Question

Solve the following lp problem using the graphical method. max x1 x2 subject to -x1 2x2 <=10 x1 2x2 <=12 2x1 x2 <=16 x1 =>0, x2=>0?

Answer 1

The optimal solution for the given linear programming problem is x1 = 4, x2 = 4, with a maximum objective function value of 8.

Step-by-step explanation:

In order to find the optimal solution using the graphical method, we first graph the feasible region determined by the system of inequalities. The inequalities are:

1.
`\( -x_1 + 2x_2 \leq 10 \)`

2.
`\( x_1 + 2x_2 \leq 12 \)`

3.
`\( 2x_1 + x_2 \leq 16 \)`

4.
`\( x_1 \geq 0 \)`

5.
`\( x_2 \geq 0 \)`

Graphing these inequalities on a coordinate plane, the feasible region is the overlapping shaded area. The corner points of this region are the potential optimal solutions. Calculating the objective function
`\( z = x_1 + x_2 \)` at each corner point, we find that the maximum value occurs at
`\( x_1 = 4, x_2 = 4 \)`, resulting in a maximum objective function value of 8.

In the given linear programming problem, we sought to maximize the objective function
`\( z = x_1 + x_2 \)` subject to certain constraints. The graphical method allowed us to visually identify the feasible region and determine the optimal solution. The solution x1 = 4, x2 = 4 yields the maximum value for the objective function within the specified constraints. This method provides a geometric insight into the problem, helping to understand the interplay between the objective function and the constraints in a two-dimensional space.

In summary, the graphical method efficiently solved the linear programming problem by visually identifying the optimal solution within the feasible region. The coordinates x1 = 4, x2 = 4 represent the values that maximize the objective function, meeting all the given constraints.