solution is obtained, these artificial variables are eliminated from the problem before proceeding to the second phase of the method.

Question 5

Coefficients in a nonbasic column in a simplex tableau indicate the amount of decrease in the current basic variables when the value of the nonbasic variable is increased from 0 to 1.

Accepted Answer

Coefficients in a nonbasic column represent the contribution of each nonbasic variable in the objective function. However, if we assume that a nonbasic variable enters the basis and replaces a basic variable, then the coefficient of the nonbasic variable in the entering column of the simplex tableau represents the amount by which the objective function value will increase if we increase the value of the nonbasic variable by 1. Similarly, the coefficients of the basic variables in the leaving row of the simplex tableau indicate the amount by which the basic variables will decrease if the value of the entering variable is increased by 1.

Question 6

We recognize infeasibility when one or more of the artificial variables do not remain in the solution at a positive value.

Accepted Answer

Infeasibility occurs when no feasible solution exists, not when artificial variables are eliminated from the solution.

Question 7

What is an artificial variable? Why is it necessary?

Accepted Answer

The answer of What is an artificial variable? Why is...

Question 8

If a variable is not in the basis, its value is 0.

Accepted Answer

If a variable is not in the basis, its corresponding coefficient in the basis matrix is 0, which means its value in the optimal solution is 0.

Question 9

For the special cases of infeasibility, unboundedness, and alternate optimal solutions, tell what you would do next with your linear programming model if the case occurred.

Accepted Answer

The answer of For the special cases of infeasibility, unboundedness,...

Question 10

What is the criterion for entering a new variable into the basis?

Accepted Answer

The answer of What is the criterion for entering a...

Question 11

A portion of a simplex tableau is $$\begin{array} { c c | c } && \mathrm { x } _ { 1 }\ \text { B asis } & \mathrm { C } _ { \mathrm { B } } & 20 \ \hline \mathrm { x } _ { 2 } & 25 & .2\ \mathrm {~s} _ { 2 } & 0 & 3\ \hline& \mathrm { z } _ { \mathrm { j } } & 5 \ & \mathrm { C } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 15 \end{array}$$ Give a complete explanation of the meaning of the z₁ = 5 value as it relates to x₂ and s₂.

Accepted Answer

The answer of A portion of a simplex tableau is...

Question 12

Solve the following problem by the simplex method. Max 14x₁ + 14.5x₂ + 18x₃ s.t. x₁ + 2x₂ + 2.5x₃ $\le$ 50 x₁ + x₂ + 1.5x₃ $\le$ 30 x₁ , x₂ , x₃ $\ge$ 0

Accepted Answer

The answer of Solve the following problem by the simplex...

Question 13

A simplex table is shown below. $$\begin{array} { c c | c c c c c c | c } & & x _ { 1 } & x _ { 2 } & x _ { 3 } & s _ { 1 } & s _ { 2 } & s _ { 3 } & \ \text { Basis } & c _ { B } & 5 & 4 & 8 & 0 & 0 & 0 & \ \hline s _ { 1 } & 0 & 2 / 5 & - 3 / 5 & 0 & 1 & - 2 / 5 & 0 & 4 \ \mathrm { x } _ { 3 } & 8 & 4 / 5 & 4 / 5 & 1 & 0 & 1 / 5 & 0 & 8 \ s _ { 3 } & 0 & 4 / 5 & 9 / 5 & 0 & 0 & 1 / 5 & 1 & 10 \ \hline & z _ { j } & 32 / 5 & 32 / 5 & 8 & 0 & 8 / 5 & 0 & 64 \ & c _ { j } - z _ { j } & - 7 / 5 & - 12 / 5 & 0 & 0 & - 8 / 5 & 0 & \end{array}$$ a.What is the current complete solution? b.The 32/5 for z₁ is composed of 0 + 8(4/5) + 0.Explain the meaning of this number. c.Explain the meaning of the -12/5 value for c₂ - z₂.

Accepted Answer

The answer of A simplex table is shown below. \[\begin{array}...

Question 14

The coefficient of an artificial variable in the objective function is zero.

