. Solve The Following Linear Programming Problem Graphically:9Minimise Z=2x +ysubject To The Constraints:3x+y≥9 X+y≥7 X+2y≥8 X,y≥0(LPP)​

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Introduction

Linear Programming (LP) is a method used to optimize a linear objective function, subject to a set of linear constraints. It is a powerful tool used in various fields such as economics, engineering, and operations research. In this article, we will solve a linear programming problem graphically, using the given constraints to find the optimal solution.

Problem Statement

The given linear programming problem is:

Minimise Z = 2x + y

Subject to the constraints:

  1. 3x + y ≥ 9
  2. x + y ≥ 7
  3. x + 2y ≥ 8
  4. x, y ≥ 0

Understanding the Constraints

To solve this problem graphically, we need to understand the constraints and plot them on a coordinate plane. The constraints are:

  1. 3x + y ≥ 9
  2. x + y ≥ 7
  3. x + 2y ≥ 8
  4. x, y ≥ 0

We will plot these constraints on a coordinate plane, using the x-axis and y-axis as the variables.

Plotting the Constraints

Constraint 1: 3x + y ≥ 9

To plot this constraint, we need to find the boundary line. We can do this by setting the inequality to an equation:

3x + y = 9

Solving for y, we get:

y = -3x + 9

This is the boundary line for the first constraint. We will plot this line on the coordinate plane.

Constraint 2: x + y ≥ 7

To plot this constraint, we need to find the boundary line. We can do this by setting the inequality to an equation:

x + y = 7

Solving for y, we get:

y = -x + 7

This is the boundary line for the second constraint. We will plot this line on the coordinate plane.

Constraint 3: x + 2y ≥ 8

To plot this constraint, we need to find the boundary line. We can do this by setting the inequality to an equation:

x + 2y = 8

Solving for y, we get:

y = -x/2 + 4

This is the boundary line for the third constraint. We will plot this line on the coordinate plane.

Constraint 4: x, y ≥ 0

This constraint is a non-negativity constraint, which means that both x and y must be greater than or equal to 0.

Plotting the Constraints on the Coordinate Plane

Now that we have plotted the constraints on the coordinate plane, we can see the feasible region. The feasible region is the area where all the constraints are satisfied.

Finding the Optimal Solution

To find the optimal solution, we need to minimize the objective function Z = 2x + y. We can do this by finding the point in the feasible region that minimizes the objective function.

Using the Corner Points

The corner points of the feasible region are the points where the constraints intersect. We can use these corner points to find the optimal solution.

Corner Point 1: (0, 9)

This corner point is found by setting x = 0 in the first constraint:

3(0) + y = 9

Solving for y, we get:

y = 9

This is the first corner point.

Corner Point 2: (0, 7)

This corner point is found by setting x = 0 in the second constraint:

0 + y = 7

Solving for y, we get:

y = 7

This is the second corner point.

Corner Point 3: (0, 3.5)

This corner point is found by setting x = 0 in the third constraint:

0 + 2y = 8

Solving for y, we get:

y = 4

However, this point does not satisfy the first constraint. We need to find another point that satisfies all the constraints.

Corner Point 4: (3, 3)

This corner point is found by setting y = 3 in the first constraint:

3x + 3 = 9

Solving for x, we get:

x = 2

However, this point does not satisfy the second constraint. We need to find another point that satisfies all the constraints.

Corner Point 5: (2, 5)

This corner point is found by setting x = 2 in the second constraint:

2 + 5 = 7

Solving for y, we get:

y = 5

This is the fifth corner point.

Corner Point 6: (1, 6)

This corner point is found by setting x = 1 in the third constraint:

1 + 2(6) = 8

Solving for y, we get:

y = 6

However, this point does not satisfy the first constraint. We need to find another point that satisfies all the constraints.

Corner Point 7: (1, 4)

This corner point is found by setting y = 4 in the third constraint:

1 + 2(4) = 8

Solving for x, we get:

x = 1

This is the seventh corner point.

Corner Point 8: (4, 2)

This corner point is found by setting x = 4 in the first constraint:

3(4) + 2 = 9

Solving for y, we get:

y = 2

This is the eighth corner point.

Evaluating the Corner Points

Now that we have found the corner points, we need to evaluate them to find the optimal solution. We can do this by substituting the values of x and y into the objective function Z = 2x + y.

Corner Point 1: (0, 9)

Z = 2(0) + 9 = 9

Corner Point 2: (0, 7)

Z = 2(0) + 7 = 7

Corner Point 3: (0, 3.5)

Z = 2(0) + 3.5 = 3.5

Corner Point 4: (3, 3)

Z = 2(3) + 3 = 9

Corner Point 5: (2, 5)

Z = 2(2) + 5 = 9

Corner Point 6: (1, 6)

Z = 2(1) + 6 = 8

Corner Point 7: (1, 4)

Z = 2(1) + 4 = 6

Corner Point 8: (4, 2)

Z = 2(4) + 2 = 10

Finding the Optimal Solution

The corner points with the minimum value of Z are (1, 4) and (0, 7). However, we need to check if these points satisfy all the constraints.

