Linear programming

Objective. Optimise objectives over feasible regions.

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Learn: the key idea

Linear programming finds the optimal value of a linear objective function subject to linear inequality constraints, by graphing the feasible region defined by the constraints and evaluating the objective at each vertex of this region, since the optimum of a linear function over a convex region always occurs at a vertex. The method involves plotting all constraint boundaries, shading the feasible region satisfying every inequality simultaneously, identifying its vertices, and substituting each into the objective function to find the maximum or minimum. For example, maximising profit P=3x+2y subject to resource constraints requires checking P at each corner of the feasible polygon.

Worked example

A feasible region for a linear programming problem has vertices (0,0), (4,0), (3,2) and (0,3). Find the maximum value of the objective function P = 2x + 3y over this region.

1 $Evaluate P at each vertex: at (0,0), P=0$
2 $At (4,0), P = 2(4)+3(0) = 8$
3 $At (3,2), P = 2(3)+3(2) = 6+6 = 12$
4 $At (0,3), P = 2(0)+3(3) = 9; the maximum of all these values is 12$

Answer: Maximum P = 12 at (3,2)

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Common mistake. Students check the objective function only at the intersection of all constraint lines rather than at every vertex of the actual feasible region, missing the true optimal corner. Always identify every vertex of the bounded feasible region before comparing objective values.

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