Question 1

Binary decision variables:

Accepted Answer

A) are either 0 or 1 
B) must be integers 
C) may be continuous 
D) may be negative 
A) are either 0 or 1 
B) must be integers 
C) may be continuous 
D) may be negative

Question 2

A practical way of dealing with the complexity of IP problems is to:

Accepted Answer

A) stop searching for a better solution when the current best solution is within specified suboptimality tolerances 
B) continue searching for the optimal solution until no further objective function improvement can be achieved 
C) use a B&B technique 
D) use LP relaxation and round the values of the decision variables to the nearest integers 
A) stop searching for a better solution when the current best solution is within specified suboptimality tolerances 
B) continue searching for the optimal solution until no further objective function improvement can be achieved 
C) use a B&B technique 
D) use LP relaxation and round the values of the decision variables to the nearest integers

Question 3

A production company wants to ensure that if Product 1 is produced,production of Product 1 not exceed production of Product 2.Which of the following constraints enforce this condition?

Accepted Answer

A) X1 ≥ M2Y2 
B) X1 ≤ M2X2 
C) X1 ≤ M1Y1,X1 ≤ Y1X2 
D) X1 ≤ X2 
A) X1 ≥ M2Y2 
B) X1 ≤ M2X2 
C) X1 ≤ M1Y1,X1 ≤ Y1X2 
D) X1 ≤ X2

Question 4

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.
 $$\begin{array}{c}
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text { To zone }\
\begin{array} { c c c c c c } 
\text { No. tall buildings } & \text { Fram zone } & 1 & 2 & 3 & 4 \
\hline 50 & 1 & 0 & 2 & 1 & 6 \
90 & 2 & 2 & 0 & 4 & 5 \
60 & 3 & 1 & 4 & 0 & 1 \
70 & 4 & 6 & 5 & 1 & 0
\end{array}
\end{array}$$ 
Based on this ILP formulation of the problem what values should go in cells B5:G24 of the following Excel spreadsheet?
Let Xi = 1 if truck located in zone i,0 otherwise
 $$\begin{array} { c c c } 
\text { Zone } & \text { Covers these zones } & \text { With the many building } \
\hline 1 & 1,2,3 & 200 \
2 & 1,2 & 140 \
3 & 1,3,4 & 180 \
4 & 3,4 & 130
\end{array}$$ 
MAX: 200 X1 + 140 X2 + 180 X3 + 130 X4
Subject to: X1 + X2 + X3 + X4 = 2
X1 + X2 ? 1
Xi = 0,1

Accepted Answer

Question 5

A vendor offers 5 different prices per unit depending on the quantity purchased.How many binary variables are needed to model this discounting scheme?

Accepted Answer

A) 4 
B) 2 
C) 3 
D) 5 
A) 4 
B) 2 
C) 3 
D) 5

Question 6

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:
Tower Sites
 $\begin{array}{rrrrr}
\text { Region } & 1 & 2 & 3 & 4 \
\hline \text { A } & & 1 & & 1 \
\text { B } & 1 & & 1 & 1 \
\text { C } & 1 & 1 & 1 &\
\text { D } & 1 & & & 1 \
\hline \text { COST } \$ 000 \text { s) } & 200 & 150 & 190 & 250
\end{array}$ 
MIN: 200 X1 + 150 X2 + 190 X3 + 250 X4
Subject to: X2 + X4 ? 1
X1 + X3 + X4 ? 1 X1 + X2 + X3 ? 1 X1 + X4 ? 1
X1 + X2 + X3 + X4 = 2 Xi = 0,1
Based on this ILP formulation of the problem what formulas should go in cells F6:F14 of the following Excel spreadsheet?

