Raj was given a linear programming problem to solve in an interview. The constraints of the problems are x ≥ 0, y ≥ 0, 3x+5y ≤ 15, 5x+2y ≤ 10. The interviewer told him one of the four points given below is the optimal solution for the problem and theorem 1 in linear programming would help him to find the answer. Which of the following points is the optimal solution?
Raj was given a linear programming problem to solve in an interview. The constraints of the problems are x ≥ 0, y ≥ 0, 3x+5y ≤ 15, 5x+2y ≤ 10. The interviewer told him one of the four points given below is the optimal solution for the problem and theorem 1 in linear programming would help him to find the answer. Which of the following points is the optimal solution?
The correct option is D (2019,4519)
To find the optimal solution, we need the objective function. We can find the feasible region without knowing the objective function, because it is determined by the constraints alone.
If we plot the common region for the constraints x ≥ 0, y ≥ 0, 3x+5y ≤ 15, 5x+2y ≤ 10, we get the following shaded region.
How does theorem 1 help in finding optimal solution when we don’t even know the objective function? Let’s see what theorem 1 says.
Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
It means for the given problem, O or A or B or C is the optimal solution, since they are the corner points(figure).
Only C (2019,4519) is given in the option and the interviewer told one of the options is the optimal solution. So the answer is (2019,4519)
(2019,4519)
To find the optimal solution, we need the objective function. We can find the feasible region without knowing the objective function, because it is determined by the constraints alone.
If we plot the common region for the constraints x ≥ 0, y ≥ 0, 3x+5y ≤ 15, 5x+2y ≤ 10, we get the following shaded region.
How does theorem 1 help in finding optimal solution when we don’t even know the objective function? Let’s see what theorem 1 says.
Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
It means for the given problem, O or A or B or C is the optimal solution, since they are the corner points(figure).
Only C (2019,4519) is given in the option and the interviewer told one of the options is the optimal solution. So the answer is (2019,4519)
