There is an alternative method for solving problems with mixed inequality constraints, which can result in computational savings, called the "Big M" method. Instead of introducing an artificial variable into every constraint asPhase 1 does, introduce it only into the " \(\geq\) " constraints, i.e., those with surplus variables. Then maximize the original objective minus a large but unspecified number \(M\) times the sum of the artificial variables.

(a) Solve Exercise 1 by the Big M method.

(b) Argue that if the original problem has an optimal solution, then at the end of the Big M procedure all artificial variables will be zero and the final system will represent the solution. (Which variables should be basic in the initial system?)

(c) Examine what happens to this procedure when it is applied to the following infeasible problem:

maximize: fx1 + x2 2.x + X2 4 subject to: x1 + x2 X2 5 X1, X2 0