Linear Programming

Extract

I feel that many of the descriptions of linear programming are either too elementary or else expressed in terms of n variables with m constraints using matrices and innumerable Σ's. This is an attempt to arrive at a description somewhere between the two and in it I take n = 7, m = 3.

Suppose that the function to be minimized is given by

and that in addition to x_q ≥ 0 for all q, the equations of constraint are

Solving equations (2) for any three of the variables, say x₁, x₂, x₃, gives equations like

Suppose that solving for the variables x₁, x₂, x₃ gives non-negative values of β₁, β₂, β₃; if it does not, three other variables must be chosen until positive numbers are obtained on the right-hand side.