Accepted Answer

In the objective function of a linear programming problem, the coefficient of an artificial variable is typically set to a very large positive or negative value (depending on whether the problem is a minimization or maximization problem) to ensure that these variables are driven to zero in the optimal solution, as they do not represent actual decision variables in the problem.

Question 15

The purpose of row operations is to create a unit column for the entering variable while maintaining unit columns for the remaining basic variables.

Accepted Answer

Row operations are used in the simplex method to transform the current basic feasible solution into a new basic feasible solution that improves the objective function value. The purpose of row operations is to create a unit column for the entering variable, which is the variable that will become basic in the next iteration, while maintaining unit columns for the remaining basic variables, which are the variables that will stay basic in the next iteration. This is done to ensure that the basic variables form a basis for the constraints and the objective function.

Question 16

At each iteration of the simplex procedure, a new variable becomes basic and a currently basic variable becomes nonbasic, preserving the same number of basic variables and improving the value of the objective function.

Accepted Answer

This statement accurately describes the behavior of the simplex procedure.

Question 17

The variable to enter into the basis is the variable with the largest positive c_j - z_j value.

Accepted Answer

This is true according to the simplex method for optimization problems. The variable with the highest positive coefficient in the objective function (c) is chosen to enter the basis. In this case, the coefficient is multiplied by the value of the variable in the row with the smallest non-negative ratio of solution value to coefficient (z). The variable with the largest resulting value becomes the entering variable.

Question 18

A simplex tableau is shown below. $$\begin{array} { c c | c c c c c c | c } 
& & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \
\text { Basis } & \mathrm { c } _ { \mathrm { B } } & 3 & 5 & 8 & 0 & 0 & 0 & \
\hline \mathrm { s } _ { 1 } & 0 & 3 & 6 & 0 & 1 & 0 & - 9 & 126 \
\mathrm {~s} _ { 2 } & 0 & - 5 / 2 & - 1 / 2 & 0 & 0 & 1 & - 9 / 2 & 45 \
\mathrm { x } _ { 3 } & 8 & 1 / 2 & 1 / 2 & 1 & 0 & 0 & 1 / 2 & 18 \
\hline & \mathrm { z } _ { \mathrm { j } } & 4 & 4 & 8 & 0 & 0 & 4 & 144 \
& \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & - 1 & 1 & 0 & 0 & 0 & - 4 &
\end{array}$$ 
a.Do one more iteration of the simplex procedure.
b.What is the current complete solution?
c.Is this solution optimal? Why or why not?

Accepted Answer

The answer of A simplex tableau is shown below. \[\begin{array}...

Question 19

Write the following problem in tableau form. Which variables would be in the initial basic solution? Min Z = 3x₁ + 8x₂ s.t. x₁ + x₂ $\le$200 x₁ $\le$ 80 x₂ $\le$ 60

Accepted Answer

The answer of Write the following problem in tableau form....

Question 20

Describe and illustrate graphically the special cases that can occur in a linear programming solution. What clues for these cases does the simplex procedure supply?

Accepted Answer

The answer of Describe and illustrate graphically the special cases...

Quiz 17: Linear Programming: Simplex Method

Solve the following problem by the simplex method. Max 100x₁ + 120x₂ + 85x₃ s.t. 3x₁ + 1x₂ + 6x₃ $\le$ 120 5x₁ + 8x₂ + 2x₃ $\le$ 160 x₁ , x₂ , x₃ $\ge$ 0

A solution is optimal when all values in the c_j -z_j row of the simplex tableau are either zero or positive.

The variable to remove from the current basis is the variable with the smallest positive c_j - z_j value.

Artificial variables are added for the purpose of obtaining an initial basic feasible solution.

Coefficients in a nonbasic column in a simplex tableau indicate the amount of decrease in the current basic variables when the value of the nonbasic variable is increased from 0 to 1.

We recognize infeasibility when one or more of the artificial variables do not remain in the solution at a positive value.

What is an artificial variable? Why is it necessary?

If a variable is not in the basis, its value is 0.

For the special cases of infeasibility, unboundedness, and alternate optimal solutions, tell what you would do next with your linear programming model if the case occurred.

What is the criterion for entering a new variable into the basis?

