Final answer:
To maximize z=x₁+2x₂ subject to given constraints, we can use either graphical method or simplex method. Graphical method involves plotting constraints and finding the maximum value of z within the feasible region. Simplex method is an analytical technique that systematically evaluates iterations to find the optimal solution.
Step-by-step explanation:
Solution using graphical method:
To solve the problem graphically, we need to plot the feasible region and find the maximum value of z. The feasible region is the region that satisfies all the given constraints.
The constraints are:
- x₁≤80
- x₂≤60
- 5x₁+6x₂≤600
- x₁+2x₂≤160
- x₁,x₂≥0
By plotting these constraints on a graph and finding the intersection points, we can determine the feasible region.
Once the feasible region is obtained, we can find the maximum value of z=x₁+2x₂ within that region. The maximum value will occur at one of the corner points of the feasible region.
As for the type of solution, it would be the maximum value of z and the corresponding values of x₁ and x₂ at that maximum point.
Solution using simplex method:
The simplex method is an analytical technique used to solve linear programming problems. It involves creating a tableau and performing iterations to find the optimal solution.
The optimal solution will be the maximum value of z and the corresponding values of x₁ and x₂ at that maximum point.
The analytical technique is potentially more accurate than the graphical technique because it considers all possible combinations of variables and constraints. It systematically evaluates each iteration to find the optimal solution.