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Let S1 be the selling price of wheat, and S2 the price of barley. If we denote the area planted with wheat and barley with x1 and x2 respectively, then the optimal number of square kilometres to plant with wheat vs barley can be expressed as a linear programming problem:

maximize S1x1 + S2x2 (maximize the revenue - this is the "objective function")

subject to (limit on total area)

(limit on fertilizer)

(limit on insecticide)

(cannot plant a negative area)

Which in matrix form becomes:

maximize

subject to

[edit]

Augmented form (slack form)

Linear programming problems must be converted into augmented form before being solved by the simplex algorithm...

This form introduces non-negative slack variables to replace non-equalities with equalities in the constraints. The problem can then be written on the following form:

Maximize Z in:

where xs are the newly introduced slack variables, and Z is the variable to be maximized.

[edit]

Example

The example above becomes as follows when converted into augmented form:

maximize S1x1 + S2x2 (objective function)

subject to x1 + x2 + x3 = A (augmented constraint)

F1x1 + F2x2 + x4 = F (augmented constraint)

P1x1 + P2x2 + x5 = P (augmented constraint)

where x3,x4,x5 are (non-negative) slack variables...

General points of the essay

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