Exam 6: Integer Linear Programming

The following ILP is being solved by the branch and bound method. You have been given the initial relaxed IP solution. Complete the entries for the 3 nodes and label the arcs when you branch on X₁. Initial solution X₁ = 4.6X₂ = 1.6 Obj = 233.9

In the B & B algorithm, B & B stands for

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 B & B algorithm solves ILP problems

One approach to solving integer programming problems is to ignore the integrality conditions and solve the problem with continuous decision variables. This is referred to as

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 and the indicated optimal solution what values should go in cells B5:G12 of the following Excel spreadsheet?

The branch-and-bound algorithm starts by

A wedding caterer has several wine shops from which it can order champagne. The caterer needs 100 bottles of champagne on a particular weekend for 2 weddings. The first supplier can supply either 40 bottles or 90 bottles. The relevant decision variable is defined as X₁ = the number of bottles supplied by supplier 1 Which set of constraints reflects the fact that supplier 1 can supply only 40 or 90 bottles?

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?

A company must invest in project 1 in order to invest in project 2. Which of the following constraints ensures that project 1 will be chosen if project 2 is invested in?

A company wants to select no more than 2 projects from a set of 4 possible projects. Which of the following constraints ensures that no more than 2 will be selected?

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. What is the appropriate formula to use in cell B15 of the following Excel implementation of the ILP model for this problem?

A sub-problem in a B & B is solved and found infeasible. Should the B & B algorithm continue further analysis on this candidate problem?

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?

How are binary variables specified in the Risk Solver Platform (RSP)?

What does the Risk Solver Platform (RSP) default integer tolerance factor of 0 accomplish?

The optimal relaxed solution for an ILP has X₁ = 3.6 and X₂ = 2.9. If we branch on X₁, what constraints must be added to the two resulting LP problems?

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 What is the appropriate value for M₁ in the linking constraint for product A?

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. Based on this ILP formulation of the problem what formulas should go in cells E5:E12 of the following Excel spreadsheet?

For maximization problems, the optimal objective function value to the LP relaxation provides what for the optimal objective function value of the ILP problem?