Exam 6: Integer Linear Programming

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

The setup cost incurred in preparing a machine to produce a batch of product is an example of a

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: $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\underline{\text { Tower Sites }}$ Region 1 2 3 4 A 1 1 B 1 1 1 C 1 1 1 D 1 1 COST (\ 000) 200 150 190 250 Formulate the ILP for this problem.

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. Variable Facility Usage Cost ( C1,000) Land Swimming pool 400 100 2 Recreation center 500 200 3 Basketball caut 300 150 4 Baseball Eield 200 100 5 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? MAX: $\quad 400 x _ { 1 } + 500 x _ { 2 } + 300 x _ { 3 } + 200 x _ { 4 }$ $\text {Subject to:}$ 100+200+150+100\leq400 budget 2+3+4+5\leq14 land -\leq0 pool and recreation center +\leq1 basketball and baseball =0.1 \text {Solution: \left( x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \right) = ( 1,1,0,1 )} A 1 2 3 Facilities 4 Pool Rec center Basketball Baseball Total usage: 5 Select (0=no, 1=yes) 6 Usage 7 8 Resources Used Available 9 Cost 10 Land 11 Pool \& Rec center 12 Basket or Baseball

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. $\quad$ $\quad$ $\quad$$\text { Hours required by }$ Operation A B Hours Cutting 3 4 48 Welding 2 1 36 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 objective function for this problem?

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

How is the integer tolerance factor set in the Risk Solver Platform (RSP)?

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

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 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. Formulate the ILP for this problem. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\underline{\text { To zone }}$ No. tall building From zone 1 2 3 4 50 1 0 2 1 6 90 2 2 0 4 5 60 3 1 4 0 1 70 4 6 5 1 0

A company produces three products which must be painted, assembled, and inspected. The machinery must be cleaned and adjusted before each batch is produced. They want to maximize their profits for the amount of operating time they have. The unit profit and setup cost per batch are: Product Profit per unit Setup cost per batch 1 9 500 2 10 600 3 12 650 The operation time per unit and total operating hours available are: $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\underline{\text {Operating Time per Unit}}$$\quad$$\quad$$\quad$$\quad$$\quad$$\text {Operating}$ Operation Product 1 Product 2 Product 3 Hours Available Paint 1 2 2 400 Assemble 2 3 2 600 Inspection 2 4 3 540 Formulate the ILP for this problem.

A manufacturing company has costs associated with production preparation and with per unit production. The per unit production costs are referred to as