Determine Graphically The Minimum Value Of The Following Objective Function:$\begin{aligned} z = & \, 500x + 400y \\ \text{subject To Constraints:} \\ & \rightarrow X + Y \leq 200 \\ & \sim X \geq 20 \\ & \Rightarrow Y \geq 4x \\ & \rightarrow Y

Introduction

In linear programming, the objective function is a mathematical expression that needs to be optimized, either maximized or minimized, subject to certain constraints. The constraints are usually represented by linear inequalities, which limit the possible values of the variables in the objective function. In this article, we will determine the minimum value of the objective function graphically, using the given constraints.

The Objective Function

The objective function is given by:

$$z = 500x + 400y$$

This is a linear function of two variables, x and y. The coefficients of x and y are 500 and 400, respectively. The objective function represents the total cost or profit, depending on the context, that we want to minimize or maximize.

Constraints

The constraints are given by:

1. $x + y \leq 200$
2. $x \geq 20$
3. $y \geq 4x$
4. $y \leq 200$

These constraints limit the possible values of x and y. The first constraint represents a budget constraint, where the sum of x and y cannot exceed 200. The second and third constraints represent lower bounds on x and y, respectively. The fourth constraint is similar to the first one, but it represents an upper bound on y.

Graphical Representation

To determine the minimum value of the objective function graphically, we need to plot the constraints on a coordinate plane. The x-axis represents the variable x, and the y-axis represents the variable y.

Plotting the Constraints

1. Constraint 1: $x + y \leq 200$

This constraint can be plotted as a line with a slope of -1 and a y-intercept of 200. The line is dashed, indicating that it is an inequality.

2. Constraint 2: $x \geq 20$

This constraint can be plotted as a vertical line at x = 20. The line is solid, indicating that it is an equality.

3. Constraint 3: $y \geq 4x$

This constraint can be plotted as a line with a slope of 4 and a y-intercept of 0. The line is dashed, indicating that it is an inequality.

4. Constraint 4: $y \leq 200$

This constraint can be plotted as a horizontal line at y = 200. The line is dashed, indicating that it is an inequality.

Finding the Feasible Region

The feasible region is the area where all the constraints are satisfied. To find the feasible region, we need to plot the constraints and identify the area where they intersect.

The feasible region is a polygon with vertices at (20, 160), (40, 80), (60, 40), and (20, 200).

Determining the Minimum Value

To determine the minimum value of the objective function, we need to find the point in the feasible region that minimizes the objective function.

The minimum value of the objective function occurs at the vertex (40, 80). This is because the objective function is a linear function, and the minimum value occurs at the point where the function is tangent to the feasible region.

Calculating the Minimum Value

To calculate the minimum value of the objective function, we need to substitute the values of x and y at the vertex (40, 80) into the objective function.

$$z = 500(40) + 400(80)$$

$$z = 20000 + 32000$$

$$z = 52000$$

Therefore, the minimum value of the objective function is 52000.

Conclusion

In this article, we determined the minimum value of the objective function graphically, using the given constraints. We plotted the constraints on a coordinate plane, identified the feasible region, and determined the minimum value of the objective function at the vertex (40, 80). The minimum value of the objective function is 52000.

References

• [1] Chvatal, V. (1983). Linear Programming. W.H. Freeman and Company.
• [2] Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. McGraw-Hill Education.
• [3] Winston, W. L. (2018). Operations Research: Applications and Algorithms. Cengage Learning.
  

Introduction

In our previous article, we determined the minimum value of an objective function graphically, using the given constraints. In this article, we will answer some frequently asked questions (FAQs) related to determining the minimum value of an objective function graphically.

Q1: What is the objective function?

A1: The objective function is a mathematical expression that needs to be optimized, either maximized or minimized, subject to certain constraints. In this article, the objective function is given by:

$$z = 500x + 400y$$

Q2: What are the constraints?

A2: The constraints are given by:

1. $x + y \leq 200$
2. $x \geq 20$
3. $y \geq 4x$
4. $y \leq 200$

These constraints limit the possible values of x and y.

Q3: How do I plot the constraints on a coordinate plane?

A3: To plot the constraints on a coordinate plane, you need to identify the type of constraint and plot it accordingly. For example, the first constraint is a linear inequality, so you can plot it as a line with a slope of -1 and a y-intercept of 200. The line is dashed, indicating that it is an inequality.

Q4: How do I find the feasible region?

A4: To find the feasible region, you need to plot the constraints and identify the area where they intersect. The feasible region is the area where all the constraints are satisfied.

Q5: How do I determine the minimum value of the objective function?

A5: To determine the minimum value of the objective function, you need to find the point in the feasible region that minimizes the objective function. In this article, the minimum value of the objective function occurs at the vertex (40, 80).

Q6: How do I calculate the minimum value of the objective function?

A6: To calculate the minimum value of the objective function, you need to substitute the values of x and y at the vertex (40, 80) into the objective function.

$$z = 500(40) + 400(80)$$

$$z = 20000 + 32000$$

$$z = 52000$$

Q7: What is the significance of the minimum value of the objective function?

A7: The minimum value of the objective function represents the minimum cost or profit that can be achieved, subject to the given constraints.

Q8: Can I use this method to determine the maximum value of the objective function?

A8: Yes, you can use this method to determine the maximum value of the objective function. However, you need to change the direction of the inequality signs in the constraints.

Q9: Can I use this method to determine the minimum value of a non-linear objective function?

A9: No, you cannot use this method to determine the minimum value of a non-linear objective function. This method is only applicable to linear objective functions.

Q10: What are some common applications of determining the minimum value of an objective function?

A10: Some common applications of determining the minimum value of an objective function include:

• Resource allocation: Determining the minimum cost of allocating resources to different projects or activities.
• Production planning: Determining the minimum cost of producing a certain quantity of goods or services.
• Inventory management: Determining the minimum cost of holding inventory and meeting customer demand.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to determining the minimum value of an objective function graphically. We hope that this article has provided you with a better understanding of the method and its applications.