Question 1

Consider the following linear programming problem:
MIN Z =&emsp; &emsp; 3x1 + 2x2
Subject to:&emsp; &emsp; 2x1 + 3x2 ? 12
&emsp; &emsp;&emsp;5x1 + 8x2 ? 37
&emsp; &emsp;&emsp;x1, x2 ? 0
At the optimal solution point, the objective function value is 18. If the constraints are changed from greater than to less than constraints and the objective function is changed from minimize to maximize, what happens to the optimal solution? Demonstrate whether it falls at the same optimal point.

Accepted Answer

The answer of Consider the following linear programming problem:
MIN Z...

Question 2

A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function.  
What would the be the new slope of the objective function if multiple optimal solutions occurred along line segment AB?

Accepted Answer

The answer of A graphical representation of a linear program...

Question 3

In a linear programming problem, a valid objective function can be represented as:

Accepted Answer

In linear programming, the objective function must be linear, meaning it can only have variables to the first power and cannot include multiplication or division of variables. Option C, "Max 3x + 3y + 1/3 z," is the only choice that meets this criterion, as it is a linear combination of the variables x, y, and z.

Question 4

Consider the following linear programming problem:
MIN Z =&emsp; &emsp;& 2x1 + 3x2
Subject to:&emsp; &emsp; x1 + 2x2 ? 20
&emsp; &emsp;&emsp;5x1 + x2 ? 40
&emsp; &emsp;&emsp;4x1 +6x2 ? 60
&emsp; &emsp;&emsp;x1 , x2 ? 0
What is the optimal solution?

Accepted Answer

The answer of Consider the following linear programming problem:
MIN Z...

Question 5

The minimization of cost or maximization of profit is the

Accepted Answer

The objective of linear programming is to either minimize cost or maximize profit. Constraints are the limitations or restrictions in the decision-making process, while the goal of management science is to use scientific techniques to improve decision-making. The assumption of financiality is the assumption that the objective of a company is to maximize profit.

Question 6

In a linear programming problem, the binding constraints for the optimal solution are:
5x1 + 3x2 ≤ 30
2x1 + 5x2 ≤ 20
Which of these objective functions will lead to the same optimal solution?

Accepted Answer

The optimal solution will occur at the intersection of the two binding constraints, which is (x1, x2) = (3, 2). Plugging this into the objective function options, we can calculate the optimal value:
Option A: 2x1 + 1x2 = 2(3) + 1(2) = 8
Option B: 7x1 + 8x2 = 7(3) + 8(2) = 35
Option C: 80x1 + 60x2 = 80(3) + 60(2) = 420
Option D: 25x1 + 15x2 = 25(3) + 15(2) = 105

Option D gives the smallest optimal value, which means it will lead to the same optimal solution as the objective function in the original problem.

Question 7

A linear programming model consists of

Accepted Answer

A linear programming model consists of decision variables, an objective function, and constraints.

Question 8

In a linear programming problem, the binding constraints for the optimal solution are:
5x1 + 3x2 ≤ 30
2x1 + 5x2 ≤ 20
As long as the slope of the objective function stays between ________ and ________, the current optimal solution point will remain optimal.

Accepted Answer

The answer of In a linear programming problem, the binding...

Question 9

Consider the following linear programming problem:
Max Z = $15x + $20y
Subject to: &emsp;8x + 5y ? 40
&emsp; &emsp;&emsp;  0.4x + y ? 4
&emsp; &emsp;&emsp;x, y ? 0
Determine the values for x and y that will maximize revenue. Given this optimal revenue, what is the amount of slack associated with the first constraint?

Accepted Answer

The answer of Consider the following linear programming problem:
Max Z...

Question 10

Consider the following linear programming problem:
Max Z =&emsp; &emsp; 3x1 + 3x2
Subject to: &emsp; &emsp;10x1 + 4x2 ? 60
&emsp; &emsp;&emsp; 25x1 + 50x2 ? 200
&emsp; &emsp;&emsp;x1, x2 ? 0
Find the optimal profit and the values of x1 and x2 at the optimal solution.

Accepted Answer

The answer of Consider the following linear programming problem:
Max Z...

Question 11

Given this set of constraints, for what objective function is the point x=5, y=3 in the feasible region?
s.t 3x + 6y ≤ 30
10x + 10y ≤ 60
10x + 15y ≤ 90

Accepted Answer

The answer of Given this set of constraints, for what...

Question 12

Consider the following linear programming problem:
Max Z =&emsp; &emsp; $3x + $9y
Subject to: &emsp; &emsp;20x + 32y ? 1600
&emsp; &emsp;&emsp; &emsp;4x + 2y ? 240
&emsp; &emsp;&emsp; &emsp;y ? 40
&emsp; &emsp;&emsp; &emsp;x, y ? 0
Solve for the quantities of x and y which will maximize Z. What is the value of the slack variable associated with constraint 2?

Accepted Answer

The answer of Consider the following linear programming problem:
Max Z...

