Question 1

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 function value for a one-unit increase in the right-hand side of the constraint equation.

Question 2

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 with a unit increase in a resource is represented by the shadow price in Solver's sensitivity report. The shadow price is the amount by which the objective function value changes when the resource availability is increased by one unit. Therefore, choice C is the correct answer.

Question 3

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.

Question 4

If the objective function has the equation {4x₁+ 2x₂},then the slope of the objective function line is −2.

Accepted Answer

The answer of If the objective function has the equation...

Question 5

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 existence of a unique solution is not guaranteed for every solvable linear programming problem.

Question 6

If a constraint has the equation 5x₁+ 2x₂≤ 60,then the constraint line passes through the points (0,12)and (30,0).

Accepted Answer

The points (0,12) and (30,0) do not satisfy the given equation 5x + 2y ≤ 60 when substituted back into the equation. For (0,12), 5(0) + 2(12) = 24 ≤ 60 is true, but for (30,0), 5(30) + 2(0) = 150 ≤ 60 is false, indicating (30,0) does not lie on the constraint line defined by 5x + 2y ≤ 60.

Question 7

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 variable cells.

Accepted Answer

The designated cells that play the role of decision variables in a linear programming spreadsheet model are known as changing variable cells, and these cells can be adjusted to optimize the solution.

Question 8

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

Accepted Answer

Constraints that have been omitted or incorrect inputs can cause unboundedness in LP models. This means that the model does not have a finite solution and can continue to improve indefinitely in certain directions. It is important to carefully review and validate all constraints and inputs in an LP model to ensure that it is well-defined and has a bounded solution.

Question 9

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 area of the graph where all constraints are satisfied, and the constraints are typically represented by lines or inequalities. When graphed, these lines or inequalities will create a polygonal feasible region with straight sides.

Question 10

If a manufacturing process takes 4 hours per unit of x1and 2 hours per unit of x2 and a maximum of 100 hours of manufacturing process time are available,then an algebraic formulation of this constraint is:
A) 4x1S1U1B0 + 2xS1U1B12S1U1B0 ≥ 100 
B) 4x1S1U1B0 − 2xS1U1B12S1U1B0 ≤ 100 
C) 4x1S1U1B0 + 2xS1U1B12S1U1B0 ≤ 100 
D) 4x1S1U1B0 − 2xS1U1B12 S1U1B0≥ 100

Accepted Answer

The manufacturing process time constraint can be written as:
4x1 + 2x2 <= 100
where x1 is the units of product x1 produced and x2 is the units of product x2 produced. 
To put it in the given format: 
4x1S1U1B0 + 2x2S1U1B1 <= 100
And simplifying:
4x1S1U1B0 + 2x2S1U1B1 + 0S1U1B0 ≤ 100
Hence, the correct choice is C.

Question 11

Suppose a company sells two different products,x1and x2,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 answer of Suppose a company sells two different products,x1and...

Question 12

Consider the following linear programming problem:The above linear programming problem:Maximize4x₁ + 2x₂ Subject to: 4x₁ + 2x₂ ≤ 40 2x₁ + x₂≥ 20 x₁,x2 ≥ 0 A) has only one feasible solution B) has more than one optimal solution C) exhibits infeasibility D) exhibits unboundedness

Accepted Answer

The problem has conflicting constraints: 2x₁ + x₂≥ 20 4x₁ + 2x₂ ≤ 40 Together, these constraints imply: 2x₁ + x₂≥ 10 2x₁ + x₂< 10 which is impossible. Therefore, the problem is infeasible.

Question 13

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

This is true. The proportionality property states that the relationship between the objective function and the decision variables, as well as the relationship between the decision variables and the constraints, is linear and proportional. Therefore, the sum of the contributions from the various activities to a particular constraint must equal the total contribution to that constraint.

Question 14

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

Accepted Answer

The changing variable cells, also known as decision variables, represent the values that we want to find in a linear programming problem. These are the quantities that we are trying to optimize or find the best value for, subject to certain constraints. Therefore, choice C is the correct answer. The other options are not accurate descriptions of what the changing variable cells represent.

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 defined by hyperplanes, which are geometric representations of the constraints in the problem. These hyperplanes form the boundaries of the feasible region. Corner points and objective lines are important in solving linear programming problems but do not bound the feasible region themselves.

Question 16

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

The answer of Linear programming models have three important properties.They...

Question 17

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

The condition of nonnegativity requires that the decision variables cannot be less than zero. This is because decision variables represent quantities that cannot be negative, such as the number of units produced or the amount of resources used. The other answer choices are not correct because the objective function, constraints, and reduced costs can be negative in some cases.

Question 18

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

Accepted Answer

There can be multiple decision variables and constraints in an optimization model, but there can only be one objective function. The objective function defines what is being optimized or minimized in the model.

Question 19

The equation of the line representing the constraint 4x₁+ 2x₂ ≤ 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 equation given is 4x + 2y ≤ 100. To find the line that represents this constraint, we set the inequality to equality: 4x + 2y = 100. Solving for y when x = 0 gives y = 50, and solving for x when y = 0 gives x = 25. Therefore, the line passes through the points (25,0) and (0,50).

Question 20

Suppose the shadow price for a constraint is $12 and the allowable increase and decrease for the right hand side of the constraint are 25 (increase)and 10 (decrease).If the right hand side of that constraint were to increase by 10 the shadow price changes.

Accepted Answer

The shadow price remains constant as long as the change in the right-hand side of the constraint is within the allowable increase and decrease range.

Quiz 3: Introduction to Optimization Modeling

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.

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:

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.

If the objective function has the equation {4x1+ 2x2},then the slope of the objective function line is −2.

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

If a constraint has the equation 5x1+ 2x2 ≤ 60,then the constraint line passes through the points (0,12)and (30,0).

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 variable cells.

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

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

If a manufacturing process takes 4 hours per unit of x1and 2 hours per unit of x2 and a maximum of 100 hours of manufacturing process time are available,then an algebraic formulation of this constraint is:

Suppose a company sells two different products,x1and x2,for net profits of $6 per unit and $3 per unit,respectively.The slope of the line representing the objective function is:

Consider the following linear programming problem:The above linear programming problem:Maximize4x1 + 2x2 Subject to: 4x1 + 2x2 ≤ 40 2x1 + x2≥ 20 x1,x2 ≥ 0

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.

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

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

Linear programming models have three important properties.They are:

The condition of nonnegativity requires that:

In an optimization model,there can only be one:

The equation of the line representing the constraint 4x1+ 2x2 ≤ 100 passes through the points:

Suppose the shadow price for a constraint is $12 and the allowable increase and decrease for the right hand side of the constraint are 25 (increase)and 10 (decrease).If the right hand side of that constraint were to increase by 10 the shadow price changes.

If the objective function has the equation {4x₁+ 2x₂},then the slope of the objective function line is −2.

If a constraint has the equation 5x₁+ 2x₂≤ 60,then the constraint line passes through the points (0,12)and (30,0).

Consider the following linear programming problem:The above linear programming problem:Maximize4x₁ + 2x₂ Subject to: 4x₁ + 2x₂ ≤ 40 2x₁ + x₂≥ 20 x₁,x2 ≥ 0

The equation of the line representing the constraint 4x₁+ 2x₂ ≤ 100 passes through the points: