Exam 7: Using Binary Integer Programming to Deal With Yes-Or-No Decisions

Note: This problem requires the use of a linear programming application such as Solver or Analytic Solver. A firm has prepared the following binary integer program to evaluate a number of potential new capital projects. The firm's goal is to maximize the net present value of their decision while not spending more than their currently available capital. Max 100x1 + 120x2 + 90x3 + 135x4 s.t. 150x1 + 200x2 + 225x3 + 175x4 ? 500 {Constraint 1} X1 + x2 + x3 + x4 ? 2 {Constraint 2} X2 + x4 ? 1 {Constraint 3} X2 + x3 ? 1 {Constraint 4} X1 = x4 {Constraint 5} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if project } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. What is the expected net present value of the optimal solution?

Note: This problem requires the use of a linear programming application such as Solver or Analytic Solver. The university is scheduling cleaning crews for its ten buildings. Each crew has a different cost and is qualified to clean only certain buildings. There are eight possible crews to choose from in this case. The goal is to minimize costs while making sure that each building is cleaned. The management science department formulated the following linear programming model to help with the selection process. Min 200x1 + 250x2 + 225x3 + 190x4 +215x5 + 245x6 + 235x7 + 220x8 s.t. x1 + x2 + x5 + x7 ? 1 {Building A constraint} X1 + x2 + x3 ? 1 {Building B constraint} X6 + x8 ? 1 {Building C constraint} X1 + x4 + x7 ? 1 {Building D constraint} X2 + x7 ? 1 {Building E constraint} X3 + x8 ? 1 {Building F constraint} X2 + x5 + x7 ? 1 {Building G constraint} X1 + x4 + x6 ? 1 {Building H constraint} X1 + x6 + x8 ? 1 {Building I constraint} X1 + x2 + x7 ? 1 {Building J constraint} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if crew } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. Which crews are selected?

In a BIP problem with 2 mutually exclusive alternatives, x1 and x2, the following constraint needs to be added to the formulation if one alternative must be chosen:

The algorithms available for solving BIP problems are much more efficient than those for linear programming which is one of the advantages of formulating problems this way.

In a BIP problem with 2 mutually exclusive alternatives, x1 and x2, the following constraint needs to be added to the formulation:

In a crew scheduling problem there is no need for a set covering constraint.

Note: This problem requires the use of a linear programming application such as Solver or Analytic Solver. The university is scheduling cleaning crews for its ten buildings. Each crew has a different cost and is qualified to clean only certain buildings. There are eight possible crews to choose from in this case. The goal is to minimize costs while making sure that each building is cleaned. The management science department formulated the following linear programming model to help with the selection process. Min 200x1 + 250x2 + 225x3 + 190x4 +215x5 + 245x6 + 235x7 + 220x8 s.t. x1 + x2 + x5 + x7 ? 1 {Building A constraint} X1 + x2 + x3 ? 1 {Building B constraint} X6 + x8 ? 1 {Building C constraint} X1 + x4 + x7 ? 1 {Building D constraint} X2 + x7 ? 1 {Building E constraint} X3 + x8 ? 1 {Building F constraint} X2 + x5 + x7 ? 1 {Building G constraint} X1 + x4 + x6 ? 1 {Building H constraint} X1 + x6 + x8 ? 1{Building I constraint} X1 + x2 + x7 ? 1 {Building J constraint} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if crew } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. What is the cost of the optimal set of locations?

To model a situation where a setup cost will be charged if a certain product is produced, the best approach is to include and Excel "IF" function.

Note: This problem requires the use of a linear programming application such as Solver or Analytic Solver. A manufacturer has the capability to produce both chairs and tables. Both products use the same materials (wood, nails and paint) and both have a setup cost ($100 for chairs, $200 for tables). The firm earns a profit of $20 per chair and $65 per table and can sell as many of each as it can produce. The daily supply of wood, nails and paint is limited. To manage the decision-making process, an analyst has formulated the following linear programming model: Max 20x1 + 65x2 - 100y1 - 200y2 s.t. 5x1 + 10x2 ? 100 {Constraint 1} 20x1 + 50x2 ? 250 {Constraint 2} 1x1 + 1.5x2 ? 10 {Constraint 3} My1 ? x1 {Constraint 4} My2 ? x2 {Constraint 5} $y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { otherwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. What is the maximum profit possible?

