Exam 6: Integer Linear Programming

Exhibit 6.1 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A research director must pick a subset of research projects to fund over the next five years. He has five candidate projects, not all of which cover the entire five-year period. Although the director has limited funds in each of the next five years, he can carry over unspent research funds into the next year. Additionally, up to $30K can be carried out of the five-year planning period. The following table summarizes the projects and budget available to the research director. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\underline{\text { Project Funds Required (in \$ 000 )}}$$\quad$$\quad$$\quad$$\text { Benefit}$ Project 1 2 3 4 5 (in \ 000 s) 1 \ 70 \ 40 \ 30 \ 15 \ 15 \ 160 2 \ 82 \ 35 \ 20 \ 20 \ 10 \ 190 3 \ 55 \ 10 \ 10 \ 5 \ 125 4 \ 69 \ 17 \ 15 \ 12 \ 8 \ 139 5 \ 75 \ 20 \ 25 \ 30 \ 45 \ 174 Budget \ 225 \ 60 \ 60 \ 50 \ 50 The following is the ILP formulation and a spreadsheet model for the problem. $\mathrm{X}_{\mathrm{i}}=0$ if project 1 not selected, 1 if project 1 selected for $1=1,2,3,4,5$ , $C_{j}=\text { amount carried out of year } j, j=1,2,3,4,5$ MAX 160+190+125+139+174 Subject to: 70+82+55+69+75+=225 40+35+10+17+20+=60+ 30+20+10+15+25+=60+ 15+20+5+12+30+=50+ 15+10+8+45+=50+ \leq30 binary ,\geq0 -Refer to Exhibit 6.1. What formula should go in cell D15 of the above Excel spreadsheet?

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?

How are general integrality requirements indicated in the Excel Risk Solver Platform (RSP)?

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 $16 million to invest. Variable Decision NPV ( \ million ) Cost ( \ million ) Factory in Columbia 3 10 Factory in Atlanta 4 8 Warehouse in Columbia 2 0 Warehouse in Atlanta 1 5 Based on this ILP formulation of the problem and the indicated optimal solution what formulas should go in cells F6:F14 of the following Excel spreadsheet? $\mathrm { MAX } : \quad 3 \mathbf { X } _ { 1 } + 4 \mathbf { X } _ { \mathbf { 2 } } + 2 \mathbf { X } _ { \mathbf { 3 } } + \mathbf { X } _ { 4 }$ Subject to: 10++6+5\leq15 +=1 +\leq1 -\leq0 -\leq0 =0 Solutian: $\left( \mathbf { X } _ { 1 } , \mathbf { X } _ { 2 } \mathbf { X } _ { \mathbf { y } } , \mathbf { X } _ { 4 } \right) = ( 0,1 ,0 , 1 )$

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 ( \1 ,000) Land Swimming pool 400 100 2 Recreation center 500 200 3 Basketball caurt 300 150 4 Baseball field 200 100 5 Based on this ILP formulation of the problem what formulas should go in cells F5:F12 of the following Excel spreadsheet? MAX: $\quad 400 x _ { 1 } + 500 x _ { 2 } + 300 x _ { 3 } + 200 x _ { 4 }$ 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 A B C 1 2 3 Facilities 4 Pool Rec center Basketball Baseball Total usage: 5 Select (0=no, 1=yes) 6 Usage 400 500 300 200 7 8 Resources Used Available 9 Cost 100 200 150 100 400 10 Land 2 3 4 5 14 11 Pool \& Rec center 1 -1 0 12 Basket or Baseball 1 1 1

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. How many decision variables are required in the formulation of this problem?

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 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. $\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 Based on this ILP formulation of the problem what formulas should go in cells B13:B16, B20:E20, F20, and F23:F24 of the following Excel spreadsheet? Let X_i = 1 if truck located in zone i, 0 otherwise Zone Covers these zones With this many buldings 1 1,2,3 200 2 1,2 140 3 1,3,4 180 4 3,4 130 $\text { MAX: } \quad 200 \mathrm{X}_{1+}+140 \mathrm{X}_{2}+180 \mathrm{X}_{3}+130 \mathrm{X}_{4}$ Subject to: +++=2 +\leq1 =0,1

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. MAX: 35+45 Subject to: 35+55\leq250 65+25\leq340 ,\geq0 and integer Initial solution X₁ = 4.6X₂ = 1.6 Obj = 233.9

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. Formulate the ILP for this problem. Variable Decision NPV ( \million ) Cost ( \ millon ) Factory in Columbia 3 10 Factory in Atlanta 4 8 Warehouse in Columbia 2 6 Warehouse in Atlanta 1 5

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. Profit Cost Number Variable Investment (\ 1,000) (\ 1,000) Available Condos 6 50 10 Apartments 12 90 5 Houses 9 100 7 Based on this ILP formulation of the problem and the indicated optimal integer solution values what values should go in cells B5:F12 of the following Excel spreadsheet? MAX: $\quad 6 \mathbf { X } _ { 1 } + 12 \mathbf { X } _ { \mathbf { 2 } } + 9 \mathbf { X } _ { \mathbf { 3 } }$ Subject to: 50+90+100\leq500 \leq10 \leq5 \leq7 \geq0 and integer Salution: $\left( \mathbf { X } _ { 1 } , \mathbf { X } _ { 2 } \mathbf { X } _ { \mathbf { 3} } \right) = ( 1,5, 0)$

The branch-and-bound algorithm starts by

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. $\quad$ $\quad$ $\quad$ $\quad$$\text { Hours required by }$ Operation A B Hours Cutting 3 4 48 Welding 2 1 36 What is the appropriate formula to use in cell E8 of the following Excel implementation of the ILP model for this problem?

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

Exhibit 6.2 The following questions pertain to the problem, formulation, and spreadsheet implementation below. A certain military deployment requires supplies delivered to four locations. These deliveries come from one of three ports. Logistics planners wish to deliver the supplies in an efficient manner, in this case by minimizing total ton-miles. The port-destination data, along with destination demand is provided in the following table. Destination 1 2 3 4 75 88 103 56 105 76 101 85 43 80 95 62 Demand 500 600 450 700 The ports are supplied by one of two supply ships. These ships travel to a particular port where their supplies are off-loaded and shipped to the requesting destinations. Ship S1 carries 1200 tones of supplies while Ship S2 carries 1120 tons of supplies. These ships can only go to a single port and each port can only accommodate one ship. Assume the costs for a ship to travel to a port are not part of the objective function. The following is the ILP formulation and a spreadsheet model for the problem. $Y_{i j}$ be 1 if ship 1 travels to port $j$ , for $1=S 1, S 2$ and $j=A, B, C$ $\mathrm{X}_{\mathbf{k}}$ be the tons shipped from poit $j=A, B, C$ to Destination $k=D 1, D 2, D 3$ , D4 Minimize 75+88+103+56+105+76+101+85+43 +80+95+62 Subject to: +\leq1 +\leq1 +\leq1 ++=1 ++=1 +++\leq1200+1120 +++\leq1200+1120 +++\leq1200+1120 ++\geq500 ++\geq600 ++\geq450 ++\geq700 \geq0 -Refer to Exhibit 6.2. What formula would go into cell E14?

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?

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

Any integer variable in an ILP that assumes a fractional value in the optimal solution to the relaxed LP problem can be designated

The B & B algorithm solves ILP problems

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?