Question 1

Nimble Automotive uses linear programming to produce a monthly production schedule for their manufacturing plant.Although the number of cars built is obviously an integer, the fractional part of a non-integer decision variable could be considered "work in progress" at the end of the month.

Accepted Answer

This statement is true. In linear programming, decision variables can take on non-integer values, which can represent "work in progress" or partial units produced. However, in the context of the production schedule, the final output of cars must be an integer value.

Question 2

The objective function coefficient for X₁ is currently $18 and for X₂ is $29, and the ranges of optimality for these coefficients are between $15 and $20 and between $25 and $35, respectively.If the objective function coefficients for X₁ and X₂ decline by $2 each, since both coefficients are still within their ranges of optimality, the optimal solution is guaranteed to remain the same.

Accepted Answer

While the new coefficients are within their ranges of optimality, this does not guarantee the optimal solution remains the same because changes in coefficients can alter the relative profitability of variables, potentially leading to a different optimal mix of variables.

Question 3

A linear programming model has a constraint that reflects a budget restriction of $100,000.The range of feasibility for this amount, reflected on the sensitivity report, is $85,000 to $325,000.Thus if the budget restriction is changed to $90,000, the optimal solution will not change.

Accepted Answer

False

Question 4

If at most 3 of 7 projects are to be performed, this can be modeled by X₁ + X₂ + X₃ + X₄ + X₅ + X₆ + X₇ $\le$ 3, where X₁, X₂, X₃, X₄, X₅, X₆, and X₇ are all restricted to be non-negative, have an upper bound of 1, and be integer-valued.

Accepted Answer

The constraint X₁ + X₂ + X₃ + X₄ + X₅ + X₆ + X₇ F1F1F1S1≤F1F1F10 3 is equivalent to saying that the sum of at most 3 out of 7 variables X₁, X₂, X₃, X₄, X₅, X₆, and X₇ must be less than or equal to 3. Therefore, the answer is true.

Question 5

The optimal solution to a supply chain management model can be found by solving the standalone separate components of the process.

Accepted Answer

The optimal solution to a supply chain management model requires an integrated approach that considers the entire process, rather than just separate components in isolation. This is because there are interdependencies and trade-offs between different parts of the supply chain that need to be taken into account for the best overall performance.

Question 6

Relaxing the integer restrictions to an integer linear model produces an optimal solution of X₁ = 23 and X₂ = 15.This must also be the optimal solution to the integer linear model.

Accepted Answer

If relaxing the integer restrictions produces the same optimal solution as the original integer linear model, then it follows that the optimal solution must be the same for both. Therefore, the statement is true.

Question 7

The optimal solution value of an integer linear programming problem with a minimization objective function may not be __________ the optimal solution value if integer requirements are ignored.

Accepted Answer

When integer requirements are ignored, it may be possible to find a non-integer solution that is better than the optimal integer solution. Therefore, the optimal solution value of the ILP problem may be less than the optimal solution value obtained by ignoring the integer requirements.

Question 8

Which of the following is true when using summation variables?

Accepted Answer

When using summation variables, the left side of the constraints will have one non-zero coefficient, which leads to fewer non-zero input coefficients on the left side of the constraints. The number of constraints and variables may or may not change, depending on the specific problem. Percentage constraints can be formulated without the use of summation variables by properly setting up the problem.

Question 9

Hong Securities has $300,000 to invest in four stocks and three bonds.X₁, X₂, X₃, and X₄ denote the amounts invested in each of the stocks, and Y₁, Y₂, and Y₃ equal the amounts invested in each of the three bonds.Which of the following shows that at least 40% of the investment in stocks must be in stock 1?

Accepted Answer

We want to show that at least 40% of the investment in stocks must be in stock 1. Let's assume that the total investment in stocks is S and the investment in stock 1 is X. We know that S + Y₁ + Y₂ + Y₃ = $300,000 And we want to show that X/S >= 0.4 or X >= 0.4S From the given options, option C satisfies this condition: 0.6X₁ - 0.4X₂ - 0.4X₃ - 0.4X₄ >= 0 Multiplying by 5/3, we get: X₁ - 0.67(S - X₁) >= 0 Simplifying, 1.67X₁ - 0.67S >= 0 X₁ >= 0.4S Therefore, at least 40% of the investment in stocks must be in stock 1.

