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 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} $y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { otherwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. What is the optimal production schedule?

A BIP problem considers one yes-or-no decision at a time with the objective of choosing the best alternative.

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} $y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { otherwise }\end{array} \right.$ Which of the constraints limit the amount of raw materials that can be consumed?

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 of the constraints enforces a contingent relationship?

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 of the constraints enforces a contingent relationship?

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 within 5 minutes of the science, music, math, and business buildings?

Note: This problem requires the use of a linear programming application such as Solver or Analytic Solver. 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} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if location } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. Which locations are selected?

Which of the following techniques or tools can be used to perform sensitivity analysis for an integer programming problem? I. The sensitivity report. II. Trial-and-error. III. A parameter analysis report.

Binary variables can have the following values:

If activities A and B are mutually exclusive, the constraint xA ≤ xB will enforce this relationship in a linear program.

Binary variables are variables whose only possible values are 0 or 1.

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 of the constraints enforces a mutually exclusive relationship?

It is possible to have a constraint in a BIP that excludes the possibility of choosing none of the alternatives available.

Note: This problem requires the use of a linear programming application such as Solver or Analytic Solver. 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} $y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { ot herwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. What is the maximum profit possible?

Variables whose only possible values are 0 and 1 are called integer variables.

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} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if loaction } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Which of the constraints ensures that at least two of the potential sites will be selected?

The Excel sensitivity report can be used to perform sensitivity analysis for integer programming problems.

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 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} $x _ { j } = \left\{ \begin{array} { l } 1 , \text { if location } j \text { is selected } \\0 , \text { otherwise }\end{array} \right.$ Set up the problem in Excel and find the optimal solution. Which locations are selected?

In a BIP problem, 1 corresponds to a yes decision and 0 to a no decision. If there are 4 projects under consideration (A, B, C, and D ) and at most 2 can be chosen then the following constraint needs to be added to the formulation:

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. Which projects are selected?