Accepted Answer

\[\begin{array}{l}
\begin{array} { l l }

Question 7

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
 $$\begin{array} { r r r r r } 
\text { Port } & \text { D1 } & \text { D2 } & \text { D3 } & \text { D4 } \
\hline \text { A } & 75 & 88 & 103 & 56 \
\mathrm {~B} & 105 & 76 & 101 & 85 \
\mathrm { C } & 43 & 80 & 95 & 82 \
\hline \text { Demand } & 500 & 600 & 450 & 700
\end{array}$$ 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.
Let S1U1P1YS1S1P0S1U1B1ij S1U1B0S1U1P1be 1 if ship i travels to port j,for i = S1,S2 and j = A,B,CS1S1P0
XS1U1B1jk S1U1B0be the tons shipped from port j = A,B,C to Destination k = D1,D2,D3,D4
Minimize S1U1P175XS1S1P0S1U1B111 S1U1B0S1U1P1+ 88XS1S1P0S1U1B112 S1U1B0S1U1P1+ 103XS1S1P0S1U1B113 S1U1B0S1U1P1+ 56XS1S1P0S1U1B114 S1U1B0S1U1P1+ 105XS1S1P0S1U1B121 S1U1B0S1U1P1+ 76XS1S1P0S1U1B122 S1U1B0S1U1P1+ 101XS1S1P0S1U1B123 S1U1B0S1U1P1+ 85XS1S1P0S1U1B124S1U1B0
+ 43XS1U1B131 S1U1B0+ 80XS1U1B132 S1U1B0+ 95XS1U1B133 S1U1B0+ 62XS1U1B134S1U1B0
Subject to: S1U1P1YS1S1P0S1U1B111 S1U1B0S1U1P1+ YS1S1P0S1U1B121 S1U1B0S1U1P1? 1S1S1P0
YS1U1B112 S1U1B0+ YS1U1B122 S1U1B0? 1 YS1U1B113 S1U1B0+ YS1U1B123 S1U1B0? 1
YS1U1B111 S1U1B0+ YS1U1B112 S1U1B0+ YS1U1B113 S1U1B0= 1 YS1U1B121 S1U1B0+ YS1U1B122 S1U1B0+ YS1U1B123 S1U1B0= 1
XS1U1B114 S1U1B0+ XS1U1B115 S1U1B0+ XS1U1B116 S1U1B0+ XS1U1B117 S1U1B0? 1200YS1U1B111 S1U1B0+ 1120YS1U1B121 S1U1B0XS1U1B124 S1U1B0+ XS1U1B125 S1U1B0+ XS1U1B126 S1U1B0+ XS1U1B127 S1U1B0? 1200YS1U1B112 S1U1B0+ 1120YS1U1B122 S1U1B0XS1U1B134 S1U1B0+ XS1U1B135 S1U1B0+ XS1U1B136 S1U1B0+ XS1U1B137 S1U1B0? 1200YS1U1B113 S1U1B0+ 1120YS1U1B123 S1U1B0XS1U1B114 S1U1B0+ XS1U1B124 S1U1B0+ XS1U1B134 S1U1B0? 500
XS1U1B115 S1U1B0+ XS1U1B125 S1U1B0+ XS1U1B135 S1U1B0? 600 XS1U1B116 S1U1B0+ XS1U1B126 S1U1B0+ XS1U1B136 S1U1B0? 450 XS1U1B117 S1U1B0+ XS1U1B127 S1U1B0+ XS1U1B137 S1U1B0? 700
YS1U1B1ijS1U1B0.XS1U1B1jk S1U1B0? 0
  
-Refer to Exhibit 6.2.What formula would go into cells B11:E11 and cells F8:F10?

Accepted Answer

The answer of Exhibit 6.2
The following questions pertain to the...

Question 8

ILP formulations can be used to model:

Accepted Answer

A) production problems 
B) personnel scheduling problems 
C) investment allocation problems 
D) all of the above 
A) production problems 
B) personnel scheduling problems 
C) investment allocation problems 
D) all of the above

Question 9

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

Accepted Answer

A) An upper bound. 
B) A lower bound. 
C) An alternative optimal solution. 
D) An additional constraint for the ILP problem. 
A) An upper bound. 
B) A lower bound. 
C) An alternative optimal solution. 
D) An additional constraint for the ILP problem.

