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.

Accepted Answer

In a simplex tableau, a variable is associated with each column, but only one or zero constraints are associated with each row, and only one basic variable is associated with each row.

Question 2

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 3

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 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.

Accepted Answer

The purpose of row operations is indeed to create a unit column for the entering variable, which means that the coefficient for that variable becomes 1, while maintaining unit columns for the remaining basic variables, which means that the coefficients for those variables stay at 0 except for the row corresponding to the exiting variable, which becomes 1. This process allows us to transform a system of linear equations into an equivalent system that is easier to solve or analyze.

Question 5

A basic feasible solution is a basic solution that

Accepted Answer

A basic feasible solution is a basic solution that satisfies the nonnegativity conditions, i.e., all variables are nonnegative. A basic solution is a solution where a certain number of variables are set to zero and the remaining variables are determined by the equations defining the constraints. It corresponds to an extreme point of the feasible region. Therefore, the option A, which mentions that a basic feasible solution also satisfies the nonnegativity conditions is correct. Option B is incorrect because a basic feasible solution need not contain any 0 variables. Option C is incorrect as a basic feasible solution corresponds to an extreme point. Option D is incorrect as option A is correct.

Question 6

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

Accepted Answer

When a system of simultaneous linear equations has more variables than equations, we can expect an infinite number of solutions. This is because there are not enough equations to determine a unique solution for each variable, leaving room for free variables to take on any values.

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.

Accepted Answer

In linear programming, a basic solution is found by setting n - m variables to zero (where n is the total number of variables and m is the number of linearly independent constraints) and solving the system of equations for the remaining m variables. This process ensures that the solution satisfies the constraints and is located at a vertex of the feasible region.

Question 8

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 in the simplex method. The variable with the largest positive coefficient in the objective function represents the variable that can increase the most, leading to an improved objective value in the next iteration. Therefore, this variable is chosen to enter the basis.

Question 9

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

Accepted Answer

This statement is true for any variable that is not in the basis, as its coefficient in the equation representing the basis will be 0.

Question 10

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, it means there are more unknowns than constraints. This creates an underdetermined system, which means there are infinitely many solutions, not a unique one.

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.

Accepted Answer

Infeasibility in linear programming problems is indicated when, after applying the simplex method or similar optimization techniques, the final solution includes artificial variables with positive values. These artificial variables are introduced to find an initial feasible solution, and their presence in the final solution with positive values means that no feasible solution exists that satisfies all the original constraints.

Question 12

Unit columns are used to identify the

Accepted Answer

Unit columns in the context of linear programming and simplex tables are used to identify the basic variables. These are the variables that are currently being used to solve the system of equations, and each basic variable will have a column in the simplex tableau that is a unit column (a column where one entry is 1 and all others are 0) corresponding to it.

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.

Accepted Answer

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

Question 14

A basic solution and a basic feasible solution

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 15

Algebraic methods such as the simplex method are used to solve

Accepted Answer

Algebraic methods such as the simplex method are used to solve any size linear programming problem. Nonlinear programming problems require more advanced techniques, while programming problems under uncertainty require the use of probabilistic or stochastic methods. Graphical models may be used as a visual aid for linear programming problems, but algebraic methods are still needed to find the optimal solution. Therefore, the best choice would be B, as it is the only option that correctly identifies the scope of algebraic methods for solving problems.

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

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 create an initial basic feasible solution for the simplex method.

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.

Accepted Answer

This statement is true. In a simplex tableau, the coefficients in a nonbasic column indicate the impact on the objective function and the amount by which the basic variables will decrease if the value of the nonbasic variable increases from 0 to 1. This information is used in the simplex method to determine the optimal solution.

Question 19

A basic feasible solution also satisfies the nonnegativity restriction.

Accepted Answer

A basic feasible solution is a solution to a linear programming problem where a subset of the variables have been assigned values, and these values satisfy the constraints of the problem and allow all other variables to take on their nonnegative lower bounds. Since the nonnegativity restriction is a lower bound on all variables in a linear programming problem, a basic feasible solution must also satisfy this restriction. Therefore, the statement is true.

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.

Accepted Answer

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

Quiz 17: Linear Programming: Simplex Method