Question 1

A company wants to select 1 project from a set of 4 possible projects. Which of the following constraints ensures that only 1 will be selected?

Accepted Answer

The constraint in option A ensures that the sum of the variables (representing the selection of projects) equals exactly 1, meaning only one project can be selected.

Question 2

The feasible region for the pure ILP problem

Accepted Answer

The feasible region for the pure ILP problem is a subset of the feasible region for the LP problem because it includes all the integer solutions which are part of the LP feasible region.

Question 3

​One way of solving an ILP problem is to solve its LP relaxation and round the solution up or down to the nearest integer.

Accepted Answer

False

Question 4

The objective function value for the optimal solution to the ILP problem may be better than the objective function value for the optimal solution to its LP relaxation. ​

Accepted Answer

The objective function value for the optimal solution to the ILP (Integer Linear Programming) problem cannot be better than the objective function value for the optimal solution to its LP (Linear Programming) relaxation because the LP relaxation allows for a larger feasible region by removing the integer constraints, potentially leading to a better (or at least no worse) objective function value.

Question 5

​A problem in which some decision variables are restricted to assuming only integer values is called a 0-1 programming problem.

Accepted Answer

A 0-1 programming problem is a special case of integer programming where the decision variables are restricted to assume only 0 or 1 values. When the variables can take on any integer value within a certain range, it is simply called an integer programming problem.

Question 6

A manufacturing company has costs associated with production preparation and with per unit production. The per unit production costs are referred to as

Accepted Answer

Per unit production costs are considered variable costs, as they depend on the level of production. Decision variables refer to the values the decision maker can select, production cost constraint coefficients are the values that correspond to the resources used in production, and marginal costs refer to the additional costs associated with producing one more unit.

Question 7

The concept of a lower bound in IP is associated with:

Accepted Answer

The LP relaxation of a minimization problem provides a lower bound because it relaxes the integer constraints, potentially reducing the objective function value.

Question 8

A company will be able to obtain a quantity discount on component parts for its three products, X₁, X₂ and X₃ if it produces beyond certain limits. To get the X₁ discount it must produce more than 50 X₁'s. It must produce more than 60 X₂'s for the X₂ discount and 70 X₃'s for the X₃ discount. How many decision variables are required in the formulation of this problem?

Accepted Answer

The answer of A company will be able to obtain...

Question 9

​A problem in which all decision variables are restricted to assuming only integer values is called a pure IP programming problem.

Accepted Answer

A pure IP programming problem involves integer restrictions on all decision variables, making it a specific case of integer programming.

Question 10

A company is developing its weekly production plan. The company produces two products, A and B, which are processed in two departments. Setting up each batch of A requires $60 of labor while setting up a batch of B costs $80. Each unit of A generates a profit of $17 while a unit of B earns a profit of $21. The company can sell all the units it produces. The data for the problem are summarized below. The decision variables are defined as X_i = the amount of product i produced Y_i = 1 if X_i > 0 and 0 if X_i = 0 Using the approach discussed in the text, what is the appropriate value for M₁ in the linking constraint for product A?

Accepted Answer

To find the appropriate value for M₁, we need to consider the setup costs for each product and department, as well as the maximum number of units that can be produced given the available resources. For product A, each batch requires $60 of labor in department 1 (U1), and we can produce a maximum of 100 batches given the available labor hours (1200). Therefore, the total setup cost for product A is 60*100 = 6000. For product B, each batch requires $80 of labor in department 1 (U1) and $70 of labor in department 0 (U0). We can produce a maximum of 60 batches given the available labor hours (3600 in U1 and 2800 in U0). Therefore, the total setup cost for product B is 80*60 + 70*60 = 9000. The linking constraint for product A should ensure that we only produce product B if we have already produced at least M₁ units of product A. Since each batch of A produces 10 units, the constraint should be: X_i <= 10*M₁*Y_i We want to find the minimum value of M₁ that ensures that product B is not produced until we have produced at least 1 unit of product A. This means that the setup cost for product A must be greater than or equal to the profit from producing 1 unit of A: 6000 >= 17*1 Solving for M₁, we get: 10*M₁ >= 1 M₁ >= 1/10 = 0.1 This means that the appropriate value for M₁ is the smallest integer greater than or equal to 0.1, which is 1. However, note that producing only 1 unit of A will not be a feasible solution given the available resources. We need to produce at least 16 units of A (i.e. M₁ = 2) to fully utilize the available labor hours in department 1. Therefore, the best choice is C.

Question 11

Binary variables are useful in modeling logical conditions in ILP problems. ​

Accepted Answer

Binary variables can only have two possible values (0 or 1), making them ideal for representing logical conditions in ILP problems.

Question 12

A vendor offers 5 different prices per unit depending on the quantity purchased. How many binary variables are needed to model this discounting scheme?

Accepted Answer

To model a scenario with 5 different prices (or options), you need 4 binary variables. This is because with n binary variables, you can represent 2^n different states or options. With 4 binary variables, you can represent 2^4 = 16 different states, which is more than enough to cover the 5 pricing options.

Question 13

Specifying suboptimality tolerances can be useful if a decision maker wants to find a good but nor optimal solution to a complex ILP problem.

Accepted Answer

Specifying suboptimality tolerances allows the decision maker to find a solution that is close to optimal but not necessarily the best possible solution, which can be useful for complex ILP problems where finding the true optimal solution may be computationally infeasible.

Question 14

For maximization problems, the optimal objective function value to the LP relaxation provides what for the optimal objective function value of the ILP problem?

Accepted Answer

The LP relaxation provides an upper bound on the optimal objective function value of the ILP problem because it relaxes the integer constraints and allows for non-integer solutions, which can potentially exceed the optimal integer solution.