Solve the following problem by the simplex method. Max 14x₁ + 14.5x₂ + 18x₃ s.t. x₁ + 2x₂ + 2.5x₃ $\le$ 50 x₁ + x₂ + 1.5x₃ $\le$ 30 x₁ , x₂ , x₃ $\ge$ 0

The coefficient of an artificial variable in the objective function is zero.

The purpose of row operations is to create a unit column for the entering variable while maintaining unit columns for the remaining basic variables.

At each iteration of the simplex procedure, a new variable becomes basic and a currently basic variable becomes nonbasic, preserving the same number of basic variables and improving the value of the objective function.

The variable to enter into the basis is the variable with the largest positive c_j - z_j value.

Write the following problem in tableau form. Which variables would be in the initial basic solution? Min Z = 3x₁ + 8x₂ s.t. x₁ + x₂ $\le$ 200 x₁ $\le$ 80 x₂ $\le$ 60

Describe and illustrate graphically the special cases that can occur in a linear programming solution. What clues for these cases does the simplex procedure supply?

Quiz 17: Linear Programming: Simplex Method

Solve the following problem by the simplex method. Max 100x1 + 120x2 + 85x3 s.t. 3x1 + 1x2 + 6x3 ≤\le≤ 120 5x1 + 8x2 + 2x3 ≤\le≤ 160 x1 , x2 , x3 ≥\ge≥ 0

A solution is optimal when all values in the cj -zj row of the simplex tableau are either zero or positive.

The variable to remove from the current basis is the variable with the smallest positive cj - zj value.

Artificial variables are added for the purpose of obtaining an initial basic feasible solution.

Coefficients in a nonbasic column in a simplex tableau indicate the amount of decrease in the current basic variables when the value of the nonbasic variable is increased from 0 to 1.

We recognize infeasibility when one or more of the artificial variables do not remain in the solution at a positive value.

What is an artificial variable? Why is it necessary?

If a variable is not in the basis, its value is 0.

For the special cases of infeasibility, unboundedness, and alternate optimal solutions, tell what you would do next with your linear programming model if the case occurred.

What is the criterion for entering a new variable into the basis?

Solve the following problem by the simplex method. Max 14x1 + 14.5x2 + 18x3 s.t. x1 + 2x2 + 2.5x3 ≤\le≤ 50 x1 + x2 + 1.5x3 ≤\le≤ 30 x1 , x2 , x3 ≥\ge≥ 0

The coefficient of an artificial variable in the objective function is zero.

The purpose of row operations is to create a unit column for the entering variable while maintaining unit columns for the remaining basic variables.

At each iteration of the simplex procedure, a new variable becomes basic and a currently basic variable becomes nonbasic, preserving the same number of basic variables and improving the value of the objective function.

The variable to enter into the basis is the variable with the largest positive cj - zj value.

Write the following problem in tableau form. Which variables would be in the initial basic solution? Min Z = 3x1 + 8x2 s.t. x1 + x2 ≤\le≤ 200 x1 ≤\le≤ 80 x2 ≤\le≤ 60

Describe and illustrate graphically the special cases that can occur in a linear programming solution. What clues for these cases does the simplex procedure supply?

Solve the following problem by the simplex method. Max 100x₁ + 120x₂ + 85x₃ s.t. 3x₁ + 1x₂ + 6x₃ $\le$ 120 5x₁ + 8x₂ + 2x₃ $\le$ 160 x₁ , x₂ , x₃ $\ge$ 0

A solution is optimal when all values in the c_j -z_j row of the simplex tableau are either zero or positive.

The variable to remove from the current basis is the variable with the smallest positive c_j - z_j value.

Solve the following problem by the simplex method. Max 14x₁ + 14.5x₂ + 18x₃ s.t. x₁ + 2x₂ + 2.5x₃ $\le$ 50 x₁ + x₂ + 1.5x₃ $\le$ 30 x₁ , x₂ , x₃ $\ge$ 0

The variable to enter into the basis is the variable with the largest positive c_j - z_j value.

Write the following problem in tableau form. Which variables would be in the initial basic solution? Min Z = 3x₁ + 8x₂ s.t. x₁ + x₂ $\le$ 200 x₁ $\le$ 80 x₂ $\le$ 60