Solve the following LP problem graphically
MAX 2X + 7Y
Subject to:
5X + 9Y >= 90
9X + 8Y <= 144
Y <= 8
X, Y >= 0
Use f(x) = 1
2
x and f -1(x) = 2x to solve the problems.
f(2) =
f−1(1) =
f−1(f(2)) =
see more recent post.
Use f(x) = 1
2
x and f -1(x) = 2x to solve the problems.
To solve the given Linear Programming (LP) problem graphically, follow these steps:
Step 1: Convert the inequalities into equations:
Rewrite the given inequalities as equations by adding slack variables (S_1, S_2, S_3) for the inequalities (1), (2), and (3) respectively:
1) 5X + 9Y + S_1 = 90
2) 9X + 8Y - S_2 = 144
3) -Y + S_3 = 8
Step 2: Plot the feasible region:
To determine the feasible region, plot the equations on a graph. For each equation, assign arbitrary values to X and solve for Y to obtain the line. Plot the lines for the three equations and shade the feasible region, considering the constraints.
For equation (1):
When X = 0, 9Y = 90, Y = 10
When Y = 0, 5X = 90, X = 18
Plotting these two points and drawing a straight line passing through them, we get a line labeled as L_1.
For equation (2):
When X = 0, 8Y = 144, Y = 18
When Y = 0, 9X = 144, X = 16
Plotting these two points and drawing a straight line passing through them, we get a line labeled as L_2.
For equation (3):
When Y = 0, S_3 = 8, X can have any value
Plotting a horizontal line labeled as L_3 at Y = 0, located at 8 units on the Y-axis.
Shade the region where all the lines intersect, as this is the feasible region that satisfies all the given constraints.
Step 3: Identify the corner points of the feasible region:
The corner points of the feasible region are the intersections of the lines. These points are potential solutions to the LP problem.
Step 4: Calculate the objective function value:
Evaluate the objective function (2X + 7Y) at each corner point to find the maximum value.
For example, at the corner point:
X = 16, Y = 0
Substituting the values, the objective function value = 2(16) + 7(0) = 32
Similarly, calculate the objective function value at the other corner points.
Step 5: Identify the optimal solution:
The optimal solution is the corner point that results in the maximum objective function value. Compare the objective function values calculated at each corner point and determine the optimal solution.
In this case, the solution with the maximum objective function value is the optimal solution.
Note: The LP problem may have multiple optimal solutions or may be unbounded. However, for this specific problem, we are assuming a bounded and unique optimal solution.