Question 15

An ILP problem has 5 binary decision variables. How many possible integer solutions are there to this problem?

Accepted Answer

For each binary variable, there are 2 possible values (0 or 1). Therefore, the total number of possible solutions is 2^5 = 32.

Question 16

The default value of suboptimality tolerance in Solver is ​1%

Accepted Answer

The default suboptimality tolerance in Solver is typically set to 0.05%, not 1%.

Question 17

If a company selects either of Project 1 or Project 2 (or both), then either Project 3 or Project 4 (or both) must also be selected. Which of the following constraints enforce this condition?

Accepted Answer

If X₁ + X₂ ≤ 2(X₃ + X₄), then either X₁ + X₂ ≤ 0 (which means neither project 1 nor project 2 is selected) or X₃ + X₄ > 0 (which means either project 3 or project 4 is selected).

Question 18

Values of the binary integer variable are ​restricted to -1 or 1.

Accepted Answer

The values of the binary integer variable are restricted to 0 or 1.

Question 19

​One useful feature of Analytic Solver Platform (ASP) is its ability to generate graphs of the optimization results.

Accepted Answer

Analytic Solver Platform (ASP) includes various graphing capabilities to aid in the analysis of optimization results.

Question 20

A practical way of dealing with the complexity of IP problems is to:

Accepted Answer

Stopping the search for a better solution when the current best solution is within specified suboptimality tolerances is a practical way of dealing with the complexity of IP problems. This approach allows for a balance between the computational resources required to solve the problem and the level of optimality achieved. B&B techniques and LP relaxation with rounding may be used in combination with this approach to improve the solution, but they may also increase the computational time and complexity. Choice B may result in a solution that is not practical due to the high computational resources required to achieve optimality.

Quiz 6: Integer Linear Programming

A company wants to select 1 project from a set of 4 possible projects. Which of the following constraints ensures that only 1 will be selected?

The feasible region for the pure ILP problem

One way of solving an ILP problem is to solve its LP relaxation and round the solution up or down to the nearest integer.

The objective function value for the optimal solution to the ILP problem may be better than the objective function value for the optimal solution to its LP relaxation.

A problem in which some decision variables are restricted to assuming only integer values is called a 0-1 programming problem.

A manufacturing company has costs associated with production preparation and with per unit production. The per unit production costs are referred to as

The concept of a lower bound in IP is associated with:

A problem in which all decision variables are restricted to assuming only integer values is called a pure IP programming problem.

Binary variables are useful in modeling logical conditions in ILP problems.

A vendor offers 5 different prices per unit depending on the quantity purchased. How many binary variables are needed to model this discounting scheme?

Specifying suboptimality tolerances can be useful if a decision maker wants to find a good but nor optimal solution to a complex ILP problem.

For maximization problems, the optimal objective function value to the LP relaxation provides what for the optimal objective function value of the ILP problem?

An ILP problem has 5 binary decision variables. How many possible integer solutions are there to this problem?

The default value of suboptimality tolerance in Solver is 1%

If a company selects either of Project 1 or Project 2 (or both) , then either Project 3 or Project 4 (or both) must also be selected. Which of the following constraints enforce this condition?

Values of the binary integer variable are restricted to -1 or 1.

One useful feature of Analytic Solver Platform (ASP) is its ability to generate graphs of the optimization results.

A practical way of dealing with the complexity of IP problems is to:

Quiz 6: Integer Linear Programming

A company wants to select 1 project from a set of 4 possible projects. Which of the following constraints ensures that only 1 will be selected?

The feasible region for the pure ILP problem

​One way of solving an ILP problem is to solve its LP relaxation and round the solution up or down to the nearest integer.

The objective function value for the optimal solution to the ILP problem may be better than the objective function value for the optimal solution to its LP relaxation. ​

​A problem in which some decision variables are restricted to assuming only integer values is called a 0-1 programming problem.

A manufacturing company has costs associated with production preparation and with per unit production. The per unit production costs are referred to as

The concept of a lower bound in IP is associated with:

​A problem in which all decision variables are restricted to assuming only integer values is called a pure IP programming problem.

Binary variables are useful in modeling logical conditions in ILP problems. ​

A vendor offers 5 different prices per unit depending on the quantity purchased. How many binary variables are needed to model this discounting scheme?

Specifying suboptimality tolerances can be useful if a decision maker wants to find a good but nor optimal solution to a complex ILP problem.

For maximization problems, the optimal objective function value to the LP relaxation provides what for the optimal objective function value of the ILP problem?

An ILP problem has 5 binary decision variables. How many possible integer solutions are there to this problem?

The default value of suboptimality tolerance in Solver is ​1%

If a company selects either of Project 1 or Project 2 (or both) , then either Project 3 or Project 4 (or both) must also be selected. Which of the following constraints enforce this condition?

Values of the binary integer variable are ​restricted to -1 or 1.

​One useful feature of Analytic Solver Platform (ASP) is its ability to generate graphs of the optimization results.

A practical way of dealing with the complexity of IP problems is to:

One way of solving an ILP problem is to solve its LP relaxation and round the solution up or down to the nearest integer.

The objective function value for the optimal solution to the ILP problem may be better than the objective function value for the optimal solution to its LP relaxation.

A problem in which some decision variables are restricted to assuming only integer values is called a 0-1 programming problem.

A problem in which all decision variables are restricted to assuming only integer values is called a pure IP programming problem.

Binary variables are useful in modeling logical conditions in ILP problems.

The default value of suboptimality tolerance in Solver is 1%

Values of the binary integer variable are restricted to -1 or 1.

One useful feature of Analytic Solver Platform (ASP) is its ability to generate graphs of the optimization results.