Corner Point (1, 4)

This point satisfies all the constraints:

  1. 3(1) + 4 = 7 ≥ 9 (false)
  2. 1 + 4 = 5 ≥ 7 (true)
  3. 1 + 2(4) = 9 ≥ 8 (true)
  4. 1, 4 ≥ 0 (true)

However, this point does not satisfy the first constraint.

Corner Point (0, 7)

This point satisfies all the constraints:

  1. 3(0) + 7 = 7 ≥ 9 (false)
  2. 0 + 7 = 7 ≥ 7 (true)
  3. 0 + 2(7) = 14 ≥ 8 (true)
  4. 0, 7 ≥ 0 (true)

However, this point does not satisfy the first constraint.

Conclusion

The given linear programming problem is:

Minimise Z = 2x + y

Subject to the constraints:

  1. 3x + y ≥ 9
  2. x + y ≥ 7
  3. x + 2y ≥ 8
  4. x, y ≥ 0

We have plotted the constraints on the coordinate plane and found the corner points. We have evaluated the corner points to find the optimal solution. The corner points with the minimum value of Z are (1, 4) and (0, 7). However, these points do not satisfy all the constraints.

Therefore, the optimal solution to the given linear programming problem is not found among the corner points. We need to find another point that satisfies all the constraints and minimizes the objective function Z = 2x + y.

Final Answer

Introduction

Linear Programming (LP) is a method used to optimize a linear objective function, subject to a set of linear constraints. It is a powerful tool used in various fields such as economics, engineering, and operations research. In this article, we will provide a Q&A guide to help you understand how to solve linear programming problems graphically.

Q: What is a linear programming problem?

A: A linear programming problem is a mathematical problem that involves finding the optimal solution to a linear objective function, subject to a set of linear constraints.

Q: What are the key components of a linear programming problem?

A: The key components of a linear programming problem are:

  1. Objective function: This is the function that we want to minimize or maximize.
  2. Constraints: These are the limitations on the variables that we are trying to optimize.
  3. Variables: These are the values that we are trying to optimize.

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

A: To plot the constraints on a coordinate plane, you need to follow these steps:

  1. Identify the constraints: Write down the constraints in the form of inequalities.
  2. Plot the boundary lines: Plot the boundary lines for each constraint on the coordinate plane.
  3. Shade the feasible region: Shade the area where all the constraints are satisfied.

Q: How do I find the corner points of the feasible region?

A: To find the corner points of the feasible region, you need to follow these steps:

  1. Identify the intersection points: Find the points where the boundary lines intersect.
  2. Check if the points satisfy the constraints: Check if the points satisfy all the constraints.
  3. List the corner points: List the points that satisfy all the constraints.

Q: How do I evaluate the corner points?

A: To evaluate the corner points, you need to follow these steps:

  1. Substitute the values into the objective function: Substitute the values of the variables into the objective function.
  2. Calculate the value of the objective function: Calculate the value of the objective function for each corner point.
  3. Compare the values: Compare the values of the objective function for each corner point.

Q: How do I find the optimal solution?

A: To find the optimal solution, you need to follow these steps:

  1. Identify the corner points with the minimum or maximum value: Identify the corner points with the minimum or maximum value of the objective function.
  2. Check if the points satisfy the constraints: Check if the points satisfy all the constraints.
  3. Select the optimal solution: Select the point that satisfies all the constraints and has the minimum or maximum value of the objective function.

Q: What are some common mistakes to avoid when solving linear programming problems graphically?

A: Some common mistakes to avoid when solving linear programming problems graphically are:

  1. Not plotting the boundary lines correctly: Make sure to plot the boundary lines correctly and shade the feasible region.
  2. Not identifying the corner points correctly: Make sure to identify the corner points correctly and list them.
  3. Not evaluating the corner points correctly: Make sure to evaluate the corner points correctly and compare the values.
  4. Not selecting the optimal solution correctly: Make sure to select the optimal solution correctly and check if it satisfies all the constraints.

Conclusion

Solving linear programming problems graphically can be a challenging task, but with practice and patience, you can become proficient in solving these problems. Remember to follow the steps outlined in this article and avoid common mistakes to ensure that you find the optimal solution.

Final Tips

  1. Practice, practice, practice: The more you practice solving linear programming problems graphically, the more comfortable you will become with the process.
  2. Use visual aids: Use visual aids such as graphs and charts to help you understand the problem and find the optimal solution.
  3. Check your work: Always check your work to ensure that you have found the optimal solution and that it satisfies all the constraints.

By following these tips and practicing regularly, you will become proficient in solving linear programming problems graphically and be able to apply this skill to real-world problems.