Question 10

A maximization integer linear model is solved by first relaxing the integer restrictions, giving an optimal solution to the resulting linear model of X₁ = 6, X₂ = 11.The shadow price for the first constraint is $9, and the range of feasibility has a maximum increase of 20 and a maximum decrease of 5.Then for the integer model, X₁ = 6, X₂ = 11 is the optimal solution.If there is an increase of 3 units of the first resource, the optimal value of the objective function will increase by $27.

Accepted Answer

The answer of A maximization integer linear model is solved...

Question 11

The shadow price for a constraint that expresses that the availability of wood is 3000 board-feet is $0.50, and the range of feasibility is between 2800 and 4000 board-feet.Which of the following is not correct?

Accepted Answer

The shadow price of $0.50 means that for every additional board-foot of wood available, the optimal objective function value will increase by $0.50. The range of feasibility suggests that if less than 2800 or more than 4000 board-feet of wood are available, the problem is infeasible.

Option A is correct because all 3000 board-feet of wood will be used because it is feasible and is at the optimal point.

Option B is correct because if there is a reduction in the availability of wood to 2900 board-feet, then it is below the current value of 3000 board-feet, therefore the optimal objective function value will be reduced by $0.50 * (3000 - 2900) = $50.

Option D is correct because if there is an increase in the availability of wood to 6000 board-feet, then the increase in the objective function will be at least $0.50 * (6000 - 3000) = $1500.

Question 12

It takes two pounds of steel and three pounds of copper to make a particular product.If there are 100 pounds of steel and 100 pounds of cooper available, one constraint will be 2X₁ + 3X₂$\le$ 200.

Accepted Answer

The constraint seems to be incorrectly formatted and includes symbols that do not represent a clear mathematical constraint. A correct constraint for using 2 pounds of steel and 3 pounds of copper per product, given 100 pounds of each available, would typically look like 2x + 3y ≤ 100 for each material, assuming x and y represent quantities of products made from steel and copper respectively.

Question 13

Joe Chan is modeling the installation of smoke alarms.The constraint Y₁ - Y₂ $\ge$ 0 uses the binary variables Y₁ for upstairs installation and Y₂ for downstairs installation.The constraint implies that if the first installation is performed, the second must also be performed.

Accepted Answer

The constraint Y₁ - Y₂ ≥ 0 implies that if the upstairs installation (Y₁) is performed, the downstairs installation (Y₂) must also be performed, not the other way around.

Question 14

If project 1 is performed then project 2 will not be performed.This can be modeled by the constraint X₁ - X₂ $\le$ 1, where X₁ and X₂ are binary variables.

Accepted Answer

The constraint should be X1 + X2 ≤ 1, not X1 - X2 ≤ 0, to ensure that if project 1 is performed (X1=1), project 2 (X2) cannot be performed, and vice versa.

Question 15

In problem 2, let A = the total amount invested in stocks and B = the total amount invested in bonds.To state that at least 40% of the investment in stocks must be in stock 1, two constraints in the model would be:

Accepted Answer

Constraint X₁ - .4AF1F1F1S1 >= F1F1F10 0 states that at least 40% of the investment in stocks must be in stock 1. Constraint X₁ + X₂ + X₃ + X₄- A = 0 ensures that the total amount invested in stocks (X₁ + X₂ + X₃+X₄) adds up to the total amount invested in stocks (A). Together, these two constraints ensure that at least 40% of the investment in stocks is in stock 1 and the total amount invested in stocks is equal to A.

Question 16

In a fixed charge integer linear model where there are variable profits of $45 and $80 for producing products 1 and 2, and a fixed charge of $1000 if any of product 2 is produced, the objective function can be modeled by MAX 45X₁ + 80X₂ - 1000Y₂, where Y₂ is a binary variable.

Accepted Answer

This is the correct formulation for the given scenario. The variable Y₂ is the binary variable that indicates whether or not product 2 is produced, and if it is, incurs the fixed charge of $1000. The objective function maximizes the profits from producing products 1 and 2, while taking into account the fixed cost of producing product 2.

