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

An auxiliary binary variable is an additional binary variable that is introduced into a model to represent additional yes-or-no decisions.

Binary integer programming problems are those where all the decision variables restricted to integer values are further restricted to be binary variables.

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. What is the cost of the optimal set of locations?

Binary integer programming can be used for:

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

If a firm wishes to choose at least 2 of 4 possible activities (A, B, C and D), the constraint xA + xB + xC + xD ≥ 2 will enforce this relationship in a linear program.

A manufacturer produces both widgets and gadgets. Widgets generate a profit of $50 each and gadgets have a profit margin of $35 each. To produce each item, a setup cost is incurred. This setup cost of $500 for widgets and $400 for gadgets. Widgets consume 4 units of raw material A and 5 units of raw material B. Gadgets consume 6 units of raw material A and 2 units of raw material B. Each day, the manufacturer has 500 units of each raw material available. Set up the problem in Excel and find the optimal solution. What is the maximum profit possible?

A parameter analysis report can be used to perform sensitivity analysis for integer programming problems.

When binary variables are used in a linear program, the Solver Sensitivity Report is not available.

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 +75x 5 + 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 within 15 minutes of neighborhoods C, H, and I?

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 constraints limit the amount of raw materials that can be consumed?

Binary integer programming problems can answer which types of questions?

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 ensures that at least two of the potential projects will be selected?

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 +215x 5 + 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 is the least expensive?

If a firm wishes to choose at most 2 of 4 possible activities (A, B, C and D), the constraint xA + xB + xC + xD ≥ 2 will enforce this relationship in a linear program.

If choosing one alternative from a group excludes choosing all of the others then these alternatives are called mutually exclusive.