minimize p= 15x +18y
subject to x+2y<=20
3x + 2y>= 36
x>=0
y>=0
To minimize the objective function p = 15x + 18y, we need to find the values of x and y that satisfy the given constraints.
1. The first constraint is x + 2y ≤ 20. This represents the inequality for the maximum value of the left-hand side to be 20.
2. The second constraint is 3x + 2y ≥ 36. This represents the inequality for the minimum value of the left-hand side to be 36.
3. The third constraint is x ≥ 0, which means x must be greater than or equal to zero.
4. The fourth constraint is y ≥ 0, which means y must be greater than or equal to zero.
To solve this problem, we will use the graphical method:
Step 1: Convert the inequality equations into equations of lines.
For x + 2y = 20:
By rearranging the equation, we get y = (20 - x)/2.
For 3x + 2y = 36:
By rearranging the equation, we get y = (36 - 3x)/2.
Step 2: Plot the lines on a graph.
Draw a graph with the x-axis and y-axis intersecting at the origin (0, 0). Plot the line x + 2y = 20 and the line 3x + 2y = 36. Extend both lines in both directions.
Step 3: Shade the feasible region.
Shade the area that satisfies all the constraints: x + 2y ≤ 20, 3x + 2y ≥ 36, x ≥ 0, and y ≥ 0. The shaded region represents the feasible region.
Step 4: Identify the corner points of the feasible region.
The corner points of the feasible region are the intersection points of the lines and the boundaries. Calculate the coordinates of these points.
Step 5: Substitute the corner points into the objective function.
Substitute the values of x and y for each corner point into the objective function p = 15x + 18y, and calculate the value of p for each point.
Step 6: Identify the point that gives the minimum value of p.
Compare the values of p obtained for each corner point and identify the point that gives the minimum value of p. This point represents the optimal solution.
And that's it! By following these steps, you can find the minimum value of p = 15x + 18y that satisfies the given constraints.
To minimize the objective function p = 15x + 18y, subject to the given constraints, we can use linear programming. Linear programming is a mathematical technique used to optimize a linear objective function given a set of linear constraints.
Step 1: Write the objective function and constraints in standard form.
The objective function:
p = 15x + 18y
Constraints:
x + 2y ≤ 20
3x + 2y ≥ 36
x ≥ 0
y ≥ 0
To convert the second constraint to the standard form, multiply both sides by -1:
-3x - 2y ≤ -36
Step 2: Graph the feasible region.
To graph the feasible region, we need to identify the region that satisfies all the constraints.
First, graph the line x + 2y = 20:
To graph this line, we can find two points that satisfy the equation. For example, we can use (0, 10) and (20, 0). Plot these points and draw a line connecting them.
Next, graph the line -3x - 2y = -36:
Similarly, find two points that satisfy the equation, such as (0, -18) and (-12, 0). Plot these points and draw a line connecting them.
Now, shade the region that satisfies all the constraints. In this case, it is the region below or on the line x + 2y = 20 and above or on the line -3x - 2y = -36. Also, since x ≥ 0 and y ≥ 0, the feasible region is confined to the first quadrant.
Step 3: Find the vertices of the feasible region.
The vertices of the feasible region are the intersection points of the lines graphed in Step 2. In this case, the feasible region is a polygon with corners at the intersection points of the lines.
Step 4: Evaluate the objective function at each vertex.
Calculate the value of the objective function p = 15x + 18y at each vertex of the feasible region. This will give you the value of the objective function at each corner point.
Step 5: Determine the minimum value.
Identify the vertex that yields the lowest value for the objective function p. This vertex corresponds to the minimum value.
In this case, follow these steps to solve the problem graphically. However, for larger systems or more complex constraints, you may need to use computer software or linear programming techniques to find the solution more efficiently.