Question 13

Consider the following linear programming problem:
MIN Z =&emsp; &emsp; 3x1 + 2x2
Subject to: &emsp; &emsp;2x1 + 3x2 ? 12
&emsp; &emsp;&emsp;5x1 + 8x2 ? 37
&emsp; &emsp;&emsp;x1, x2 ? 0
What is minimum cost and the value of x1 and x2 at the optimal solution?

Accepted Answer

The answer of Consider the following linear programming problem:
MIN Z...

Question 14

Which of the following could not be a linear programming problem constraint?

Accepted Answer

In linear programming, all constraints must be linear equations (or inequalities). Option A is not a linear equation, as it involves a inequality using the "not equals" sign.

Question 15

Consider the following linear programming problem:
MIN Z =&emsp; &emsp; 10x1 + 20x2
Subject to:&emsp; &emsp; x1 + x2 ? 12
&emsp; &emsp;&emsp;2x1 + 5x2 ? 40
&emsp; &emsp;&emsp;x2 ? 13
&emsp; &emsp;&emsp;x1, x2 ? 0
At the optimal solution, what is the value of surplus associated with constraint 1 and constraint 3, respectively?

Accepted Answer

The answer of Consider the following linear programming problem:
MIN Z...

Question 16

The poultry farmer decided to make his own chicken scratch by combining alfalfa and corn in rail car quantities. A rail car of corn costs $400 and a rail car of alfalfa costs $200. The farmer's chickens have a minimum daily requirement of vitamin K (500 milligrams) and iron (400 milligrams), but it doesn't matter whether those elements come from corn, alfalfa, or some other grain. A unit of corn contains 150 milligrams of vitamin K and 75 milligrams of iron. A unit of alfalfa contains 250 milligrams of vitamin K and 50 milligrams of iron. Formulate the linear programming model for this situation.

Accepted Answer

The answer of The poultry farmer decided to make his...

Question 17

Consider the following linear programming problem:
MIN Z =&emsp; &emsp; 10x1 + 20x2
Subject to:&emsp; &emsp; x1 + x2 ? 12
&emsp; &emsp;&emsp;2x1 + 5x2 ? 40
&emsp; &emsp;&emsp;x2 ? 13
&emsp; &emsp;&emsp;x1, x2 ? 0
What is minimum cost and the value of x1 and x2 at the optimal solution?

Accepted Answer

The answer of Consider the following linear programming problem:
MIN Z...

Question 18

Consider the following linear programming problem:
Max Z = &emsp; &emsp;5x1 + 3x2
Subject to:&emsp; &emsp; 6x1 + 2x2 ? 18
&emsp; &emsp;&emsp; &emsp;15x1 + 20x2 ? 60
&emsp; &emsp;&emsp; &emsp;x1 , x2 ? 0
Find the optimal profit and the values of x1 and x2 at the optimal solution.

Accepted Answer

The answer of Consider the following linear programming problem:
Max Z...

Question 19

Decision variables

Accepted Answer

Decision variables are the variables that measure how much or how many items to produce, purchase, hire, etc. They are the variables that a decision-maker has the control over and can be altered to achieve the optimal outcome. Therefore, option B is the correct answer.

Question 20

A company producing a standard line and a deluxe line of dishwashers has the following time requirements (in minutes) in departments where either model can be processed.
 $$\begin{array} { | l | l | l | } 
\hline & \text { Standard } &  { \text { Deluxe } } \
\hline \text { Stamping } & 3 & 6 \
\hline \text { Motor installation } & 10 & 10 \
\hline \text { Wiring } & 10 & 15 \
\hline
\end{array}$$ 
The standard models contribute $20 each and the deluxe $30 each to profits. Because the company produces other items that share resources used to make the dishwashers, the stamping machine is available only 30 minutes per hour, on average. The motor installation production line has 60 minutes available each hour. There are two lines for wiring, so the time availability is 90 minutes per hour.
Let x = number of standard dishwashers produced per hour
y = number of deluxe dishwashers produced per hour
Write the formulation for this linear program.

Accepted Answer

The answer of A company producing a standard line and...

Quiz 2: Linear Programming: Model Formulation and Graphical Solution

In a linear programming problem, a valid objective function can be represented as:

Consider the following linear programming problem: MIN Z = & 2x1 + 3x2 Subject to: x1 + 2x2 ? 20 5x1 + x2 ? 40 4x1 +6x2 ? 60 x1 , x2 ? 0 What is the optimal solution?

The minimization of cost or maximization of profit is the

In a linear programming problem, the binding constraints for the optimal solution are: 5x1 + 3x2 ≤ 30 2x1 + 5x2 ≤ 20 Which of these objective functions will lead to the same optimal solution?

A linear programming model consists of

In a linear programming problem, the binding constraints for the optimal solution are: 5x1 + 3x2 ≤ 30 2x1 + 5x2 ≤ 20 As long as the slope of the objective function stays between and , the current optimal solution point will remain optimal.

Consider the following linear programming problem: Max Z = $15x + $20y Subject to: 8x + 5y ? 40 0.4x + y ? 4 x, y ? 0 Determine the values for x and y that will maximize revenue. Given this optimal revenue, what is the amount of slack associated with the first constraint?

