Question 1

In order to choose the best solution for implementation, practitioners usually recommend re-solving the integer linear program several times with variations in the

Accepted Answer

Constraint coefficients affect the feasibility and optimality of the solution.

Question 2

The objective function for a linear optimization problem is: Max 3x + 5y, with constraints x ≥ 0, y ≥ 0 and x and y are both integers and they are also the only decisions variables. This is an example of a(n)

Accepted Answer

The problem is an all-integer linear program because both decision variables, x and y, are required to be integers, and the objective function and constraints are linear.

Question 3

In a binary integer linear program, the integer variables take only the values

Accepted Answer

Binary variables can only take the values 0 or 1. The other choices given are not binary values.

Question 4

The imposition of an integer restriction is necessary for models where

Accepted Answer

The imposition of an integer restriction means that the decision variables can only take integer values. This is necessary when the possible values of variables are restricted to particular intervals, such as the set of integers rather than the set of real numbers. Nonnegativity constraints can still be achieved by adding additional constraints. Variables taking negative values are allowed in models with no integer restrictions, as long as they satisfy the other constraints. The decision variables can take fractional values in models with no integer restrictions.

Question 5

The __________ approach to solving integer linear optimization problems breaks the feasible region of the LP Relaxation into subregions until the subregions have integer solutions or it is determined that the solution cannot be in the subregion.

Accepted Answer

The branch-and-bound approach is designed specifically for solving integer linear optimization problems. It entails breaking the feasible region of the LP relaxation into subregions and then recursively breaking these subregions further until the subregion has an integer solution, or it is determined that the solution cannot be in the subregion. Branch-and-bound is an effective method for solving integer linear optimization problems, particularly when the problem has a small number of integer variables.

Question 6

The optimal solution to the integer linear program will be an extreme point of the

Accepted Answer

An extreme point of a convex hull is a vertex of the polytope. The optimal solution to the integer linear program is a vertex of the feasible region, which is also the convex hull of the feasible points. Therefore, it is an extreme point of the convex hull.

Question 7

Which of the following approaches to solving integer linear optimization problems tries to identify the convex hull by adding a series of new constraints that do not exclude any feasible integer points?

Accepted Answer

The cutting plane approach to solving integer linear optimization problems involves adding new constraints to the problem to more closely approximate the convex hull of the feasible integer points. This method does not exclude any feasible integer points but rather aims to eliminate fractional solutions from the linear programming relaxation, thereby moving closer to the optimal integer solution.

Question 8

The objective function for an optimization problem is: Max 5x - 3y, with constraints x ≥ 0, y ≥ 0 and y must be an integer. x and y are the only decisions variables. This is an example of a(n)

Accepted Answer

The problem includes the constraint that y must be an integer, making it a mixed-integer linear program.

Question 9

The worksheet formulation for integer linear programs and linear programming problems is exactly the same except that the __________ for integer linear programs.

Accepted Answer

The key difference between integer linear programs and linear programming problems in worksheet formulation is in the constraints setup. For integer linear programs, constraints must be specifically added to identify the integer variables, and adjustments in the Tolerance value in the Integer Options dialog box might be necessary to handle the integer nature of the variables.

Question 10

The linear program that results from dropping the integer requirements for the variables in an integer linear program is known as

Accepted Answer

The LP relaxation is the linear program that results from dropping the integer requirements for the variables in an integer linear program.

Question 11

The objective function for an optimization problem is: Min 3x - 2y, with constraints x ≥ 0, y ≥ 0. x and y must be integers. Suppose that the integer restriction on the variables is removed. If so, this would be a familiar two-variable linear program; however, it would also be an example of

Accepted Answer

Removing the integer restriction on the variables converts the problem from an integer linear program (ILP) to a linear program (LP). The LP relaxation of an ILP is a technique used to solve the ILP by first solving the LP without the integer restrictions and then rounding the fractional values of the LP solution to obtain a feasible solution for the original ILP. Therefore, C is the best choice because removing the integer restriction on the variables in this case would result in an LP relaxation of the original ILP.

Question 12

A __________ problem is a binary integer programming problem that involves choosing which possible projects or activities provide the best investment return.

Accepted Answer

A capital budgeting problem involves choosing which projects or activities to invest in based on their potential return on investment. This often involves binary integer programming, where decisions must be made about whether to invest in a particular project or not. Fixed-cost, market share optimization, and location problems may also involve binary integer programming, but they are not specifically related to choosing investments based on return.

Question 13

Which of the following is true of rounding the optimized solution of a linear program to an integer?

Accepted Answer

Rounding the optimized solution of a linear program to an integer may or may not produce a feasible solution. It also may affect the value of the objective function. Therefore, option A and C are incorrect. Option B is also incorrect because rounding may result in some constraints being violated, resulting in an infeasible solution. Therefore, option D is the best choice.

Question 14

The __________ of a set of points is the smallest intersection of linear inequalities that contain the set of points.

Accepted Answer

The definition provided is that of a convex hull. A convex hull provides a concise representation of the geometry of a set of points, and is the smallest convex polygon that contains all the points. A concave hull may be a more complex shape, and slope is a term used to describe the steepness of a line or surface, and is not directly related to the definition provided.

Question 15

Which of the following is true about the sensitivity analysis for integer optimization problems?

Accepted Answer

Unlike linear programming problems, sensitivity reports are not readily available for integer optimization problems. Because of the discrete nature of the problem, Excel Solver takes much more time to calculate objective function coefficient ranges, shadow prices, and right-hand-side ranges. To determine the sensitivity of the solution to changes in model inputs for integer optimization problems, the data must be changed and the problem must be re-solved. Therefore, option D is the correct choice.

Question 16

The importance of _________ for integer linear programming problems is often intensified by the fact that a small change in one of the coefficients in the constraints can cause a relatively large change in the value of the optimal solution.

Accepted Answer

Sensitivity analysis is important for integer linear programming problems because a small change in one of the coefficients in the constraints can cause a relatively large change in the value of the optimal solution. Sensitivity analysis helps us understand the effect of such changes on the optimal solution and can help us make better decisions regarding the problem.

Question 17

The objective function for a linear optimization problem is: Max 3x + 2y, with one of the constraints being x and y both only take the values 0, 1. Also x and y are the only decision variables. This is an example of a

Accepted Answer

Since both x and y can only take the values 0 or 1 and are the only decision variables, this problem is a binary integer linear program. The objective function and constraints are linear, but the decision variables are restricted to binary values, fitting the definition of a binary integer linear program.

Question 18

Binary variables are identified with the __________ designation in the Solver Parameters dialog box.

Accepted Answer

Binary variables in the Solver Parameters dialog box are designated with "bin" to indicate that they can only take on the values 0 or 1.

Question 19

In cases where Excel Solver experiences excessive run times when solving integer linear problems, the Integer Optimality is set to

Accepted Answer

Setting the Integer Optimality to 5% allows Solver to stop when it finds a solution within 5% of the optimal, reducing run times.

Question 20

Which of the following is true of the relationship between the value of the optimal integer solution and the value of the optimal solution to the LP Relaxation?

Accepted Answer

The answer of Which of the following is true of...

Quiz 13: Integer Linear Optimization Models