Maximize P = 16x + 80y subject to these constraints:
2x + 20y ≤ 430
4x + 70y ≤ 1400
8x + 30y ≤ 980
10x + 10y ≤ 1000
4x + 30y ≤ 700
x ≥ 0, y ≥ 0
Maximum value for P = ?.
This value of P occurs when x = ? and y = ?
If u1, u2, u3, u4, and u5 represent the slack variables in the five counstraints (u1 the slack variable in the top constraint listed above, u2 for the second, etc.), then give the values for each of the four slack variables when the optimal solution is reached:
u1=?
u2=?
u3=?
u4=?
u5=?
To maximize P=16x+80y, subject to the given constraints, we can use the graphical method or the simplex method. Here, we'll use the simplex method.
Step 1: Convert the inequality constraints into equality constraints by introducing slack variables.
2x + 20y + u1 = 430
4x + 70y + u2 = 1400
8x + 30y + u3 = 980
10x + 10y + u4 = 1000
4x + 30y + u5 = 700
Step 2: Write the objective function in standard form.
P = 16x + 80y -> -16x - 80y + P = 0
Step 3: Setup the initial simplex tableau.
Coefficients:
Equation | x | y | u1 | u2 | u3 | u4 | u5 | Constants
----------------------------------------------------
-16 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
-80 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
2 | 2 | 20| 1 | 0 | 0 | 0 | 0 | 430
4 | 4 | 70| 0 | 1 | 0 | 0 | 0 | 1400
8 | 8 | 30| 0 | 0 | 1 | 0 | 0 | 980
10 |10 | 10| 0 | 0 | 0 | 1 | 0 | 1000
4 | 4 | 30| 0 | 0 | 0 | 0 | 1 | 700
Step 4: Implement the simplex algorithm to find the optimal solution.
a) Find the most negative value in the bottom row.
In this case, it is -80.
b) Select the most negative value and identify the column index.
The column index is 2, which corresponds to variable y.
c) Divide the constants column by the pivot column to determine the row for the pivot element.
We find the smallest non-negative value by dividing 430/20, 1400/70, 980/30, and 700/30.
The minimum ratio is 430/20 = 21.5, so the pivot element is 20.
d) Use row operations to make the pivot element 1 and all other elements in the pivot column 0.
We divide the equation 2 by 20, which gives us:
Equation 3: (1/10)x + y/10 + (1/20)u1 = 43/20
Row operations:
Equation | x | y | u1 | u2 | u3 | u4 | u5 | Constants
-----------------------------------------------------------
-16 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
-80 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
21.5 | 0 | 1 | 1/20 | 0 | 0 | 0 | 1 | 430/20
4 | 4 | 30| 0 | 1 | 0 | 0 | 0 | 1400
8 | 8 | 30| 0 | 0 | 1 | 0 | 0 | 980
10 |10 | 10| 0 | 0 | 0 | 1 | 0 | 1000
4 | 4 | 30| 0 | 0 | 0 | 0 | 1 | 700
e) Repeat steps a-d until there are no negative values in the bottom row or all negative values in the bottom row are greater than or equal to zero.
We perform row operations:
Row 3 - 0.1 * Row 2 -> Row 3
Equation | x | y | u1 | u2 | u3 | u4 | u5 | Constants
-----------------------------------------------------------
-16 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
-80 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
0 | 0 | 1 | 1/20| 0 | 0 | 0 | 1 | 430/20 - (21.5/20)*0
4 | 4 | 30| 0 | 1 | 0 | 0 | 0 | 1400
8 | 8 | 30| 0 | 0 | 1 | 0 | 0 | 980
10 |10 | 10| 0 | 0 | 0 | 1 | 0 | 1000
4 | 4 | 30| 0 | 0 | 0 | 0 | 1 | 700
No more negative values in the bottom row. The solution is optimal.
Step 5: Read the optimal solution from the constants column.
The optimal value for P (maximum value) = 0.
This means the objective function P = 16x + 80y is maximized at P = 0. The values of x and y are not determined as all variables have coefficients of 0 in the optimal row.
To maximize P = 16x + 80y subject to the given constraints, we can use the method of linear programming. Here's how you can solve this problem step-by-step:
Step 1: Write down the objective function and constraints:
Objective function: P = 16x + 80y
Constraints:
2x + 20y ≤ 430
4x + 70y ≤ 1400
8x + 30y ≤ 980
10x + 10y ≤ 1000
4x + 30y ≤ 700
x ≥ 0, y ≥ 0
Step 2: Graph the feasible region:
To graph the feasible region, we first convert each constraint into an equation and then plot the lines or inequalities on a coordinate plane. The feasible region is the intersection of all the shaded areas.
For example, the first constraint 2x + 20y ≤ 430 can be converted into the equation 2x + 20y = 430. Similarly, do this for the remaining constraints.
Step 3: Identify the feasible region:
The feasible region represents the area where all the constraints are satisfied simultaneously. Shade the feasible region on the graph.
Step 4: Find the corner points of the feasible region:
The corner points are the vertices of the feasible region. To find them, locate the intersection points of the lines forming the boundaries of the feasible region.
Step 5: Evaluate the objective function at each corner point:
Substitute the x and y values of each corner point into the objective function P = 16x + 80y and calculate the value of P.
Step 6: Determine the maximum value of P and the corresponding x and y values:
Check the value of P at each corner point and identify the largest value. The corresponding x and y values will give the values for x and y when the maximum value of P is reached.
Step 7: Solve for the slack variables:
To find the slack variables, subtract each constraint equation from its corresponding equation for the objective function. The slack variables will help determine the unused portion of each constraint.
Using this step-by-step method, find the maximum value of P, as well as the corresponding x and y values, and the slack variables u1, u2, u3, u4, and u5.