Deck 17: Linear Programming: Simplex Method

Question 1

In the simplex method, a tableau is optimal only if all the c_j -z_j values are

A)zero or negative.
B)zero.
C)negative and nonzero.
D)positive and nonzero.

Answer 1

zero or negative.

Question 2

When there is a tie between two or more variables for removal from the simplex tableau,

A)post-optimality analysis is required.
B)their dual prices will be equal.
C)converting the pivot element will break the tie.
D)a condition of degeneracy is present.

Answer 2

D

Question 3

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

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.

Answer 3

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.

Question 4

Every extreme point of the graph of a two variable linear programming problem is a basic feasible solution.

Question 5

Infeasibility exists when one or more of the artificial variables

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 6

An alternative optimal solution is indicated when in the simplex tableau

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

Question 7

Unit columns are used to identify

A)the tableau.
B)the c row.
C)the b column.
D)the basic variables.

Question 8

What coefficient is assigned to an artificial variable in the objective function?

A)zero.
B)one.
C)a very large negative number.
D)a very large positive number.

Question 9

Algebraic methods such as the simplex method are used to solve

A)nonlinear programming problems.
B)any size linear programming problem.
C)programming problems under uncertainty.
D)graphical models.

Question 10

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

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 11

Which is not required for a problem to be in tableau form?

A)Each constraint must be written as an equation.
B)Each of the original decision variables must have a coefficient of 1 in one equation and 0 in every other equation.
C)There is exactly one basic variable in each constraint.
D)The right-hand side of each constraint must be nonnegative.

Question 12

A basic solution and a basic feasible solution

A)are the same thing.
B)differ in the number of variables allowed to be zero.
C)describe interior points and exterior points, respectively.
D)differ in their inclusion of nonnegativity restrictions.

Question 13

When a set of simultaneous equations has more variables than constraints,

A)it is a basic set.
B)it is a feasible set.
C)there is a unique solution.
D)there are many solutions.

Question 14

The basic solution to a problem with three equations and four variables would assign a value of 0 to

A)0 variables.
B)1 variable.
C)3 variables.
D)7 variables.

Question 15

When a system of simultaneous equations has more variables than equations, there is a unique solution.

Question 16

A minimization problem with four decision variables, two greater-than-or-equal-to constraints, and one equality constraint will have

A)2 surplus variables, 3 artificial variables, and 3 variables in the basis.
B)4 surplus variables, 2 artificial variables, and 4 variables in the basis.
C)3 surplus variables, 3 artificial variables, and 4 variables in the basis.
D)2 surplus variables, 2 artificial variables, and 3 variables in the basis.

Question 17

To determine a basic solution set of n-m, the variables equal to zero and solve the m linear constraint equations for the remaining m variables.

Question 18

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

Question 19

The purpose of the tableau form is to provide

A)infeasible solution.
B)optimal infeasible solution.
C)initial basic feasible solution.
D)degenerate solution.

Question 20

A basic feasible solution satisfies the nonnegativity restriction.

Question 21

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

Question 22

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

Question 23

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

Question 24

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

Question 25

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.

Question 26

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

Question 27

What is an artificial variable? Why is it necessary?

Question 28

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

Question 29

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.

Question 30

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

Question 31

A portion of a simplex tableau is Give a complete explanation of the meaning of the z₁ = 5 value as it relates to x₂ and s₂.

Question 32

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

Question 33

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.

Question 34

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

Question 35

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

Question 36

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.

Question 37

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

Question 38

A simplex tableau is shown below.
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?

Question 39

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

Question 40

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?

Question 41

Write the following problem in tableau form. Which variables would be in the initial basis?
Max
x₁ + 2x₂
s.t.
3x₁ + 4x₂ $\le$ 100
2x₁ + 3.5x₂ $\ge$ 60
2x₁ -1x₂ = 4
x₁ , x₂ $\ge$ 0

Question 42

Comment on the solution shown in this simplex tableau.

Question 43

Comment on the solution shown in this simplex tableau.

Question 44

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

Question 45

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

Question 46

Comment on the solution shown in this simplex tableau.