Can anyone help me with the Optimisation problem of the big M method. Thanks!

5. It is possible to combine the two phases of the two-phase simplex algorithm into a single proce- dure by the bigM method (see Exercise 8 in Week 4 Tutorial). Let c E R", 0 S b E Rm and A be an m x 11 matrix. Given the LP problem in the form Maximize Z = ch subject to Ax = b with x Z 0, x E R" we form the approximating LP problem Maximize Z(M)=chMZzi i=1 subject to Ax+z = b with 1:20.220, XERn,zElRm In the approximating problem 2 = (21,22, ..., 2...\") is a vector of articial variables and M > 1 is a large constant. The term M 21:1 21- is a penalty term for nonzero articial variables 21-. If this problem is solved by the simplex method, show the following: (a) If an optimal solution is found with z = 0, then the corresponding 3: is an optimal basic feasible solution to the original problem. (b) If for every M > 0 an optimal solution is found with z 95 0, then the original problem is infeasible. (c) If for every M > 0 the approximating problem is unbounded, then the original problem is either unbounded or infeasible. In questions (d)(f) suppose that the original problem has a nite optimal value Z' = Z100). (d) Let Z\" (M ) be the optimal value of the approximating problem. Show that Z*(M) 2 Z*(oo). (e) Show that for M1 5 M2 we have Z*(M1) Z Z*(M2). (f) Show that there is a value Mu such that for M 2 Mo, then Z*(M) = Z100), and hence conclude that the big-M method will produce the right solution for large enough values of M