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Linear Programming

Question

Maximize z = 4x + 1y such that x + 2y ≤ 20, x + y ≤ 15, x ≥ 0, y ≥ 0.

Solution

We are to maximize
z = 4x + 7y
subject to the constraints
x + 2y ≤ 20
x + y ≤ 15
x ≥ 0, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
Now we draw the graph of line
x + 2y = 20.
For x = 0, 2y = 20 or y = 10
For y = 0, x = 20
∴ line meets OX in A (20, 0) and OY in L (0, 10).
Let us draw the graphs of line
x + y = 15.
For x = 0, y = 15
For y = 0, x = 15
∴ line meets OX in B (15, 0) and OY in M (0, 15).

Since feasible region is the region which satisfies all the constraints
∴ OBCL is the feasible region, which is bounded.
The corner points are O (0, 0), B (15, 0), C (10, 5), L (0, 10)
At O (0, 0), z = 0 + 0 = 0
At B (15, 0), z = 4 (15) + 7 (0) = 60 + 0 = 60
At C(10, 5), z = 4 (10) + 7 (5) = 40 + 35 = 75
At L (0, 10), z = 4 (0) + 7 (10) = 0 + 70 = 70
∴ maximum value = 75 at the point (10, 5).

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