Deck 17: Linear Programming: Simplex Method

Question 1

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

Answer 1

True

Question 2

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

Answer 2

False

Question 3

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

Answer 3

False

Question 4

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 5

A basic feasible solution is a basic solution that

A)also satisfies the nonnegativity conditions.
B)contains 0 variables.
C)corresponds to no extreme points.
D)None of these are correct.

Question 6

Whenever a system of simultaneous linear equations has more variables than equations,

A)it is a basic set.
B)it is a feasible set.
C)there is a unique solution.
D)we can expect an infinite number of solutions.

Question 7

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

Question 8

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

Question 9

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

Question 10

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

Question 11

Infeasibility can be recognized when the optimality criterion indicates that an optimal solution has been obtained,and one or more of the artificial variables remain in the solution at a positive value.

Question 12

Unit columns are used to identify the

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

Question 13

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

Question 14

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 15

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 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.

Question 17

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

Question 18

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 19

A basic feasible solution also satisfies the nonnegativity restriction.

Question 20

The simplex method is an iterative procedure for moving from one basic feasible solution (an extreme point)to another until the optimal solution is reached.

Question 21

A simplex tableau is shown below. ​
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 22

Determine from a review of the following tableau whether the linear programming problem has multiple optimal solutions.

Question 23

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

A)value of the objective function.
B)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)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)values of the decision variables.

Question 24

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 25

What type of solution is shown in this simplex tableau?

Question 26

A simplex tableau is shown below. ​
a.Perform one more iteration of the simplex procedure.
b.What is the current complete solution?
c.Is this solution optimal? Why or why not?

Question 27

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.

Question 28

Jackie Quinn developed the following LP formulation for a problem she is working on and now needs to create an initial simplex tableau.Create the initial simplex tableau.​
Min 75x_l + 45x₂
s.t.3x_l + 2x₂ > 10
x_l + 6x₂ > 15
​

Question 29

Which of the following steps is included in the preparation of a linear programming problem for a solution using the simplex method?

A)Formulate the problem.
B)Set up the standard form by adding slack and/or subtracting surplus variables.
C)Set up the tableau form.
D)All of these are correct.

Question 30

Write the following problem in tableau form.Which variables would be in the initial basic solution?

Question 31

Write the following problem in tableau form.Which variables would be in the initial basis?

Question 32

Joe Forrester,an operations analyst for a manufacturing company,developed the following LP formulation.From it,create an initial simplex tableau.​
Max 40x_l + 30x₂ + 50x₃
s.t.2x_l + 3x₂ + 4x₃ < 200
x₁ + 2x₂ + 2x₃ < 300
3x_l + x₂ + 5x₃ < 500

Question 33

Write the following problem in tableau form.Which variables would be in the initial basic solution?

Question 34

Solve the following problem by the simplex method.

Question 35

An alternative optimal solution is indicated when,in the simplex tableau,a

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

Question 36

The purpose of the tableau form is to provide a(n)

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

Question 37

If one or more of the basic variables in a linear program have a value of zero,

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.

Question 38

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 39

Solve the following problem by the simplex method.

Question 40

Identify the type of solution shown in this simplex tableau. ​

Question 41

A student in a Management Science class wants to perform row operations on this second tableau and complete the third tableau to see if it is optimal.If it is optimal,what is the optimal answer?
​ ​
​

Question 42

A student in a Management Science class developed this initial tableau for a maximization problem and now wants to perform row operations to create the next tableau and check for an optimal solution.Create the next tableau.Is there an optimal solution?
​ ​
​

Question 43

A manager for a food company is putting together a buffet,and she is trying to determine the best mix of crab and steak to be served.Below are variable definitions she developed including vitamin,mineral,and protein requirements and an optimal simplex tableau she obtained from her computations.She is interested in interpreting what it means.​
Variable definitions:
x_l = amount of crab (oz.)to be served per buffet batch
x₂ = amount of steak (oz.)to be served per buffet batch
s₁ = vitamin A units provided in excess of requirements
s₂ = mineral units provided in excess of requirements
s₃ = protein units provided in excess of requirements
Optimal tableau ​

Question 44

An operations research analyst for a communications company has the following LP problem and wants to solve it using the simplex method.​
Max 50x₁ + 20x₂
s.t.2x₁ + x₂ < 200
x₁ + x₂ < 350
x_l + 2x₂ < 275

Question 45

The operations research analyst for a big manufacturing firm in Oregon developed the following variable definitions for an LP maximization problem she was working on.The company was trying to determine how many consoles of each model to produce next week given each console had to go through three production departments.She obtained the following optimal simplex tableau for the problem and wanted to interpret its meaning.Variable definitions:
x_l = number of model 1 consoles produced
x₂ = number of model 2 consoles produced
s₁ = unused personnel hours in department 1
s₂ = unused personnel hours in department 2
s₃ = unused personnel hours in department 3
objective function = total profit on model 1 and model 2 consoles produced in the coming week
​
Optimal tableau ​