Solve the following Linear programming models using graphical method.
Maximize Z= 7x1+ 3x2
Subject to: 2 x1+ 6 x2 ≤ 24
6 x1 + 2 x2 ≤ 24
x1, x2 ≥ 0
Minimize C= 4 x1 + 5 x2
Subject to: 2 x1 + 7 x2 ≥ 31
5 x1 + 3 x2 ≥ 34
To solve the first linear programming model, we can graph the feasible region and find the optimal solution.
1. Graph the inequalities:
Start by graphing the lines for the equality constraints:
- 2x1 + 6x2 = 24 (or x2 = 4 - (1/3)x1)
- 6x1 + 2x2 = 24 (or x2 = 12 - 3x1)
Plot these lines on a graph.
2. Determine the feasible region:
The feasible region is the region of the graph that satisfies all the constraints. Shade the region that is below both lines since the constraints are inequalities (≤).
3. Determine the corner points:
The corner points of the feasible region will be the points where the lines intersect or where they touch the axes.
4. Evaluate the objective function:
For each corner point, evaluate the objective function Z = 7x1 + 3x2.
5. Determine the optimal solution:
Choose the corner point that gives the maximum value for the objective function. This will be the optimal solution.
To solve the second linear programming model, follow the same steps as above but use the given constraints and the objective function C = 4x1 + 5x2 instead.