Consider the linear programming problem
Maximize P=60x+50y.
x+y≤80
5x+10y≤560
50x+20y≤1600
x≥0
y≥0
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3220 at (12,50)
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maximize | function | 60 x + 50 y\ndomain | x + y<=80 && 5 x + 10 y<=560 && 50 x + 20 y<=1600 && x>=0 && y>=0
Global maximum:
max{60 x + 50 y|x + y<=80 && 5 x + 10 y<=560 && 50 x + 20 y<=1600 && x>=0 && y>=0} = 3220 at (x, y) = (12, 50)
therefore at the intersection of
5x+10y≤560
and
50x+20y≤1600
If you want to do it manually, on graph paper, plot these lines:
x+y=80
5x+10y=560
50x+20y=1600
x=0 ; y=0
The acceptable solution region will be on the border or in the enclosed area. But there is a nice theorem in Math that tells us the maximum will lie on the border, at one of the intersecting border lines. So test the corners with P=60x+50y, and you will find the maximum in a jiffy.
To solve this linear programming problem, we can use the graphical method. Here's how we can find the solution:
Step 1: Graph the constraints:
We start by graphing each constraint using the inequalities given.
The first constraint: x + y ≤ 80
To graph this inequality, rewrite it as y ≤ -x + 80. The graph of this constraint will be a straight line with a slope of -1, passing through the point (80, 0). Shade the region below this line.
The second constraint: 5x + 10y ≤ 560
Rewrite it as y ≤ -0.5x + 56. This inequality represents another straight line with a slope of -0.5, passing through the point (112, 0). Shade the region below this line.
The third constraint: 50x + 20y ≤ 1600
Rewrite it as y ≤ -2.5x + 80. This inequality represents a straight line with a slope of -2.5, passing through the point (64, 0). Shade the region below this line.
Lastly, verify that x ≥ 0 and y ≥ 0. This will determine the quadrant that satisfies the constraints.
Step 2: Identify the feasible region:
The feasible region is the area where all the shaded regions overlap. In this case, it will be a triangular region.
Step 3: Identify the objective function:
The objective function is the function to be maximized or minimized. In this case, the objective function is P = 60x + 50y.
Step 4: Identify the corner points of the feasible region:
The corners of the feasible region represent the values of x and y that maximize or minimize the objective function. To find these points, determine the coordinates of the intersections of the lines representing the constraints.
Step 5: Evaluate the objective function at each corner point:
For each corner point, substitute the values of x and y into the objective function and calculate the value of P.
Step 6: Select the optimal solution:
From the values obtained for P at each corner point, choose the largest value. This will be the maximum value of the objective function, P, and it corresponds to the optimal solution.
Remember that in this problem, x and y are both non-negative (x ≥ 0 and y ≥ 0).
I hope this helps you understand the process of solving a linear programming problem.