To maximize the objective function P = 4x + 20y, subject to the given constraints, we can use the method of linear programming.
Step 1: Convert the inequalities to equations and introduce slack variables:
2x + 15y + u1 = 700
5x + 10y + u2 = 1150
5x + 5y + u3 = 1000
4x + 15y + u4 = 980
Step 2: Write the objective function in terms of the variables and slack variables:
P = 4x + 20y
Step 3: Create a table to solve the system of equations using the Simplex method:
| x | y | u1 | u2 | u3 | u4 | RHS |
-----------------------------------------------------------
Eq1 | 2 | 15 | 1 | 0 | 0 | 0 | 700 |
Eq2 | 5 | 10 | 0 | 1 | 0 | 0 | 1150 |
Eq3 | 5 | 5 | 0 | 0 | 1 | 0 | 1000 |
Eq4 | 4 | 15 | 0 | 0 | 0 | 1 | 980 |
Obj | -4 | -20 | 0 | 0 | 0 | 0 | 0 |
Step 4: Apply the Simplex method to find the optimal solution:
Perform row operations to make the objective function row the pivot row. Divide the pivot row by the pivot element (in this case, -4).
| x | y | u1 | u2 | u3 | u4 | RHS |
-----------------------------------------------------------
Obj | 1 | 5 | 0 | 0 | 0 | 0 | 0 |
Eq2 | 5/4 | 9/4 | 0 | 1 | 0 | 0 | 1150 |
Eq3 | 5/4 | 1/4 | 0 | 0 | 1 | 0 | 1000 |
Eq4 | 1 | 3 | 0 | 0 | 0 | 1 | 980 |
Eq1 | 1/2 | 15/4| 1 | 0 | 0 | 0 | 700 |
Perform row operations to make all other elements in the pivot column zero:
| x | y | u1 | u2 | u3 | u4 | RHS |
-----------------------------------------------------------
Obj | 0 | 17/4 | 0 | 0 | 0 | 0 | 2125 |
Eq2 | 0 | 1 | 0 | 3/4 | 0 | 0 | 230 |
Eq3 | 0 |-3/4 | 0 | -1/4| 1 | 0 | 800 |
Eq4 | 0 | 18 | 0 | -3 | 0 | 1 | 180 |
Eq1 | 1 | 15/8 | 2 | 15/8 | 0 | 0 | 650 |
Step 5: Repeat Steps 4 until the objective function coefficients are all nonnegative:
| x | y | u1 | u2 | u3 | u4 | RHS |
--------------------------------------------------------------
Obj | 0 | 0 | 3/2 | 0 | 0 | 875/3 | 6375/2 |
Eq2 | 0 | 0 | -1 | 0 | 0 | 230 | 2740 |
Eq3 | 0 | 0 | -1/2| 0 | 1 |-375/2 | 1250 |
Eq4 | 1 | 0 | 1/6 | 0 | 0 | 60 | 280 |
Eq1 | 0 | 1 |-1/16 | 1 | 0 |-325/8 | 75 |
Step 6: The maximum value for P = 2125 and occurs when x = 280 and y = 75.
Step 7: The values for the slack variables when the optimal solution is reached are:
u1 = 325/8
u2 = 0
u3 = 0
u4 = 0