The university administration would like to add some additional parking locations. To make everyone happy, they would like each building to be within a 5 minute walk of one set of new parking spaces (the spaces will be added in blocks of 10 parking spaces). The university is considering six locations for the new parking spaces, but would like to minimize the overall cost of the project. In addition to the walking time requirement, the university would like to add at least 40 new parking spaces (at least 4 blocks of 10). To help with the decision, the management science department formulated the following linear programming model: Min 400x1 + 375x2 + 425x3 + 350x4 +410x5 + 500x6 s.t. x1 + x2 + x5 + x6 ? 1 {Residence Hall A constraint} X1 + x2 + x3 ? 1 {Residence Hall B constraint} X4 + x5 + x6 ? 1 {Science building constraint} X1 + x4 + x5 ? 1 {Music building constraint} X2 + x3 + x4 ? 1 {Math building constraint} X3 + x4 + x5 ? 1 {Business building constraint} X2 + x5 + x6 ? 1 {Auditorium constraint} X1 + x4 + x6 ? 1 {Arena constraint} X1 + x2 + x3 + x4 + x5 + x6 ? 4 {Total locations constraint} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if location } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Which of the locations is the most expensive?

Note: This problem requires the use of a linear programming application such as Solver or Analytic Solver. A manufacturer has the capability to produce both chairs and tables. Both products use the same materials (wood, nails and paint) and both have a setup cost ($100 for chairs, $200 for tables). The firm earns a profit of $20 per chair and $65 per table and can sell as many of each as it can produce. The daily supply of wood, nails and paint is limited. To manage the decision-making process, an analyst has formulated the following linear programming model: Max 20x1 + 65x2 - 100y1 - 200y2 s.t. 5x1 + 10x2 ? 100 {Constraint 1} 20x1 + 50x2 ? 250 {Constraint 2} 1x1 + 1.5x2 ? 10 {Constraint 3} My1 ? x1 {Constraint 4} My2 ? x2 {Constraint 5} $y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { otherwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. What is the optimal production schedule?

A firm has prepared the following binary integer program to evaluate a number of potential locations for new warehouses. The firm's goal is to maximize the net present value of their decision while not spending more than their currently available capital. Max 20x1 + 30x2 + 10x3 + 15x4 s.t. 5x1 + 7x2 + 12x3 + 11x4 ? 21 {Constraint 1} X1 + x2 + x3 + x4 ? 2 {Constraint 2} X1 + x2 ? 1 {Constraint 3} X1 + x3 ? 1 {Constraint 4} X2 = x4 {Constraint 5} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if location } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Which constraint ensures that the firm will not spend more capital than it has available (assume that each potential location has a different cost)?

A manufacturer has the capability to produce both chairs and tables. Both products use the same materials (wood, nails and paint) and both have a setup cost ($100 for chairs, $200 for tables). The firm earns a profit of $20 per chair and $65 per table and can sell as many of each as it can produce. The daily supply of wood, nails and paint is limited. To manage the decision-making process, an analyst has formulated the following linear programming model: Max 20x1 + 65x2 - 100y1 - 200y2 s.t. 5x1 + 10x2 ? 100 {Constraint 1} 20x1 + 50x2 ? 250 {Constraint 2} 1x1 + 1.5x2 ? 10 {Constraint 3} My1 ? x1 {Constraint 4} My2 ? x2 {Constraint 5} $y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { otherwise }\end{array} \right.$ Which of the following would be a reasonable value for the variable "M"?

The university administration would like to add some additional parking locations. To make everyone happy, they would like each building to be within a 5 minute walk of one set of new parking spaces (the spaces will be added in blocks of 10 parking spaces). The university is considering six locations for the new parking spaces, but would like to minimize the overall cost of the project. In addition to the walking time requirement, the university would like to add at least 40 new parking spaces (at least 4 blocks of 10). To help with the decision, the management science department formulated the following linear programming model: Min 400x1 + 375x2 + 425x3 + 350x4 +410x5 + 500x6 s.t. x1 + x2 + x5 + x6 ? 1 {Residence Hall A constraint} X1 + x2 + x3 ? 1 {Residence Hall B constraint} X4 + x5 + x6 ? 1 {Science building constraint} X1 + x4 + x5 ? 1 {Music building constraint} X2 + x3 + x4 ? 1 {Math building constraint} X3 + x4 + x5 ? 1 {Business building constraint} X2 + x5 + x6 ? 1 {Auditorium constraint} X1 + x4 + x6 ? 1 {Arena constraint} X1 + x2 + x3 + x4 + x5 + x6 ? 4 {Total locations constraint} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if location } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Which of the locations is NOT within 5 minutes of the Arena?

