Solve the following LPP by graphical method:

Minimize Z = 20x + 10y

Subject to

• x + 2y ≤ 40
• x + y ≥ 30
• 4x + 3y ≥ 60
• x, y ≥ 0

To solve the linear programming problem (LPP) using the graphical method, we need to follow several steps. The goal is to minimize the objective function Z = 20x + 10y while satisfying the given constraints.

Step 1: Identify the Constraints

The constraints provided are:

• x + 2y ≤ 40
• x + y ≥ 30
• 4x + 3y ≥ 60
• x, y ≥ 0

Step 2: Graph the Constraints

To graph these inequalities, we first convert them into equations:

• For x + 2y = 40, rearranging gives y = 20 - 0.5x.
• For x + y = 30, rearranging gives y = 30 - x.
• For 4x + 3y = 60, rearranging gives y = 20 - (4/3)x.

Next, plot these lines on a graph. The feasible region is where all the constraints overlap, considering only the first quadrant (x, y ≥ 0).

Step 3: Determine the Feasible Region

After plotting the lines, shade the areas that satisfy each inequality. The feasible region will be a polygon formed by the intersection of these shaded areas.

Step 4: Identify Corner Points

Find the corner points of the feasible region. These points are where the lines intersect. Calculate the coordinates of these points by solving the equations of the lines:

• Intersection of x + 2y = 40 and x + y = 30
• Intersection of x + 2y = 40 and 4x + 3y = 60
• Intersection of x + y = 30 and 4x + 3y = 60

Step 5: Evaluate the Objective Function

Calculate Z = 20x + 10y for each corner point. The point that gives the lowest value of Z will be the optimal solution.

Step 6: Conclusion

After evaluating the objective function at all corner points, the point with the minimum Z value is your solution. This point will provide the values of x and y that minimize the objective function while satisfying all constraints.