Artificial variables

A set of nonnegative variables added temporarily to a linear program to obtain an initial basic (artificial) feasible solution. If the original constraints are Axv = b, xv ≥ 0, than adding an artificial variable y_i to each equation yields the system Axv + Iyv = b, xv ≥ 0, y ≥ 0, where yv is a column vector of artificial variables. Assuming the vector bv ≥ 0, this system has an obvious basic (artificial) feasible solution, with y_i = b_i being the basic variables and the x_i the nonbasic variables. To obtain a basic solution to the original constraints, the artificial variables must be driven to zero. One way to do this is to solve an auxiliary linear program (known as Phase I) where the objective is to minimize the sum of the artificial variables. If the new system has no solution with all artificial variables equal to zero, then the original constraints are infeasible. Big M method; Phase I; Phase II.