The Minimum Value Of The Objective Function $Z = 3x + 5y$ Subject To:${ \begin{cases} x + 3y \geq 3 \ x + Y \geq 2 \ x \geq 0, Y \geq 0 \end{cases} }$A. 5 B. 6 C. 7 D. 8
Introduction
Linear programming is a method to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. In this article, we will explore the minimum value of the objective function Z = 3x + 5y subject to certain constraints. The constraints are given as:
{ \begin{cases} x + 3y \geq 3 \\ x + y \geq 2 \\ x \geq 0, y \geq 0 \end{cases} \}
Understanding the Problem
To find the minimum value of the objective function Z = 3x + 5y, we need to understand the constraints given. The constraints are:
- x + 3y ≥ 3
- x + y ≥ 2
- x ≥ 0, y ≥ 0
These constraints represent the feasible region, which is the set of all possible solutions that satisfy the constraints. Our goal is to find the minimum value of the objective function within this feasible region.
Graphical Method
One way to solve this problem is to use the graphical method. We can plot the constraints on a graph and find the feasible region. The feasible region is the area where all the constraints are satisfied.
Plotting the Constraints
Let's plot the constraints on a graph.
- The first constraint is x + 3y ≥ 3. We can plot this constraint by drawing a line with equation x + 3y = 3 and shading the area above the line.
- The second constraint is x + y ≥ 2. We can plot this constraint by drawing a line with equation x + y = 2 and shading the area above the line.
- The third constraint is x ≥ 0 and y ≥ 0. We can plot this constraint by drawing the x-axis and y-axis and shading the area above both axes.
Finding the Feasible Region
The feasible region is the area where all the constraints are satisfied. We can find the feasible region by finding the intersection of the shaded areas.
Finding the Minimum Value
To find the minimum value of the objective function, we need to find the point in the feasible region that minimizes the objective function Z = 3x + 5y.
Corner Points
The minimum value of the objective function can occur at one of the corner points of the feasible region. The corner points are the points where the constraints intersect.
Let's find the corner points of the feasible region.
- The first constraint intersects the x-axis at (3, 0).
- The second constraint intersects the x-axis at (2, 0).
- The first constraint intersects the y-axis at (0, 1).
- The second constraint intersects the y-axis at (0, 2).
Evaluating the Corner Points
We need to evaluate the objective function at each of the corner points to find the minimum value.
- At (3, 0), the objective function is Z = 3(3) + 5(0) = 9.
- At (2, 0), the objective function is Z = 3(2) + 5(0) = 6.
- At (0, 1), the objective function is Z = 3(0) + 5(1) = 5.
- At (0, 2), the objective function is Z = 3(0) + 5(2) = 10.
Conclusion
The minimum value of the objective function Z = 3x + 5y subject to the given constraints is 6. This occurs at the point (2, 0).
Answer
The correct answer is B. 6.
Discussion
This problem is a classic example of a linear programming problem. The objective function is a linear function, and the constraints are linear inequalities. The feasible region is the area where all the constraints are satisfied, and the minimum value of the objective function occurs at one of the corner points of the feasible region.
Related Topics
- Linear Programming
- Graphical Method
- Corner Points
- Feasible Region
References
- [1] Chvatal, V. (1983). Linear Programming. W.H. Freeman and Company.
- [2] Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press.
- [3] Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. McGraw-Hill Education.
Introduction
In our previous article, we explored the minimum value of the objective function Z = 3x + 5y subject to certain constraints. We used the graphical method to find the feasible region and the corner points, and we evaluated the objective function at each of the corner points to find the minimum value. In this article, we will answer some frequently asked questions related to the minimum value of the objective function.
Q&A
Q: What is the minimum value of the objective function Z = 3x + 5y subject to the given constraints?
A: The minimum value of the objective function Z = 3x + 5y subject to the given constraints is 6. This occurs at the point (2, 0).
Q: How do you find the feasible region?
A: To find the feasible region, we need to plot the constraints on a graph and find the area where all the constraints are satisfied.
Q: What are the corner points of the feasible region?
A: The corner points of the feasible region are the points where the constraints intersect. In this case, the corner points are (3, 0), (2, 0), (0, 1), and (0, 2).
Q: How do you evaluate the objective function at each of the corner points?
A: To evaluate the objective function at each of the corner points, we need to substitute the values of x and y into the objective function and calculate the result.
Q: What is the significance of the corner points in linear programming?
A: The corner points are the points where the constraints intersect, and they represent the possible solutions to the linear programming problem. The minimum value of the objective function occurs at one of the corner points.
Q: Can you explain the graphical method in linear programming?
A: The graphical method is a technique used to solve linear programming problems by plotting the constraints on a graph and finding the feasible region. It is a visual method that helps to understand the problem and find the solution.
Q: What are the advantages of the graphical method in linear programming?
A: The advantages of the graphical method are that it is a visual method, it is easy to understand, and it helps to find the feasible region and the corner points.
Q: What are the disadvantages of the graphical method in linear programming?
A: The disadvantages of the graphical method are that it is limited to two variables, it is not suitable for large-scale problems, and it requires a good understanding of graph theory.
Conclusion
In this article, we answered some frequently asked questions related to the minimum value of the objective function Z = 3x + 5y subject to certain constraints. We explained the graphical method, the corner points, and the significance of the corner points in linear programming. We also discussed the advantages and disadvantages of the graphical method.
Related Topics
- Linear Programming
- Graphical Method
- Corner Points
- Feasible Region
References
- [1] Chvatal, V. (1983). Linear Programming. W.H. Freeman and Company.
- [2] Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press.
- [3] Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. McGraw-Hill Education.
Frequently Asked Questions
- What is the minimum value of the objective function Z = 3x + 5y subject to the given constraints?
- How do you find the feasible region?
- What are the corner points of the feasible region?
- How do you evaluate the objective function at each of the corner points?
- What is the significance of the corner points in linear programming?
- Can you explain the graphical method in linear programming?
- What are the advantages of the graphical method in linear programming?
- What are the disadvantages of the graphical method in linear programming?
Glossary
- Linear Programming: A method to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.
- Graphical Method: A technique used to solve linear programming problems by plotting the constraints on a graph and finding the feasible region.
- Corner Points: The points where the constraints intersect, and they represent the possible solutions to the linear programming problem.
- Feasible Region: The area where all the constraints are satisfied, and it represents the possible solutions to the linear programming problem.