Question 10

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.
Hours required by
$$\begin{array} { r c c c } 
\text { Operation } & \text { A } & \text { B } & \text { Hours } \
\hline \text { Cutting } & 3 & 4 & 48 \
\text { Welding } & 2 & 1 & 36
\end{array}$$
What is the appropriate formula to use in cell B15 of the following Excel implementation of the ILP model for this problem?

Accepted Answer

A) =B5 ? MIN$E$11/B11,$E$11/C11)*B14 
B) =B5 ? MIN$E$11/B11,$E$12/B12) 
C) =B5 ? $E$12/B12*B14 
D) =B5 ? MIN$E$11/B11,$E$12/B12)*B14 
A) =B5 ? MIN$E$11/B11,$E$11/C11)*B14 
B) =B5 ? MIN$E$11/B11,$E$12/B12) 
C) =B5 ? $E$12/B12*B14 
D) =B5 ? MIN$E$11/B11,$E$12/B12)*B14

Question 11

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.
Formulate the ILP for this problem.
 $$\begin{array} { r r r r r } 
\text { Variable } & \text { Facilty } & \text { Usage } & \text { Cost \$1,000) } & \text { Land } \
\hline \mathbf { x } _ { 1 } & \text { Swimruine pool } & 400 & 100 & 2 \
\mathbf { x } _ { 2 } & \text { Recreation center } & 500 & 200 & 3 \
\mathbf { x } _ { 3 } & \text { Basketball court } & 300 & 150 & 4 \
\mathbf { x } _ { 4 } & \text { Baseball field } & 200 & 100 & 5
\end{array}$$

Accepted Answer

The answer of A small town wants to build some...

Question 12

An ILP problem has 5 binary decision variables.How many possible integer solutions are there to this problem?

Accepted Answer

A) 5 
B) 10 
C) 25 
D) 32 
A) 5 
B) 10 
C) 25 
D) 32

Question 13

What does the Analytic Solver Platform default integer tolerance factor of 0 accomplish?

Accepted Answer

A) Stops B & B after 100% of all solutions are examined. 
B) Stops B & B when any feasible ILP solution is 0% from the current ILP solution. 
C) Stops B & B when the true optimal integer solution has been found. 
D) Stops B & B when no more than 0% of the changing cells have integer values. 
A) Stops B & B after 100% of all solutions are examined. 
B) Stops B & B when any feasible ILP solution is 0% from the current ILP solution. 
C) Stops B & B when the true optimal integer solution has been found. 
D) Stops B & B when no more than 0% of the changing cells have integer values.

Question 14

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 X2.
MAX: 50 X1 + 40 X2
Subject to: 2 X1 + 4 X2 ≤ 40
3 X1 + 2 X2 ≤ 30
X1,X2 ≥ 0 and integer
Initial solution X1 = 5.0
X2 = 7.5
Obj = 550

Accepted Answer

Question 15

The LP relaxation of an ILP problem

Accepted Answer

A) always encompasses all the feasible integer solutions to the original ILP problem. 
B) encompasses at least 90% of the feasible integer solutions to the original ILP problem. 
C) encompasses different set of feasible integer solutions to the original ILP problem. 
D) will not contain the feasible integer solutions to the original ILP problem. 
A) always encompasses all the feasible integer solutions to the original ILP problem. 
B) encompasses at least 90% of the feasible integer solutions to the original ILP problem. 
C) encompasses different set of feasible integer solutions to the original ILP problem. 
D) will not contain the feasible integer solutions to the original ILP problem.

Question 16

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?

Accepted Answer

A) X1 + X2 = 0 
B) X1 + X2 = 1 
C) X1 − X2 ≥ 0 
D) X1 − X2 ≤ 0 
A) X1 + X2 = 0 
B) X1 + X2 = 1 
C) X1 − X2 ≥ 0 
D) X1 − X2 ≤ 0

Question 17

How are binary variables specified in the Analytic Solver Platform ASP)?

