Question 1

Infeasibility exists when one or more of the artificial variables

Accepted Answer

A) remain in the final solution as a negative value. 
B) remain in the final solution as a positive value. 
C) have been removed from the basis. 
D) remain in the basis. 
A) remain in the final solution as a negative value. 
B) remain in the final solution as a positive value. 
C) have been removed from the basis. 
D) remain in the basis.

Question 2

The values in the cj - zj , or net evaluation, row indicate

Accepted Answer

A) the value of the objective function. 
B) the decrease in value of the objective function that will result if one unit of the variable corresponding to the jth column of the A matrix is brought into the basis. 
C) the net change in the value of the objective function that will result if one unit of the variable corresponding to the jth column of the A matrix is brought into the basis. 
D) the values of the decision variables. 
A) the value of the objective function. 
B) the decrease in value of the objective function that will result if one unit of the variable corresponding to the jth column of the A matrix is brought into the basis. 
C) the net change in the value of the objective function that will result if one unit of the variable corresponding to the jth column of the A matrix is brought into the basis. 
D) the values of the decision variables.

Question 3

An alternative optimal solution is indicated when in the simplex tableau

Accepted Answer

A) a non-basic variable has a value of zero in the cj -zj row. 
B) a basic variable has a positive value in the cj -zj row. 
C) a basic variable has a value of zero in the cj - zj row. 
D) a non-basic variable has a positive value in the cj - zj row. 
A) a non-basic variable has a value of zero in the cj -zj row. 
B) a basic variable has a positive value in the cj -zj row. 
C) a basic variable has a value of zero in the cj - zj row. 
D) a non-basic variable has a positive value in the cj - zj row.

Question 4

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 simplex tableau provided represents

Question 5

In a simplex tableau, there is a variable associated with each column and both a constraint and a basic variable associated with each row.

Accepted Answer

A) True 
 B)False

Question 6

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

Accepted Answer

A) True 
 B)False

Question 7

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 8

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

Accepted Answer

In the context of linear programming, pa

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

If the linear programming model is infea

Question 10

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

In linear programming, there are several

Question 11

A basic feasible solution satisfies the nonnegativity restriction.

Accepted Answer

A) True 
 B)False

Question 12

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

\[\begin{array} { c c | c c c c c | c }

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 c 2 - z2.

Accepted Answer

a.x₁ = 0, x₂ = 0, x₃ = 8, s₁ = 4, s₂ = 0, s₃ =

Question 14

Which of the following is not a step that is necessary to prepare a linear programming problem for solution using the simplex method?

Accepted Answer

A) formulate the problem. 
B) set up the standard form by adding slack and/or subtracting surplus variables. 
C) perform elementary row and column operations. 
D) set up the tableau form. 
A) formulate the problem. 
B) set up the standard form by adding slack and/or subtracting surplus variables. 
C) perform elementary row and column operations. 
D) set up the tableau form.

Question 15

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

Accepted Answer

A) True 
 B)False

Question 16

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

A) True 
 B)False

Question 17

Comment on the solution shown in this simplex tableau. $$\begin{array} { c c | c c c c c c | c } 
& & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm { x } _ { 3 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm { a } _ { 1 } & \
\text { Basis } & \mathrm { C } _ { \mathrm { B } } & 1 & 2 & 5 & 0 & 0 & - \mathrm { M } & \
\hline \mathrm { a } _ { 1 } & - \mathrm { M } & - 3 & - 1 & 0 & - 1 & - 2 & 1 & 4 \
\mathrm { x } _ { 3 } & 5 & 1 & 1 / 2 & 1 & 0 & 1 / 2 & 0 & 4 \
\hline & \mathrm { z } _ { \mathrm { j } } & 5 + 3 \mathrm { M } & 2.5 + \mathrm { M } & 5 & \mathrm { M } & 2.5 + 2 \mathrm { M } & - \mathrm { M } & - 4 \mathrm { M } + 20 \
& \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & - 4 - 3 \mathrm { M } & - .5 - \mathrm { M } & 0 & - \mathrm { M } & - 2.5 - 2 \mathrm { M } & 0 &
\end{array}$$

Accepted Answer

The tablea

Question 18

Write the following problem in tableau form. Which variables would be in the initial basic solution?
Min Z
= -3x1 + x2 + x3
s.t.
x1 - 2x2 + x3 $\le$ 11
-4 x1 + x2 + 2x3 $\ge$ 3
2x1 -x3 $\ge$ -1

Accepted Answer

Min Z = -3x₁ + x₂ + x₃+ Ma₁ + Ma₂ S

Question 19

Determine from a review of the following tableau whether the linear programming problem has multiple optimal solutions. $$\begin{array} { c c | c c c c c | c } 
\text { Basis } & \mathrm { C } _ { \mathrm { B } } & \mathrm { x } _ { 1 } & \mathrm { x } _ { 2 } & \mathrm {~s} _ { 1 } & \mathrm {~s} _ { 2 } & \mathrm {~s} _ { 3 } & \
\hline \mathrm { s } _ { 3 } & 0 & 0 & 0 & 1 & - 1 / 5 & 8 / 6 & 6 \
\mathrm { x } _ { 2 } & 2 & 0 & 1 & 0 & 1 / 5 & - 3 / 5 & 1 \
\mathrm { x } _ { 1 } & 3 & 1 & 0 & 0 & 1 / 5 & 2 / 5 & 4 \
\hline & \mathrm { z } _ { \mathrm { j } } & 3 & 2 & 0 & 0 & 0 & 14 \
& \mathrm { c } _ { \mathrm { j } } - \mathrm { z } _ { \mathrm { j } } & 0 & 0 & 0 & - 1 & 0 &
\end{array}$$

Accepted Answer

Since non-basic variable s₃ has a relativ

Question 20

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

A) True 
 B)False

Infeasibility exists when one or more of the artificial variables

The values in the c_j - z_j , or net evaluation, row indicate

An alternative optimal solution is indicated when in the simplex tableau

A portion of a simplex tableau is B asis 20 25 .2 0 3 5 - 15 Give a complete explanation of the meaning of the z₁ = 5 value as it relates to x₂ and s₂.

