Question 1

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 2

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

Accepted Answer

This is the definition of a basic feasible solution in linear programming. Any variable not in the basis is set to 0, and the remaining variables are determined based on this.

Question 3

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.

Accepted Answer

When there are more variables than constraints, the system is underdetermined, leading to infinitely many solutions.

Question 4

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

In a simplex tableau, each column corresponds to a variable, and each row corresponds to a constraint, with one basic variable associated with each row.

Question 5

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 in simplex method aim to create a unit column for the entering variable, which helps to determine a new basic feasible solution. It is also important to maintain the unit columns for the remaining basic variables to keep the solution feasible.

Question 6

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.

Accepted Answer

The answer of The basic solution to a problem with...

Question 7

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

The answer of Coefficients in a nonbasic column in a...

Question 8

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.

Accepted Answer

A basic solution is any solution that satisfies the constraint equations with exactly n variables nonzero, where n is the number of constraints. A basic feasible solution is a basic solution that satisfies the additional requirement that all the variables are nonnegative. Therefore, the difference between a basic solution and a basic feasible solution is the inclusion of nonnegativity restrictions. Thus, the correct choice is D.

Question 9

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

Accepted Answer

Artificial variables are added to the constraints in order to obtain an initial basic feasible solution in the simplex method.

Question 10

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.

Accepted Answer

In linear programming, a basic solution is obtained by setting n-m variables to zero and solving the m linear equations for the remaining m variables, which are called basic variables.

Question 11

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

Accepted Answer

In linear programming, particularly in the simplex method, the variable to remove from the current basis is the one associated with the smallest positive ratio of the solution component to the corresponding element in the entering column (not the smallest positive $c_j - z_j$ value). The $c_j - z_j$ values are used to determine the entering variable, not the leaving one.

Question 12

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 is true and reflects the fundamental principle of the simplex method, where at each iteration a variable with a positive coefficient in the objective function is chosen to enter the basis, while a variable with a zero coefficient (or a negative coefficient in the case of a maximization problem) is chosen to leave the basis, resulting in the improvement of the objective function value. The number of basic variables remains the same throughout the method.

Question 13

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

Accepted Answer

When a system of simultaneous equations has more variables than equations, there are infinite solutions or no solutions.

Question 14

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 15

A basic feasible solution satisfies the nonnegativity restriction.

Accepted Answer

A basic feasible solution in linear programming is one that satisfies all the constraints of the problem, including the nonnegativity restrictions, which require all variables to be zero or positive.

Question 16

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

Accepted Answer

This statement is true according to the simplex method for linear programming. The variable with the largest positive coefficient in the objective function (cS1U1B1jS1U1B0 − zS1U1B1jS1U1B0) divides the value of the constant term (S1U1B0) in the corresponding row to determine which variable enters the basis.

Question 17

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

Accepted Answer

In linear programming, every extreme point of the feasible region corresponds to a basic feasible solution, as these points are where the constraints intersect and are potential candidates for the optimal solution.

Question 18

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.

Accepted Answer

In tableau form, the original decision variables do not need to have a coefficient of 1 in one equation and 0 in every other equation. This condition is specific to the identity matrix form in the simplex method, not a requirement for tableau form.

Question 19

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

Accepted Answer

Infeasibility is recognized when one or more artificial variables remain in the solution at a positive value after the optimization process, indicating that the original constraints cannot be satisfied.

Question 20

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.

Accepted Answer

Algebraic methods such as the simplex method are specifically designed to solve linear programming problems, regardless of their size, making option B the correct answer. Nonlinear programming problems require different methods, while programming problems under uncertainty may involve stochastic methods. Graphical models can often serve as a helpful visualization tool, but they cannot be used to solve programming problems in the same way that algebraic methods can.

Quiz 17: Linear Programming: Simplex Method

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

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

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

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

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

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.

A basic solution and a basic feasible solution

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

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.

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

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.

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

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

A basic feasible solution satisfies the nonnegativity restriction.

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

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

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

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

Algebraic methods such as the simplex method are used to solve

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

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