Question 17

The optimal solution obtained to a maximization integer linear programming model, where the integer requirements are at first ignored, provides a lower bound for the optimal objective function value of the integer model.

Accepted Answer

The optimal solution obtained by ignoring the integer requirements may not be feasible for the integer model and therefore may not provide a lower bound for the optimal objective function value.

Question 18

One approach for solving an integer linear programming problem is simply to enumerate all feasible points and select the one yielding the "best" value for the objective function.However, the number of feasible integer points is usually so large, even for small problems, that this approach is inefficient for solving most models even with a computer.

Accepted Answer

Enumerating all feasible points is not a practical approach for solving most integer linear programming problems due to the large number of possible integer points, even for small problems. A more efficient approach is needed, such as using specialized algorithms like branch and bound, cutting planes, or heuristics.

Question 19

A management science professional with extensive modeling experience will focus on management concerns and need not spend much time questioning accountants and front line workers.

Accepted Answer

While a management science professional may have expertise in modeling and analysis, it is important to gather input and insights from various stakeholders within the organization, including accountants and front line workers. This collaborative approach can lead to more effective solutions and implementation.

Question 20

You are currently paying $12 per hour for labor, and labor costs are included in the calculation of the objective function coefficients of a maximization problem.The shadow price for labor printed on the sensitivity analysis report is $8.It would be economically beneficial to you if you could secure extra labor for $15 per hour.

Accepted Answer

The shadow price of $8 indicates that the objective function value would increase by $8 for each additional unit of labor available. Since you can secure extra labor for $15 per hour, and the shadow price is $8, it would be economically beneficial to do so.

Quiz 3: Applications of Linear and Integer Programming Models

Nimble Automotive uses linear programming to produce a monthly production schedule for their manufacturing plant.Although the number of cars built is obviously an integer, the fractional part of a non-integer decision variable could be considered "work in progress" at the end of the month.

A linear programming model has a constraint that reflects a budget restriction of $100,000.The range of feasibility for this amount, reflected on the sensitivity report, is $85,000 to $325,000.Thus if the budget restriction is changed to $90,000, the optimal solution will not change.

If at most 3 of 7 projects are to be performed, this can be modeled by X₁ + X₂ + X₃ + X₄ + X₅ + X₆ + X₇ $\le$ 3, where X₁, X₂, X₃, X₄, X₅, X₆, and X₇ are all restricted to be non-negative, have an upper bound of 1, and be integer-valued.

The optimal solution to a supply chain management model can be found by solving the standalone separate components of the process.

Relaxing the integer restrictions to an integer linear model produces an optimal solution of X₁ = 23 and X₂ = 15.This must also be the optimal solution to the integer linear model.

The optimal solution value of an integer linear programming problem with a minimization objective function may not be __________ the optimal solution value if integer requirements are ignored.

Which of the following is true when using summation variables?

The shadow price for a constraint that expresses that the availability of wood is 3000 board-feet is $0.50, and the range of feasibility is between 2800 and 4000 board-feet.Which of the following is not correct?

It takes two pounds of steel and three pounds of copper to make a particular product.If there are 100 pounds of steel and 100 pounds of cooper available, one constraint will be 2X₁ + 3X₂ $\le$ 200.

Joe Chan is modeling the installation of smoke alarms.The constraint Y₁ - Y₂ $\ge$ 0 uses the binary variables Y₁ for upstairs installation and Y₂ for downstairs installation.The constraint implies that if the first installation is performed, the second must also be performed.

If project 1 is performed then project 2 will not be performed.This can be modeled by the constraint X₁ - X₂ $\le$ 1, where X₁ and X₂ are binary variables.

In problem 2, let A = the total amount invested in stocks and B = the total amount invested in bonds.To state that at least 40% of the investment in stocks must be in stock 1, two constraints in the model would be:

In a fixed charge integer linear model where there are variable profits of $45 and $80 for producing products 1 and 2, and a fixed charge of $1000 if any of product 2 is produced, the objective function can be modeled by MAX 45X₁ + 80X₂ - 1000Y₂, where Y₂ is a binary variable.

The optimal solution obtained to a maximization integer linear programming model, where the integer requirements are at first ignored, provides a lower bound for the optimal objective function value of the integer model.

