Infeasible linear programming reduce errors to find solution
One approach: replace each inequality $a_1 x_1 + \dots + a_n x_n \le b$ by $a_1 x_1 + \dots + a_n x_n \le b + t$, replace each inequality $a_1 x_1 + \dots + a_n x_n \ge b$ by $a_1 x_1 + \dots + a_n x_n \ge b - t$ (using the same $t$ for all of them), and then minimize $t$.