Suppose you want to maximize an objective function z=2x1+3x2.Both decision variables must be integer.The optimal solution to the LP relaxation will:

A) be smaller than the optimal IP solution B) be larger than the optimal IP solution C) can be either smaller or larger than the optimal IP solution D) will be within 5% of the optimal IP solution value A) be smaller than the optimal IP solution B) be larger than the optimal IP solution C) can be either smaller or larger than the optimal IP solution D) will be within 5% of the optimal IP solution value

Exam 6: Integer Linear Programming

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. Variable Decision NPV $million) Cost $million) ^X₁Factory in Columbia 3 10 ^X₂Factory in Atlanta 4 8 ^X₃Warehouse in Columbia 2 6 ^X₄Warehouse in Atlanta 1 5 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? MAX: ^{3 X}₁^{+ 4 X}₂^{+ 2 X}₃^{+ X}₄ Subject to: ^{10 X}₁^{+ 8 X}₂^{+ 6 X}₃^{+ 5 X}₄^{≤ 15} X₁+ X₂= 1 X₃+ X₄≤ 1 X₃− X₁≤ 0 X₄− X₂≤ 0 X_i= 0,1

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Question 21

Which of the following are potential pitfalls of using a non-zero integer tolerance factor in the Analytic Solver Platform?

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Question 22

Variables,which are not required to assume strictly integer values are referred to as

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Question 23

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. To zone No. tall buildings Fram 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

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Question 24

Suppose you want to maximize an objective function z=2x₁+3x₂.Both decision variables must be integer.The optimal solution to the LP relaxation will:

(Multiple Choice)

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Question 25

The branch & bound algorithm stops when:

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Question 26

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

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Question 27

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) ^X₁Factory in Columbia 3 10 ^X₂Factory in Atlanta 4 8 ^X₃Warehouse in Columbia 2 6 ^X₄Warehouse in Atlanta 1 5 Based on this ILP formulation of the problem what is the optimal solution to the problem? MAX: ^{3 X}₁^{+ 4 X}₂^{+ 2 X}₃^{+ X}₄ Subject to: ^{10 X}₁^{+ 8 X}₂^{+ 6 X}₃^{+ 5 X}₄^{≤ 15} X₁+ X₂= 1 X₃+ X₄≤ 1 X₃− X₁≤ 0 X₄− X₂≤ 0 X_i= 0,1

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Question 28

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. Project Funds Required in $000s)Benefit Project 1 2 3 4 5 in \ 000s) 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 $225K $60K $60K $50K $50K The following is the ILP formulation and a spreadsheet model for the problem. $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. Project Funds Required in $000s)Benefit \begin{array}{ccccccc} \text { Project } & 1 & 2 & 3 & 4 & 5 & \text { in } \$ 000 s) \\ \hline 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 \\ \hline \end{array} Budget $225K $60K $60K $50K $50K The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1.What formula should go in cell D15 of the above Excel spreadsheet?$ $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. Project Funds Required in $000s)Benefit \begin{array}{ccccccc} \text { Project } & 1 & 2 & 3 & 4 & 5 & \text { in } \$ 000 s) \\ \hline 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 \\ \hline \end{array} Budget $225K $60K $60K $50K $50K The following is the ILP formulation and a spreadsheet model for the problem. -Refer to Exhibit 6.1.What formula should go in cell D15 of the above Excel spreadsheet?$ -Refer to Exhibit 6.1.What formula should go in cell D15 of the above Excel spreadsheet?

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Question 29

The branch-and-bound algorithm starts by

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Question 30

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 Port D1 D2 D3 D4 A 75 88 103 56 B 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. Formulate the ILP for this problem capturing the ship choice of ports and the supply-to-demand transportation from the ports to the destinations.

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Question 31

How is an LP problem changed into an ILP problem?

(Multiple Choice)

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Question 32

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

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Question 33

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 Facilty Usage Cost \1 ,000) Land Swimming pool 400 100 2 Recreation center 500 200 3 Basketball court 300 150 4 Baseball field 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: 400+500+300+200 Subject ta: 100+200+150+100\leq400 budget 2+3+4+5\leq14 land -\leq0 pool and recreation center +\leq1 basketball and baseball =0.1 Solution: $\left. \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 } , \mathbf { X } _ { 4 } \right) = 1,1,0,1$ ) $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 $1,000)are shown below. \begin{array} { r r r r r } \text { Variable } & \text { Facilty } & \text { Usage } & \text { Cost \$1,000) } & \text { Land } \\ \hline \mathbf { x } _ { 1 } & \text { Swimming pool } & 400 & 100 & 2 \\ \mathbf { x } _ { 2 } & \text { Recreation center } & 500 & 200 & 3 \\ \mathbf { x } _ { 3 } & \text { Basketball court } & 300 & 150 & 4 \\ \mathbf { x } _ { 4 } & \text { Baseball field } & 200 & 100 & 5 \end{array} Based on this ILP formulation of the problem and the indicated optimal solution what values should go in cells B5:G12 of the following Excel spreadsheet? \begin{array} { l l l } \text { MAX: } & 400 \mathbf { x } _ { 1 } + 500 \mathbf { x } _ { 2 } + 300 \mathbf { x } _ { 3 } + 200 \mathbf { x } _ { 4 } & \\ \text { Subject ta: } & 100 \mathbf { x } _ { 1 } + 200 \mathbf { x } _ { 2 } + 150 \mathbf { x } _ { 3 } + 100 \mathbf { x } _ { 4 } \leq 400 & \text { budget } \\ & 2 \mathbf { x } _ { 1 } + 3 \mathbf { x } _ { 2 } + 4 \mathbf { x } _ { 3 } + 5 \mathbf { x } _ { 4 } \leq 14 & \text { land } \\ & \mathbf { x } _ { 1 } - \mathbf { x } _ { 2 } \leq 0 & \text { pool and recreation center } \\ & \mathbf { x } _ { 3 } + \mathbf { x } _ { 4 } \leq 1 & \text { basketball and baseball } \\ & \mathbf { x } _ { 1 } = 0.1 & \end{array} Solution: \left. \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 } , \mathbf { X } _ { 4 } \right) = 1,1,0,1 )$

