CLASS 12 MATHS · LINEAR PROGRAMMING

Linear programming by the corner-point method

For a bounded feasible region, the maximum and minimum of Z = px + qy occur at corner points. Evaluate Z at every corner and compare.

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Worked examples

Example 1 · level 1

The feasible region of an LPP is bounded, with corner points (0, 0), (10, 0), (9, 2), (0, 5). Find the maximum value of Z = 6x + y.

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Corner points: (0, 0), (10, 0), (9, 2), (0, 5).
Z(0, 0) = 0; Z(10, 0) = 60; Z(9, 2) = 56; Z(0, 5) = 5.
Maximum Z = 60 at (10, 0).

Answer: 60

Example 2 · level 2

Rohan runs a small workshop in Mysuru making x chairs and y tables. The constraints are 2x + y ≤ 8, x + 3y ≤ 9, x ≥ 0, y ≥ 0. Find the maximum value of profit Z = x + 8y (in ₹ hundreds).

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Corner points: (0, 0), (4, 0), (3, 2), (0, 3).
Z(0, 0) = 0; Z(4, 0) = 4; Z(3, 2) = 19; Z(0, 3) = 24.
Maximum Z = 24 at (0, 3).

Answer: 24

Example 3 · level 3

Rohan runs a small workshop in Kochi making x chairs and y tables. The constraints are 2x + y ≤ 18, x + 3y ≤ 24, x ≥ 0, y ≥ 0. Find the maximum value of profit Z = 7x + 7y (in ₹ hundreds).

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Corner points: (0, 0), (9, 0), (6, 6), (0, 8).
Z(0, 0) = 0; Z(9, 0) = 63; Z(6, 6) = 84; Z(0, 8) = 56.
Maximum Z = 84 at (6, 6).

Answer: 84