Question 1

Binary variables are:

Accepted Answer

A)  a subset of integer variables 
B)  continuous on an interval (0,1) 
C)  continuous on an interval (0,+∞) 
D)  negative 
A)  a subset of integer variables 
B)  continuous on an interval (0,1) 
C)  continuous on an interval (0,+∞) 
D)  negative

Question 2

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

Accepted Answer

Question 3

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?   ​

Accepted Answer

The answer of An investor has $500,000 to invest and...

Question 4

Mixed IP formulation requires that:

Accepted Answer

A)  all decision variables must be integer 
B)  the optimal objective function value must be integer 
C)  some decision variables must be integer 
D)  some decision variables and the optimal value of the objective function must be integer 
A)  all decision variables must be integer 
B)  the optimal objective function value must be integer 
C)  some decision variables must be integer 
D)  some decision variables and the optimal value of the objective function must be integer

Question 5

​A problem in which all decision variables are restricted to assuming only integer values is called a pure IP programming problem.

Accepted Answer

A) True 
 B)False

Question 6

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:   The operation time per unit and total operating hours available are:   Based on this ILP formulation of the problem and the optimal solution (X1, X2, X3) = (270, 0, 0), what values should appear in the shaded cells in the following Excel spreadsheet?
Xi = amount of product i produced
Yi = 1 if product i produced, 0 otherwise   ​   ​

Accepted Answer

Question 7

A popular solution technique to IP problems is called:

Accepted Answer

A)  a branch-and-bound algorithm 
B)  branching 
C)  bounding 
D)  LP relaxation 
A)  a branch-and-bound algorithm 
B)  branching 
C)  bounding 
D)  LP relaxation

Question 8

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

Accepted Answer

A)  strictly non-integer. 
B)  continuous. 
C)  discrete. 
D)  infinite. 
A)  strictly non-integer. 
B)  continuous. 
C)  discrete. 
D)  infinite.

Question 9

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 10

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

Accepted Answer

A)  No assurance the returned solution is optimal. 
B)  No assurance the returned solution is integer. 
C)  The true optimal solution may be worse than the returned solution. 
D)  There are no pitfalls to consider since the Solver will obtain solutions quicker. 
A)  No assurance the returned solution is optimal. 
B)  No assurance the returned solution is integer. 
C)  The true optimal solution may be worse than the returned solution. 
D)  There are no pitfalls to consider since the Solver will obtain solutions quicker.

Question 11

Specifying suboptimality tolerances can be useful if a decision maker wants to find a good but nor optimal solution to a complex ILP problem.

Accepted Answer

A) True 
 B)False

Question 12

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 13

Binary variables are useful in modeling logical conditions in ILP problems. ​

Accepted Answer

A) True 
 B)False

Question 14

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

Question 15

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 16

A company wants to select 1 project from a set of 4 possible projects. Which of the following constraints ensures that only 1 will be selected?

Accepted Answer

A)  X1 + X2 + X3 + X4 = 1 
B)  X1 + X2 + X3 + X4 ≤ 1 
C)  X1 + X2 + X3 + X4 ≥ 1 
D)  X1 + X2 + X3 + X4 ≥ 0 
A)  X1 + X2 + X3 + X4 = 1 
B)  X1 + X2 + X3 + X4 ≤ 1 
C)  X1 + X2 + X3 + X4 ≥ 1 
D)  X1 + X2 + X3 + X4 ≥ 0

Question 17

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

Accepted Answer

A)  a diverging variable. 
B)  a branching variable. 
C)  a bifurcating variable. 
D)  a splitting variable. 
A)  a diverging variable. 
B)  a branching variable. 
C)  a bifurcating variable. 
D)  a splitting variable.

Question 18

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:

Accepted Answer

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

Question 19

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.

Accepted Answer

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

Question 20

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 Xi = the amount of product i produced
Yi = 1 if Xi > 0 and 0 if Xi = 0
What is the objective function for this problem?

Accepted Answer

A)  MAX: 17 X1 + 21 X2 
B)  MAX: 17 X1 + 21 X2 − 60 Y1 − 80 Y2 
C)  MIN: 17 X1 + 21 X2 − 60 Y1 − 80 Y2 
D)  MIN: 60 Y1 + 80 Y2 
A)  MAX: 17 X1 + 21 X2 
B)  MAX: 17 X1 + 21 X2 − 60 Y1 − 80 Y2 
C)  MIN: 17 X1 + 21 X2 − 60 Y1 − 80 Y2 
D)  MIN: 60 Y1 + 80 Y2

Binary variables are:

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

Mixed IP formulation requires that:

A problem in which all decision variables are restricted to assuming only integer values is called a pure IP programming problem.

A popular solution technique to IP problems is called:

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

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

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

Specifying suboptimality tolerances can be useful if a decision maker wants to find a good but nor optimal solution to a complex ILP problem.

Binary decision variables:

Binary variables are useful in modeling logical conditions in 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?

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 1 project from a set of 4 possible projects. Which of the following constraints ensures that only 1 will be selected?

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

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:

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 Evolutionary Optimization

Regression Analysis

Data Mining

Time Series Forecasting

Introduction to Simulation Using Analytic Solver Platform

Queuing Theory

Decision Analysis

Project Management Online

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Exam 6: Integer Linear Programming

Binary variables are:

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

Mixed IP formulation requires that:

​A problem in which all decision variables are restricted to assuming only integer values is called a pure IP programming problem.

A popular solution technique to IP problems is called:

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

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

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

Specifying suboptimality tolerances can be useful if a decision maker wants to find a good but nor optimal solution to a complex ILP problem.

Binary decision variables:

Binary variables are useful in modeling logical conditions in 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?

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 1 project from a set of 4 possible projects. Which of the following constraints ensures that only 1 will be selected?

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

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:

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 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 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

A problem in which all decision variables are restricted to assuming only integer values is called a pure IP programming problem.

Binary variables are useful in modeling logical conditions in 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?

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: