Question 1

Consider the following linear programming problem: The above linear programming problem:
Maximize
4x1 + 2y2
Subject to:
4x1 + 2y2 ≤ 40
2x1 + y2 ≥ 20
X1, y2 ≥ 0
A)has only one feasible solution
B)has more than one optimal solution
C)exhibits infeasibility
D)exhibits unboundedness

Accepted Answer

C

Question 2

Linear programming models have three important properties. They are:
A)optimality, additivity and sensitivity
B)optimality, linearity and divisibility
C)divisibility, linearity and nonnegativity
D)proportionality, additivity and divisibility

Accepted Answer

D

Question 3

The feasible region in a graphical solution of a linear programming problem will appear as some type of polygon, with lines forming all sides.

Accepted Answer

This is true. The feasible region is the region where all constraints are satisfied, and it is typically shown as a polygon with lines forming its sides in a graphical solution of a linear programming problem.

Question 4

The proportionality property of LP models implies that the sum of the contributions from the various activities to a particular constraint equals the total contribution to that constraint.

Accepted Answer

According to the proportionality property of LP models, the sum of the contributions from the various activities to a particular constraint should equal the total contribution to that constraint. Therefore, the statement is true.

Question 5

If a constraint has the equation 5x + 2y ≤ 60, then the constraint line passes through the points (0,12) and (30,0).

Accepted Answer

The answer of If a constraint has the equation 5x...

Question 6

Common errors in LP models that exhibit unboundedness are a constraint that has been omitted or an input which is incorrect.

Accepted Answer

It is true that common errors in LP models that exhibit unboundedness are a constraint that has been omitted or an input which is incorrect.

Question 7

If the objective function has the equation {4x + 2y}, then the slope of the objective function line is −2.

Accepted Answer

The slope of the objective function line is the coefficient of the x-term, which is 4 divided by the coefficient of the y-term, which is 2. Thus, the slope is -2.

Question 8

The condition of nonnegativity requires that:
A)the objective function cannot be less that zero
B)the decision variables cannot be less than zero
C)the right hand side of the constraints cannot be greater then zero
D)the reduced cost cannot be less than zero

Accepted Answer

Nonnegativity requires that decision variables cannot be negative, as negative values do not make sense in real-world applications (for example, you cannot have a negative number of products). The other options are not correct: A refers to the objective function, which may be negative in certain situations; C refers to the right-hand side of the constraints, which can be either positive, negative, or zero; and D refers to the reduced cost, which is a property of the simplex algorithm used to find the optimal solution to a linear programming problem, but is not related to the nonnegativity condition.

Question 9

Shadow prices are associated with nonbinding constraints, and show the change in the optimal objective function value when the right side of the constraint equation changes by one unit.

Accepted Answer

Shadow prices are associated with binding constraints, not nonbinding constraints, and they show the change in the optimal objective value for a one-unit increase in the right-hand side of the constraint equation.

Question 10

Suppose the allowable increase and decrease for shadow price for a constraint are $25 (increase) and $10 (decrease). If the right hand side of that constraint were to increase by $10 the optimal solution would not change.

Accepted Answer

The optimal solution would change because the increase in the right-hand side exceeds the allowable increase for the shadow price, which is $25.

Question 11

The equation of the line representing the constraint 4x + 2y ≤ 100 passes through the points:
A)(25,0) and (0,50)
B)(0,25) and (50,0)
C)(−25,0) and (0,−50)
D)(0,−25) and (−50,0)

Accepted Answer

The answer of The equation of the line representing the...

Question 12

In an optimization model, there can only be one:
A)decision variable
B)constraint
C)objective function
D)shadow price

Accepted Answer

An optimization model must have only one objective function which defines the goal of the optimization problem. Having multiple objective functions would lead to conflicting goals and not allow for an optimal solution.

Question 13

In using Excel to solve linear programming problems, the changing cells represent the:
A)value of the objective function
B)constraints
C)decision variables
D)total cost of the model

Accepted Answer

The changing cells in Excel represent the decision variables in a linear programming model. These are the variables that we are solving for in order to optimize the objective function, subject to the constraints. The value of the objective function itself is not a changing cell, but rather a calculated result based on the values of the decision variables. The constraints are represented by fixed values or formulas in other cells of the Excel spreadsheet. The total cost of the model may be related to the objective function, but it is not a changing cell in and of itself.

Question 14

When the profit increases with a unit increase in a resource, this change in profit will be shown in Solver's sensitivity report as the:
A)add-in price
B)sensitivity price
C)shadow price
D)additional profit

Accepted Answer

The change in profit when a resource is increased by one unit is represented by the shadow price in Solver's sensitivity report.

