Simplex algorithm and initialization methods (30 pts] Consider the following optimization problems: (P) max (2) min s.t. - 21 - 22 +2.13 + 1 s.t. 2.r1 + 12 +13 +21 > 6 21 + 2x2 - 2x3 +1 0. max 21 + x2 +63 +31 (R) s.t. x1 + x2 +1> 4 x1 + x3 + x1 0 21 +.22 + x3 + 14 x1 + x2 + x1 > 4 2x1 +12 +13 +21 > 6 21,22,23,01 > 0. Solve these problems using the Simplex method seen in class (i.e., matrix form). Proceed as follows: 1. (10 pts) Find an initial basic feasible solution for problems P and Q using the two-phase method. Illustrate all your steps. For each problem, report the basic variables and the corresponding basis at the end of Phase I. What can you conclude about problem Q? Explain. 2. (15 pts) Use the basic feasible solution from Part 1 to solve problem P. For each iteration report: Basic and nonbasic variables Basis (B) Reduced costs Direction of movement O Minimum ratio test Entering and leaving variables Report the optimal solution and the optimal objective function value (if any). What can you conclude about problem P? Explain. 3. (15 pts) Use the Big-M approach to solve problem R. Report the same information as in Part 2