Consider the following linear programming problem: Max Z = 3x1 + 3x2 Subject to: 10x1 + 4x2 ? 60 25x1 + 50x2 ? 200 x1, x2 ? 0 Find the optimal profit and the values of x1 and x2 at the optimal solution.

Given this set of constraints, for what objective function is the point x=5, y=3 in the feasible region? s.t 3x + 6y ≤ 30 10x + 10y ≤ 60 10x + 15y ≤ 90

Consider the following linear programming problem: Max Z = $3x + $9y Subject to: 20x + 32y ? 1600 4x + 2y ? 240 y ? 40 x, y ? 0 Solve for the quantities of x and y which will maximize Z. What is the value of the slack variable associated with constraint 2?

Consider the following linear programming problem: MIN Z = 3x1 + 2x2 Subject to: 2x1 + 3x2 ? 12 5x1 + 8x2 ? 37 x1, x2 ? 0 What is minimum cost and the value of x1 and x2 at the optimal solution?

Which of the following could not be a linear programming problem constraint?

Consider the following linear programming problem: MIN Z = 10x1 + 20x2 Subject to: x1 + x2 ? 12 2x1 + 5x2 ? 40 x2 ? 13 x1, x2 ? 0 At the optimal solution, what is the value of surplus associated with constraint 1 and constraint 3, respectively?

Consider the following linear programming problem: MIN Z = 10x1 + 20x2 Subject to: x1 + x2 ? 12 2x1 + 5x2 ? 40 x2 ? 13 x1, x2 ? 0 What is minimum cost and the value of x1 and x2 at the optimal solution?

Consider the following linear programming problem: Max Z = 5x1 + 3x2 Subject to: 6x1 + 2x2 ? 18 15x1 + 20x2 ? 60 x1 , x2 ? 0 Find the optimal profit and the values of x1 and x2 at the optimal solution.

Decision variables

Quiz 2: Linear Programming: Model Formulation and Graphical Solution

In a linear programming problem, a valid objective function can be represented as:

Consider the following linear programming problem: MIN Z = & 2x1 + 3x2 Subject to: x1 + 2x2 ? 20 5x1 + x2 ? 40 4x1 +6x2 ? 60 x1 , x2 ? 0 What is the optimal solution?

The minimization of cost or maximization of profit is the

In a linear programming problem, the binding constraints for the optimal solution are: 5x1 + 3x2 ≤ 30 2x1 + 5x2 ≤ 20 Which of these objective functions will lead to the same optimal solution?

A linear programming model consists of

In a linear programming problem, the binding constraints for the optimal solution are: 5x1 + 3x2 ≤ 30 2x1 + 5x2 ≤ 20 As long as the slope of the objective function stays between ________ and ________, the current optimal solution point will remain optimal.

Consider the following linear programming problem: Max Z = $15x + $20y Subject to: 8x + 5y ? 40 0.4x + y ? 4 x, y ? 0 Determine the values for x and y that will maximize revenue. Given this optimal revenue, what is the amount of slack associated with the first constraint?

Consider the following linear programming problem: Max Z = 3x1 + 3x2 Subject to: 10x1 + 4x2 ? 60 25x1 + 50x2 ? 200 x1, x2 ? 0 Find the optimal profit and the values of x1 and x2 at the optimal solution.

Given this set of constraints, for what objective function is the point x=5, y=3 in the feasible region? s.t 3x + 6y ≤ 30 10x + 10y ≤ 60 10x + 15y ≤ 90

Consider the following linear programming problem: Max Z = $3x + $9y Subject to: 20x + 32y ? 1600 4x + 2y ? 240 y ? 40 x, y ? 0 Solve for the quantities of x and y which will maximize Z. What is the value of the slack variable associated with constraint 2?

Consider the following linear programming problem: MIN Z = 3x1 + 2x2 Subject to: 2x1 + 3x2 ? 12 5x1 + 8x2 ? 37 x1, x2 ? 0 What is minimum cost and the value of x1 and x2 at the optimal solution?

Which of the following could not be a linear programming problem constraint?

Consider the following linear programming problem: MIN Z = 10x1 + 20x2 Subject to: x1 + x2 ? 12 2x1 + 5x2 ? 40 x2 ? 13 x1, x2 ? 0 At the optimal solution, what is the value of surplus associated with constraint 1 and constraint 3, respectively?

Consider the following linear programming problem: MIN Z = 10x1 + 20x2 Subject to: x1 + x2 ? 12 2x1 + 5x2 ? 40 x2 ? 13 x1, x2 ? 0 What is minimum cost and the value of x1 and x2 at the optimal solution?

Consider the following linear programming problem: Max Z = 5x1 + 3x2 Subject to: 6x1 + 2x2 ? 18 15x1 + 20x2 ? 60 x1 , x2 ? 0 Find the optimal profit and the values of x1 and x2 at the optimal solution.

Decision variables

In a linear programming problem, the binding constraints for the optimal solution are: 5x1 + 3x2 ≤ 30 2x1 + 5x2 ≤ 20 As long as the slope of the objective function stays between and , the current optimal solution point will remain optimal.