A management science professional with extensive modeling experience will focus on management concerns and need not spend much time questioning accountants and front line workers.

Quiz 3: Applications of Linear and Integer Programming Models

Nimble Automotive uses linear programming to produce a monthly production schedule for their manufacturing plant.Although the number of cars built is obviously an integer, the fractional part of a non-integer decision variable could be considered "work in progress" at the end of the month.

A linear programming model has a constraint that reflects a budget restriction of $100,000.The range of feasibility for this amount, reflected on the sensitivity report, is $85,000 to $325,000.Thus if the budget restriction is changed to $90,000, the optimal solution will not change.

If at most 3 of 7 projects are to be performed, this can be modeled by X1 + X2 + X3 + X4 + X5 + X6 + X7 ≤\le≤ 3, where X1, X2, X3, X4, X5, X6, and X7 are all restricted to be non-negative, have an upper bound of 1, and be integer-valued.

The optimal solution to a supply chain management model can be found by solving the standalone separate components of the process.

Relaxing the integer restrictions to an integer linear model produces an optimal solution of X1 = 23 and X2 = 15.This must also be the optimal solution to the integer linear model.

The optimal solution value of an integer linear programming problem with a minimization objective function may not be __________ the optimal solution value if integer requirements are ignored.

Which of the following is true when using summation variables?

The shadow price for a constraint that expresses that the availability of wood is 3000 board-feet is $0.50, and the range of feasibility is between 2800 and 4000 board-feet.Which of the following is not correct?

It takes two pounds of steel and three pounds of copper to make a particular product.If there are 100 pounds of steel and 100 pounds of cooper available, one constraint will be 2X1 + 3X2 ≤\le≤ 200.

Joe Chan is modeling the installation of smoke alarms.The constraint Y1 - Y2 ≥\ge≥ 0 uses the binary variables Y1 for upstairs installation and Y2 for downstairs installation.The constraint implies that if the first installation is performed, the second must also be performed.

If project 1 is performed then project 2 will not be performed.This can be modeled by the constraint X1 - X2 ≤\le≤ 1, where X1 and X2 are binary variables.

In problem 2, let A = the total amount invested in stocks and B = the total amount invested in bonds.To state that at least 40% of the investment in stocks must be in stock 1, two constraints in the model would be:

In a fixed charge integer linear model where there are variable profits of $45 and $80 for producing products 1 and 2, and a fixed charge of $1000 if any of product 2 is produced, the objective function can be modeled by MAX 45X1 + 80X2 - 1000Y2, where Y2 is a binary variable.

The optimal solution obtained to a maximization integer linear programming model, where the integer requirements are at first ignored, provides a lower bound for the optimal objective function value of the integer model.

A management science professional with extensive modeling experience will focus on management concerns and need not spend much time questioning accountants and front line workers.

If at most 3 of 7 projects are to be performed, this can be modeled by X₁ + X₂ + X₃ + X₄ + X₅ + X₆ + X₇ $\le$ 3, where X₁, X₂, X₃, X₄, X₅, X₆, and X₇ are all restricted to be non-negative, have an upper bound of 1, and be integer-valued.

Relaxing the integer restrictions to an integer linear model produces an optimal solution of X₁ = 23 and X₂ = 15.This must also be the optimal solution to the integer linear model.

It takes two pounds of steel and three pounds of copper to make a particular product.If there are 100 pounds of steel and 100 pounds of cooper available, one constraint will be 2X₁ + 3X₂ $\le$ 200.

Joe Chan is modeling the installation of smoke alarms.The constraint Y₁ - Y₂ $\ge$ 0 uses the binary variables Y₁ for upstairs installation and Y₂ for downstairs installation.The constraint implies that if the first installation is performed, the second must also be performed.

If project 1 is performed then project 2 will not be performed.This can be modeled by the constraint X₁ - X₂ $\le$ 1, where X₁ and X₂ are binary variables.

In a fixed charge integer linear model where there are variable profits of $45 and $80 for producing products 1 and 2, and a fixed charge of $1000 if any of product 2 is produced, the objective function can be modeled by MAX 45X₁ + 80X₂ - 1000Y₂, where Y₂ is a binary variable.