In a simplex tableau, there is a variable associated with each column and both a constraint and a basic variable associated with each row.

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

A simplex tableau is shown below. Basis 3 5 8 0 0 0 0 3 6 0 1 0 -9 126 0 -5/2 -1/2 0 0 1 -9/2 45 8 1/2 1/2 1 0 0 1/2 18 4 4 8 0 0 4 144 - -1 1 0 0 0 -4 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?

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

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.

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?

A basic feasible solution satisfies the nonnegativity restriction.

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

Which of the following is not a step that is necessary to prepare a linear programming problem for solution using the simplex method?

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

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

Comment on the solution shown in this simplex tableau. Basis 1 2 5 0 0 - - -3 -1 0 -1 -2 1 4 5 1 1/2 1 0 1/2 0 4 5+3 2.5+ 5 2.5+2 - -4+20 - -4-3 -.5- 0 - -2.5-2 0

Write the following problem in tableau form. Which variables would be in the initial basic solution? Min Z = -3x₁ + x₂ + x₃ s.t. x₁ - 2x₂ + x₃ $\le$ 11 -4 x₁ + x₂ + 2x₃ $\ge$ 3 2x₁ -x₃ $\ge$ -1

Determine from a review of the following tableau whether the linear programming problem has multiple optimal solutions. Basis 0 0 0 1 -1/5 8/6 6 2 0 1 0 1/5 -3/5 1 3 1 0 0 1/5 2/5 4 3 2 0 0 0 14 - 0 0 0 -1 0

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.

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Forecasting

Markov Processes

Simplex-Based Sensitivity Analysis and Duality

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

Exam 17: Linear Programming: Simplex Method

Infeasibility exists when one or more of the artificial variables

The values in the cj - zj , or net evaluation, row indicate

An alternative optimal solution is indicated when in the simplex tableau

A portion of a simplex tableau is B asis 20 25 .2 0 3 5 - 15 Give a complete explanation of the meaning of the z1 = 5 value as it relates to x2 and s2.

In a simplex tableau, there is a variable associated with each column and both a constraint and a basic variable associated with each row.

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

A simplex tableau is shown below. Basis 3 5 8 0 0 0 0 3 6 0 1 0 -9 126 0 -5/2 -1/2 0 0 1 -9/2 45 8 1/2 1/2 1 0 0 1/2 18 4 4 8 0 0 4 144 - -1 1 0 0 0 -4 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?

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

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.

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?

A basic feasible solution satisfies the nonnegativity restriction.

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

Which of the following is not a step that is necessary to prepare a linear programming problem for solution using the simplex method?

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

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

Comment on the solution shown in this simplex tableau. Basis 1 2 5 0 0 - - -3 -1 0 -1 -2 1 4 5 1 1/2 1 0 1/2 0 4 5+3 2.5+ 5 2.5+2 - -4+20 - -4-3 -.5- 0 - -2.5-2 0

Write the following problem in tableau form. Which variables would be in the initial basic solution? Min Z = -3x1 + x2 + x3 s.t. x1 - 2x2 + x3 ≤\le≤ 11 -4 x1 + x2 + 2x3 ≥\ge≥ 3 2x1 -x3 ≥\ge≥ -1

Determine from a review of the following tableau whether the linear programming problem has multiple optimal solutions. Basis 0 0 0 1 -1/5 8/6 6 2 0 1 0 1/5 -3/5 1 3 1 0 0 1/5 2/5 4 3 2 0 0 0 14 - 0 0 0 -1 0

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.

Introduction

An Introduction to Linear Programming

Linear Programming: Sensitivity Analysis and Interpretation of Solution

Linear Programming Applications in Marketing, Finance and Operations Management

Advanced Linear Programming Applications

Distribution and Network Problems

Integer Linear Programming

Nonlinear Optimization Models

Project Scheduling: Pertcpm

Inventory Models

Waiting Line Models

Simulation

Decision Analysis

Multicriteria Decisions

Forecasting

Markov Processes

Simplex-Based Sensitivity Analysis and Duality

Solution Procedures for Transportation and Assignment Problems

Minimal Spanning Tree

Dynamic Programming

Filters

The values in the c_j - z_j , or net evaluation, row indicate

A portion of a simplex tableau is B asis 20 25 .2 0 3 5 - 15 Give a complete explanation of the meaning of the z₁ = 5 value as it relates to x₂ and s₂.

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

Write the following problem in tableau form. Which variables would be in the initial basic solution? Min Z = -3x₁ + x₂ + x₃ s.t. x₁ - 2x₂ + x₃ $\le$ 11 -4 x₁ + x₂ + 2x₃ $\ge$ 3 2x₁ -x₃ $\ge$ -1