Question 1

The improvement in the value of the optimal solution per unit increase in a constraint's right-hand side is

Accepted Answer

The improvement in the value of the optimal solution per unit increase in a constraint's right-hand side is known as the dual price or shadow price. The dual price represents the marginal value of relaxing the corresponding constraint, and it is computed as the change in the objective function value divided by the change in the right-hand side. The dual price is always non-negative and provides valuable insights into the sensitivity of the optimal solution to changes in the problem parameters. Therefore, choice B is the best answer.

Question 2

We can often avoid the process of formulating and solving a modified linear programming problem by using the range of optimality to determine whether a change in an objective function coefficient is large enough to cause a change in the optimal solution.

Accepted Answer

The statement is true. By analyzing the range of optimality of a linear programming problem, we can determine the range of values for each objective function coefficient that will not affect the optimal solution. If a coefficient falls outside this range, it will cause a change in the optimal solution. This eliminates the need to formulate and solve a modified linear programming problem.

Question 3

Dual prices and ranges for objective function coefficients and right-hand-side values are found by considering

Accepted Answer

Sensitivity analysis is used to determine the effect of changes in the problem data (such as objective function coefficients and right-hand-side values) on the optimal solution. Dual prices and ranges can be calculated as part of sensitivity analysis. Therefore, the best choice is sensitivity analysis (option D).

Question 4

The range of optimality is calculated by considering changes in the c_j − z_j value of the variable in question.

Accepted Answer

The range of optimality is calculated by considering changes in the objective function coefficients, not by the formula provided. It determines how much an objective function coefficient can change without altering the optimal solution.

Question 5

As long as the actual value of the objective function coefficient is within the range of optimality,the current basic feasible solution will remain optimal.

Accepted Answer

If the actual value of the objective function coefficient changes outside the range of optimality, then the current basic feasible solution may no longer be optimal. However, as long as the coefficient remains within the range of optimality, the current solution will remain optimal.

Question 6

A one-sided range of optimality

Accepted Answer

A one-sided range of optimality always occurs for nonbasic variables because their coefficients can only be zero or positive.

Question 7

The dual price is the improvement in value of the optimal solution per unit increase in a constraint's right-hand-side value.

Accepted Answer

The dual price, also known as shadow price, represents the increase in the objective function value per unit increase in the right-hand-side of a constraint in the primal problem. Therefore, the statement is correct.

Question 8

If the simplex tableau is from a maximization converted from a minimization,the signs and directions of the inequalities that give the objective function ranges will need to be adjusted to apply to the original coefficients.

Accepted Answer

When converting from minimization to maximization, the coefficients in the objective function are multiplied by -1 to change the direction of optimization. However, this also changes the signs of the inequalities that give the objective function ranges. Therefore, when going back to the original coefficients, the signs and directions will need to be adjusted accordingly.

Question 9

The range of optimality for a basic variable defines the objective function coefficient values for which that variable will remain part of the current optimal basic feasible solution.

Accepted Answer

This statement is true. The range of optimality for a basic variable determines the values of its corresponding objective function coefficient for which it remains a part of the optimal basic feasible solution. In other words, if the objective function coefficient of a basic variable falls outside its range of optimality, the current optimal solution will no longer remain optimal and the variable will need to be replaced by a non-basic variable.

Question 10

The ranges for which the right-hand-side values are valid are the same as the ranges over which the dual prices are valid.

Accepted Answer

This statement is true as per the duality theory in linear programming. The range of validity of dual prices is determined by the range of change that can be made to the right-hand-side values of the constraints in the primal linear program without affecting the optimality of the solution. Thus, the two ranges are equivalent and valid over the same set of values.

Question 11

The range of feasibility indicates right-hand-side values for which

Accepted Answer

The range of feasibility indicates the range of right-hand-side values for which the current basis (i.e., those variables that are in the basis) remains optimal, meaning the variables in the basis do not change. This is crucial in sensitivity analysis within linear programming.

Question 12

The entries in the associated slack column of the final tableau can also be interpreted as the changes in the values of the current basic variables corresponding to a one-unit increase in the right-hand side.​

Accepted Answer

This statement is true. The associated slack variables in the final tableau represent the amount of slack or surplus for each constraint in the optimal solution. These values also represent the changes in the values of the current basic variables if the right-hand side of each constraint is increased by one unit, as the optimal solution remains unchanged.

Question 13

The dual variable represents the

Accepted Answer

The dual variable in linear programming represents the marginal value of the constraint. It measures the impact of a small increase in the right-hand-side value of the constraint on the objective function value. Therefore, option A is the correct choice.

Question 14

The range of optimality is useful only for basic variables.

Accepted Answer

The range of optimality is useful for both basic and non-basic variables in linear programming, as it indicates how much an objective function's coefficient can change before the current solution's optimality is affected, impacting decision-making for all variables.

Question 15

As long as the objective function coefficient remains within the range of optimality,the variable values will not change although the value of the objective function could.

Accepted Answer

This statement is true. The optimal solution of a linear programming problem is determined by the values of the objective function coefficients and the constraints. As long as the objective function coefficient remains within the range of optimality, which is the range of values for which the current optimal solution is optimal, the variable values will remain the same. However, it is possible that the value of the objective function could change within this range, since it depends on the specific values of the decision variables.

Question 16

For the basic feasible solution to remain optimal,

Accepted Answer

In order for the basic feasible solution to remain optimal, we need to ensure that the values of the basic variables (i.e. the variables corresponding to the non-zero values in the optimal solution) do not change. This means that the values of the non-basic variables must change in such a way as to keep the basic variables at their optimal values. To achieve this, we need to make sure that all c_j − z_j values remain ≤ 0, as this ensures that increasing any non-basic variable will not result in a better solution. Therefore, choice A is the correct answer. Choices B and C are incorrect because they are too restrictive and do not take into account the possibility of changes to the non-basic variables that do not affect the optimal solution.

Question 17

The dual price for an equality constraint is the z_j value for its artificial variable.

Accepted Answer

This is true. When using the simplex method to solve a linear programming problem, artificial variables may be introduced in order to convert inequality constraints into equality constraints. The dual price for an equality constraint is the coefficient of the corresponding slack variable in the final tableau of the simplex method. When an artificial variable is used to convert an inequality into an equality constraint, its coefficient in the final tableau will be zero, and its dual price will be equal to the coefficient of the corresponding slack variable, which is equal to the z-value of the row containing the artificial variable (assuming maximization).

Question 18

A linear programming problem with the objective function 3x₁ + 8x₂ has the optimal solution x₁ = 5,x₂ = 6.If c₂ decreases by 2 and the range of optimality shows 5 ≤ c₂ ≤ 12,the value of Z

Accepted Answer

The objective function value Z will decrease by 12 because the coefficient c2 decreases by 2, and x2 is 6, so the change in Z is 2 * 6 = 12.

Question 19

Within the concept of duality is the original formulation of a linear programming problem known as the primal problem.

Accepted Answer

Within the concept of duality, there are two linear programming problems, the primal problem and the dual problem. The primal problem is the original formulation of the linear programming problem.

Question 20

A dual price is associated with each decision variable.

Accepted Answer

A dual price is associated with each constraint in a linear programming problem, not with each decision variable. It represents the change in the objective function value per unit increase in the right-hand side of the constraint.

Quiz 18: Simplex-Based Sensitivity Analysis and Duality