Accepted Answer

A) By replacing RHS values in constraints with 0 or 1. 
B) By specifying changing cells as INTEGER and as non-negative. 
C) By specifying changing cells as BINARY in the Variable Type/Bound area of ASP. 
D) By selecting Assume Binary Model in the ASP Options dialog box. 
A) By replacing RHS values in constraints with 0 or 1. 
B) By specifying changing cells as INTEGER and as non-negative. 
C) By specifying changing cells as BINARY in the Variable Type/Bound area of ASP. 
D) By selecting Assume Binary Model in the ASP Options dialog box.

Question 18

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

Accepted Answer

A) Brooks and Baker 
B) Best Bound 
C) Best Branch 
D) Branch and Bound 
A) Brooks and Baker 
B) Best Bound 
C) Best Branch 
D) Branch and Bound

Question 19

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

Accepted Answer

A) Yes,a feasible solution may be found when additional constraints are added. 
B) Yes,removing a constraint in further analysis may restore feasibility. 
C) No,adding more constraints will not restore problem feasibility. 
D) No,the result cannot occur so re-examine the formulation and start over. 
A) Yes,a feasible solution may be found when additional constraints are added. 
B) Yes,removing a constraint in further analysis may restore feasibility. 
C) No,adding more constraints will not restore problem feasibility. 
D) No,the result cannot occur so re-examine the formulation and start over.

Question 20

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?

Accepted Answer

A) X1 ≥ 3,X1 ≥ 4 
B) X1 = 4 
C) 3 ≤ X1,X1 ≤ 4 
D) X1 ≤ 3,X1 ≥ 4 
A) X1 ≥ 3,X1 ≥ 4 
B) X1 = 4 
C) 3 ≤ X1,X1 ≤ 4 
D) X1 ≤ 3,X1 ≥ 4

Binary decision variables:

A practical way of dealing with the complexity of IP problems is to:

A production company wants to ensure that if Product 1 is produced,production of Product 1 not exceed production of Product 2.Which of the following constraints enforce this condition?

A vendor offers 5 different prices per unit depending on the quantity purchased.How many binary variables are needed to model this discounting scheme?

ILP formulations can be used to model:

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

An ILP problem has 5 binary decision variables.How many possible integer solutions are there to this problem?

What does the Analytic Solver Platform default integer tolerance factor of 0 accomplish?

The LP relaxation of an ILP 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?

How are binary variables specified in the Analytic Solver Platform ASP)?

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

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

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?

Introduction to Modeling and Decision Analysis

Introduction to Optimization and Linear Programming

Modeling and Solving Lp Problems in a Spreadsheet

Sensitivity Analysis and the Simplex Method

Network Modeling

Goal Programming and Multiple Objective Optimization

Nonlinear Programming and Evolutionary Optimization

Regression Analysis

Data Mining

Time Series Forecasting

Introduction to Simulation Using Analytic Solver Platform

Queuing Theory

Decision Analysis

Project Management Online

Filters

Exam 6: Integer Linear Programming

Binary decision variables:

A practical way of dealing with the complexity of IP problems is to:

A production company wants to ensure that if Product 1 is produced,production of Product 1 not exceed production of Product 2.Which of the following constraints enforce this condition?

A vendor offers 5 different prices per unit depending on the quantity purchased.How many binary variables are needed to model this discounting scheme?

ILP formulations can be used to model:

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

An ILP problem has 5 binary decision variables.How many possible integer solutions are there to this problem?

What does the Analytic Solver Platform default integer tolerance factor of 0 accomplish?

The LP relaxation of an ILP 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?

How are binary variables specified in the Analytic Solver Platform ASP)?

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

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

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?

Introduction to Modeling and Decision Analysis

Introduction to Optimization and Linear Programming

Modeling and Solving Lp Problems in a Spreadsheet

Sensitivity Analysis and the Simplex Method

Network Modeling

Goal Programming and Multiple Objective Optimization

Nonlinear Programming and Evolutionary Optimization

Regression Analysis

Data Mining

Time Series Forecasting

Introduction to Simulation Using Analytic Solver Platform

Queuing Theory

Decision Analysis

Project Management Online

Filters

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?