Final answer:
The simplex method is used to perform a pivot on a tableau, and after the pivot, the basic and non-basic variables (Groups I and II) are redefined. Feasibility is determined by non-negative values of variables. The pivot increasing M the most while maintaining solution feasibility is chosen.
Step-by-step explanation:
The question refers to the process of using the simplex method in linear programming to perform a pivot operation on a tableau and determine which variables are part of the basic (Group I) and non-basic (Group II) sets after the pivot. It also requires analyzing the feasibility of the solutions and the effect on the objective function value represented by M.
To answer part (a), without the actual tableau, it is not possible to definitively name the Group I and Group II variables. Typically, Group I variables are the basic variables, which are currently solutions in the tableau, and Group II variables are the non-basic variables, not currently used as solutions.
For part (b), a pivot operation involves changing the basic and non-basic variables by making the pivot element (the element in row 2, column 2 in this case) the new basis, leading to a re-calculation of the tableau. The feasibility is determined by ensuring that all the basic variables maintain non-negative values after the pivot.
In part (c), to increase the value of M the most, we must select the pivot that will lead to the largest increase in the objective function value (M), while still maintaining the feasibility of the solution by keeping all variables (including slack and surplus variables, represented here by u and v) non-negative.