Question 15

The feasible region in all linear programming problems is bounded by:
A)corner points
B)hyperplanes
C)an objective line
D)all of these options

Accepted Answer

The feasible region in linear programming problems is bounded by hyperplanes, which are geometric representations of the constraints in the problem. These hyperplanes are essentially the lines (in two dimensions), planes (in three dimensions), or higher-dimensional analogs that define the limits of the feasible region. Corner points and an objective line are elements related to the feasible region and the optimization process, but they do not bound the feasible region themselves.

Question 16

When formulating a linear programming spreadsheet model, there is a set of designated cells that play the role of the decision variables. These are called the changing cells.

Accepted Answer

The designated cells that play the role of the decision variables in a linear programming spreadsheet model are known as changing cells or variable cells. These are the cells that we can adjust or change to optimize the objective function.

Question 17

Suppose a company sells two different products, x and y, for net profits of $6 per unit and $3 per unit, respectively. The slope of the line representing the objective function is:
A)0.5
B)−0.5
C)2
D)−2

Accepted Answer

The slope of the line representing the objective function is the negative ratio of the coefficients of x and y, which is -6/3 = -2.

Question 18

All linear programming problems should have a unique solution, if they can be solved.

Accepted Answer

Linear programming problems can have multiple solutions, a unique solution, or no solution at all. The presence of multiple optimal solutions is possible when the objective function is parallel to a constraint boundary in the feasible region.

Question 19

If a manufacturing process takes 4 hours per unit of x and 2 hours per unit of y and a maximum of 100 hours of manufacturing process time are available, then an algebraic formulation of this constraint is:
A)4x + 2y ≥ 100
B)4x − 2y ≤ 100
C)4x + 2y ≤ 100
D)4x − 2y ≥ 100

Accepted Answer

The constraint represents the limitation of the total available hours of manufacturing process time. 
The manufacturing process takes 4 hours for each unit of x and 2 hour for each unit of y. Therefore, the total manufacturing process time can be calculated as 4x + 2y. We know that the maximum available manufacturing process time is 100 hours. So, we can represent this constraint as:
4x + 2y ≤ 100

Question 20

Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value of $50 are $25 (increase) and $10 (decrease). If the coefficient were to change from $50 to $65, the optimal value of the objective function would not change.

Accepted Answer

The allowable increase for the objective coefficient is $25, meaning the coefficient can increase up to $75 ($50 + $25) without changing the optimal solution. Since the change is to $65, within the allowable increase, the optimal value of the objective function would remain unchanged, contradicting the statement.

Quiz 3: Introduction to Optimization Modeling

Consider the following linear programming problem: The above linear programming problem: Maximize 4x1 + 2y2 Subject to: 4x1 + 2y2 ≤ 40 2x1 + y2 ≥ 20 X1, y2 ≥ 0

Linear programming models have three important properties. They are:

The feasible region in a graphical solution of a linear programming problem will appear as some type of polygon, with lines forming all sides.

The proportionality property of LP models implies that the sum of the contributions from the various activities to a particular constraint equals the total contribution to that constraint.

If a constraint has the equation 5x + 2y ≤ 60, then the constraint line passes through the points (0,12) and (30,0).

Common errors in LP models that exhibit unboundedness are a constraint that has been omitted or an input which is incorrect.

If the objective function has the equation {4x + 2y}, then the slope of the objective function line is −2.

The condition of nonnegativity requires that:

Shadow prices are associated with nonbinding constraints, and show the change in the optimal objective function value when the right side of the constraint equation changes by one unit.

Suppose the allowable increase and decrease for shadow price for a constraint are $25 (increase) and $10 (decrease). If the right hand side of that constraint were to increase by $10 the optimal solution would not change.

The equation of the line representing the constraint 4x + 2y ≤ 100 passes through the points:

In an optimization model, there can only be one:

In using Excel to solve linear programming problems, the changing cells represent the:

When the profit increases with a unit increase in a resource, this change in profit will be shown in Solver's sensitivity report as the:

The feasible region in all linear programming problems is bounded by:

When formulating a linear programming spreadsheet model, there is a set of designated cells that play the role of the decision variables. These are called the changing cells.

Suppose a company sells two different products, x and y, for net profits of $6 per unit and $3 per unit, respectively. The slope of the line representing the objective function is:

All linear programming problems should have a unique solution, if they can be solved.

If a manufacturing process takes 4 hours per unit of x and 2 hours per unit of y and a maximum of 100 hours of manufacturing process time are available, then an algebraic formulation of this constraint is:

Suppose the allowable increase and decrease for an objective coefficient of a decision variable that has a current value of $50 are $25 (increase) and $10 (decrease). If the coefficient were to change from $50 to $65, the optimal value of the objective function would not change.