Constraints{x>=0 y>=0 y<=1/5x+2 5>=y+x objective function: C=7x-3y
(2.5,2.5)
(0,2)
(0,0)
(5,0)
We can find which of these points is optimal (maximizes or minimizes the objective function) by plugging each into the objective function (C = 7x-3y) and determining which yields the highest value.
For the point (2.5,2.5),
C = 7(2.5) - 3(2.5) = 10
For the point (0,2),
C = 7(0) - 3(2) = -6
For the point (0,0),
C = 7(0) - 3(0) = 0
For the point (5,0),
C = 7(5) - 3(0) = 35
The point (5,0) yields the highest value (C = 35), therefore it is the point that optimizes the objective function.
To determine the feasibility of each point with respect to the given constraints, we can substitute the values of x and y into each constraint inequality and check if the inequality holds true.
1. For the point (2.5, 2.5):
- x = 2.5, y = 2.5
- Constraints:
i. x >= 0 (2.5 >= 0) - True
ii. y >= 0 (2.5 >= 0) - True
iii. y <= 1/5x + 2 (2.5 <= 1/5(2.5) + 2) - True
- Therefore, the point (2.5, 2.5) satisfies all the constraints.
2. For the point (0, 2):
- x = 0, y = 2
- Constraints:
i. x >= 0 (0 >= 0) - True
ii. y >= 0 (2 >= 0) - True
iii. y <= 1/5x + 2 (2 <= 1/5(0) + 2) - True
- Therefore, the point (0, 2) satisfies all the constraints.
3. For the point (0, 0):
- x = 0, y = 0
- Constraints:
i. x >= 0 (0 >= 0) - True
ii. y >= 0 (0 >= 0) - True
iii. y <= 1/5x + 2 (0 <= 1/5(0) + 2) - True
- Therefore, the point (0, 0) satisfies all the constraints.
4. For the point (5, 0):
- x = 5, y = 0
- Constraints:
i. x >= 0 (5 >= 0) - True
ii. y >= 0 (0 >= 0) - True
iii. y <= 1/5x + 2 (0 <= 1/5(5) + 2) - True
- Therefore, the point (5, 0) satisfies all the constraints.
Now, let's calculate the value of the objective function for each point.
1. For the point (2.5, 2.5):
- C = 7x - 3y = 7(2.5) - 3(2.5) = 17.5 - 7.5 = 10
- The value of the objective function at (2.5, 2.5) is 10.
2. For the point (0, 2):
- C = 7x - 3y = 7(0) - 3(2) = 0 - 6 = -6
- The value of the objective function at (0, 2) is -6.
3. For the point (0, 0):
- C = 7x - 3y = 7(0) - 3(0) = 0 - 0 = 0
- The value of the objective function at (0, 0) is 0.
4. For the point (5, 0):
- C = 7x - 3y = 7(5) - 3(0) = 35 - 0 = 35
- The value of the objective function at (5, 0) is 35.
Therefore, out of the given four points, (5, 0) has the highest value of the objective function.
To determine the feasible region and the optimal solution in this linear programming problem, we need to plot the given constraints and find the corner points of the feasible region. We will then evaluate the objective function at these corner points to find the optimal solution.
Step 1: Plotting the Constraints
Plot the four given constraints on a graph:
1. y ≤ 1/5x + 2:
For this constraint, plot the line y = 1/5x + 2. To do this, plot two points, such as (0,2) and (5,3), and draw a line passing through these points. Shade the region below this line because we need y to be less than or equal to the equation.
2. x ≥ 0:
This constraint simply states that x can be any non-negative value. Since x cannot be negative, we will only plot the region on or to the right of the y-axis.
3. y ≥ 0:
Similarly, this constraint states that y can be any non-negative value. We will only plot the region above or at the x-axis.
4. x + y ≤ 5:
This constraint means that the sum of x and y must be less than or equal to 5. Plot the line x + y = 5. To do this, plot two points, such as (0,5) and (5,0), and draw a line passing through these points. Shade the region below this line.
Step 2: Identify the Feasible Region
The feasible region is the intersection of all the shaded regions from the plotted constraints. In this case, it should be a triangle-shaped region.
Step 3: Find the Corner Points
For a linear programming problem with linear constraints, the optimal solution will always be at one of the corner points of the feasible region.
- The corner points of the feasible region are (2.5,2.5), (0,2), (0,0), and (5,0).
Step 4: Evaluate the Objective Function
Now, substitute the x and y coordinates of each corner point into the objective function C = 7x - 3y to find the corresponding values.
- For (2.5,2.5): C = 7(2.5) - 3(2.5) = 17.5 - 7.5 = 10
- For (0,2): C = 7(0) - 3(2) = -6
- For (0,0): C = 7(0) - 3(0) = 0
- For (5,0): C = 7(5) - 3(0) = 35
Step 5: Determine the Optimal Solution
Compare the values of the objective function at each corner point: 10, -6, 0, and 35. The highest value is 35 at (5,0). Therefore, the optimal solution in this linear programming problem is x = 5 and y = 0, with the objective function value C = 35.