A problems where all the variables are binary variables is called a pure BIP problem.

BIP can be used in capital budgeting decisions to determine whether to invest a certain amount.

The university is scheduling cleaning crews for its ten buildings. Each crew has a different cost and is qualified to clean only certain buildings. There are eight possible crews to choose from in this case. The goal is to minimize costs while making sure that each building is cleaned. The management science department formulated the following linear programming model to help with the selection process. Min 200x1 + 250x2 + 225x3 + 190x4 +215x5 + 245x6 + 235x7 + 220x8 s.t. x1 + x2 + x5 + x7 ? 1 {Building A constraint} X1 + x2 + x3 ? 1 {Building B constraint} X6 + x8 ? 1 {Building C constraint} X1 + x4 + x7 ? 1 {Building D constraint} X2 + x7 ? 1 {Building E constraint} X3 + x8 ? 1 {Building F constraint} X2 + x5 + x7 ? 1 {Building G constraint} X1 + x4 + x6 ? 1 {Building H constraint} X1 + x6 + x8 ? 1 {Building I constraint} X1 + x2 + x7 ? 1 {Building J constraint} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if crew } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Which of the crews can be scheduled to clean buildings B and F?

A bakery produces both pies and cakes. Both products use the same materials (flour, sugar and eggs) and both have a setup cost ($100 for cakes, $200 for pies). The baker earns a profit of $10 per cake and $12 per pie and can sell as many of each as it can produce. The daily supply of flour, sugar and eggs is limited. To manage the decision-making process, an analyst has formulated the following linear programming model (assume that it is possible to produce fractional pies and cakes for this example): Max 10x1 + 12x2 - 100y1 - 200y2 s.t. 5x1 + 10x2 ? 1000 {Constraint 1} 2x1 + 5x2 ? 2500 {Constraint 2} 2x1 + 1x2 ? 300 {Constraint 3} My1 ? x1 {Constraint 4} My2 ? x2 {Constraint 5} $y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { ot herwise }\end{array} \right.$ Which of the following would be a reasonable value for the variable "M"?

A new pizza restaurant is moving into town. The owner is considering a number of potential sites and would like to minimize the initial investment involved with purchasing locations. However, the owner is very concerned about delivery time and wants to make sure that every neighborhood in the city can have a pizza delivered in 15 minutes or less. The owner has divided the city into 10 neighborhoods (A-J) and is currently considering a total of 8 different locations. To help with the decision, the owner formulated the following linear programming model: Min 100x1 + 120x2 + 90x3 + 135x4 +75x5 + 85x6 + 110x7 + 135x8 s.t. x1 + x2 + x5 + x7 ? 1 {Neighborhood A constraint} X1 + x2 + x3 ? 1 {Neighborhood B constraint} X5 + x6 + x8 ? 1 {Neighborhood C constraint} X1 + x4 + x7 ? 1 {Neighborhood D constraint} X2 + x3 + x7 ? 1 {Neighborhood E constraint} X3 + x4 + x8 ? 1 {Neighborhood F constraint} X2 + x5 + x7 ? 1 {Neighborhood G constraint} X1 + x4 + x6 ? 1 {Neighborhood H constraint} X1 + x6 + x8 ? 1 {Neighborhood I constraint} X1 + x2 + x7 ? 1 {Neighborhood J constraint} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if location } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Which of the locations is NOT within 15 minutes of neighborhood A?

A firm has prepared the following binary integer program to evaluate a number of potential new capital projects. The firm's goal is to maximize the net present value of their decision while not spending more than their currently available capital. Max 100x1 + 120x2 + 90x3 + 135x4 s.t. 150x1 + 200x2 + 225x3 + 175x4 ? 500 {Constraint 1} X1 + x2 + x3 + x4 ? 2 {Constraint 2} X2 + x4 ? 1 {Constraint 3} X2 + x3 ? 1 {Constraint 4} X1 = x4 {Constraint 5} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if project } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Which constraint ensures that the firm will not spend more capital than it has available (assume that each potential project has a different cost)?