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Question 34

Consider the constraint X₃+ X₄+ X₅+ X₆+ X₇≥ 27 Representing Air Express' Monday minimum worker requirement.Why was a "≥" used versus an "="?

(Multiple Choice)

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Question 35

Mixed IP formulation requires that:

(Multiple Choice)

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Question 36

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 $million) ^X₁Factory in Columbia 3 10 ^X₂Factory in Atlanta 4 8 ^X₃Warehouse in Columbia 2 6 ^X₄Warehouse in Atlanta 1 5

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Question 37

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?

(Multiple Choice)

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Question 38

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 1 Condos 6 50 10 2 Apartments 12 90 5 3 Houses 9 1 7 Based on this ILP formulation of the problem what formulas should go in cells E5:E12 of the following Excel spreadsheet? MAX: ^{6 X}₁^{+ 12 X}₂^{+ 9 X}₃ Subject to: ^{50 X}₁^{+ 90 X}₂^{+ 100 X}₃^{? 500} X₁? 10 X₂? 5 X₃? 7 Xi ? 0 and integer $An investor has $500,000 to invest and wants to maximize the money they will receive at the end of one year.They can invest in condos,apartments and houses.The profit after one year,the cost and the number of units available are shown below. \begin{array}{rrrrr} &&\text { Profit } & \text { Cost } & \text { Number } \\ \text { Variable } & \text { Investment } &\$ 1,000)&\$ 1,000) & \text { Available }\\ \hline \mathrm{X} 1 & \text { Condos } & 6 & 50 & 10 \\ \mathrm{X} 2 & \text { Apartments } & 12 & 90 & 5 \\ \mathrm{X} 3 & \text { Houses } & 9 & 1 \mathrm{nn} & 7 \end{array} Based on this ILP formulation of the problem what formulas should go in cells E5:E12 of the following Excel spreadsheet? MAX: 6 X1 + 12 X2 + 9 X3 Subject to: 50 X1 + 90 X2 + 100 X3 ? 500 X1 ? 10 X2 ? 5 X3 ? 7 Xi ? 0 and integer$

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Question 39

The objective function value for the ILP problem can never

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Question 40

Showing 21 - 40 of 88

Which of the following are potential pitfalls of using a non-zero integer tolerance factor in the Analytic Solver Platform?

Variables,which are not required to assume strictly integer values are referred to as

Suppose you want to maximize an objective function z=2x₁+3x₂.Both decision variables must be integer.The optimal solution to the LP relaxation will:

The branch & bound algorithm stops when:

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

The branch-and-bound algorithm starts by

How is an LP problem changed into an ILP problem?

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

Consider the constraint X₃+ X₄+ X₅+ X₆+ X₇≥ 27 Representing Air Express' Monday minimum worker requirement.Why was a "≥" used versus an "="?

Mixed IP formulation requires that:

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?

The objective function value for the ILP problem can never

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

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Exam 6: Integer Linear Programming

Which of the following are potential pitfalls of using a non-zero integer tolerance factor in the Analytic Solver Platform?

Variables,which are not required to assume strictly integer values are referred to as

Suppose you want to maximize an objective function z=2x1+3x2.Both decision variables must be integer.The optimal solution to the LP relaxation will:

The branch & bound algorithm stops when:

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

The branch-and-bound algorithm starts by

How is an LP problem changed into an ILP problem?

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

Consider the constraint X3 + X4 + X5 + X6 + X7 ≥ 27 Representing Air Express' Monday minimum worker requirement.Why was a "≥" used versus an "="?

Mixed IP formulation requires that:

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?

The objective function value for the ILP problem can never

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

Suppose you want to maximize an objective function z=2x₁+3x₂.Both decision variables must be integer.The optimal solution to the LP relaxation will:

Consider the constraint X₃+ X₄+ X₅+ X₆+ X₇≥ 27 Representing Air Express' Monday minimum worker requirement.Why was a "≥" used versus an "="?