Exam 6: Integer Linear Programming

A cellular phone company wants to locate two new communications towers to cover 4 regions. The company wants to minimize the cost of installing the two towers. The regions that can be covered by each tower site are indicated by a 1 in the following table: ​ Based on this ILP formulation of the problem what formulas should go in cells F6:F14 of the following Excel spreadsheet?

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?

A company needs to hire workers to cover a 7 day work week. Employees work 5 consecutive days with 2 days off. The demand for workers by day of the week and the wages per shift are: ​ Based on this ILP formulation of the problem and the optimal solution (X₁, X₂, X₃, X₄, X₅, X₆) = (2, 10, 16, 6, 14, 8, 6) what values should go in cells B5:J13 of the following Excel spreadsheet? ​ ​

ILP formulations can be used to model:

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 Which of the following constraints creates the link between setting up to produce A's and making some A's for this problem?

The LP relaxation of an ILP problem

A company needs to hire workers to cover a 7 day work week. Employees work 5 consecutive days with 2 days off. The demand for workers by day of the week and the wages per shift are: Formulate the ILP for this problem.

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

A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $15 million to invest. Based on this ILP formulation of the problem and the indicated optimal solution what values should go in cells B6:G14 of the following Excel spreadsheet? ​

A small town wants to build some new recreational facilities. The proposed facilities include a swimming pool, recreation center, basketball court and baseball field. The town council wants to provide the facilities which will be used by the most people, but faces budget and land limitations. The town has $400,000 and 14 acres of land. The pool requires locker facilities which would be in the recreation center, so if the swimming pool is built the recreation center must also be built. Also the council has only enough flat land to build the basketball court or the baseball field. The daily usage and cost of the facilities (in $1,000) are shown below. Based on this ILP formulation of the problem what formulas should go in cells F5:F12 of the following Excel spreadsheet? ​

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. Which of the following pair of constraints enforces the quantity discount relationship on X₃?

An investor has $500,000 to invest and wants to maximize the money they will receive at the end of one year. They can invest in condos, apartments and houses. The profit after one year, the cost and the number of units available are shown below. Formulate the ILP for this problem.

A city wants to locate 2 new fire fighting ladder trucks to maximize the number of tall buildings which they can cover within a 3 minute response time. The city is divided into 4 zones. The fire chief wants to locate no more than one of the trucks in either Zone 1 or Zone 2. The number of tall buildings in each zone and the travel time between zones is listed below. ​ Based on this ILP formulation of the problem what values should go in cells B5:G24 of the following Excel spreadsheet? Let X_i = 1 if truck located in zone i, 0 otherwise ​ ​

The feasible region for the pure ILP problem

If a company selects Project 1 then it must also select either Project 2 or Project 3. Which of the following constraints enforces this condition?

Which of the following is not a benefit of using binary variables?

What are binary integer variables?

The branch-and-bound algorithm starts by

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. ​

Rounding the LP relaxation solution up or down to the nearest integer may: