Consider the following linear program P with two resources and three activities. The resources are in the amounts of 8 and 4 and the activities are represented by the three variables. Maximize Z = 2x1 + x2 - x3 (0) subject to x1 + 2x2 + x3 ≤ 8 (1) - x1 + x2 - 2x3 ≤ 4 (2) and x1, x2, x3 ≥ 0 Let x4 and x5 denote the slack variable for functional constraint (1) and (2), respectively. Answer the following questions. 

(a) Provide an upper bound on the number of basic solutions to this problem. 

(b) Find the basic solution of P in which x2 and x3 are the basic variables. (i) Characterize the solution for feasibility (feasible or infeasible) and non-degeneracy (non-degenerate or degenerate) and justify your answer. (ii) Write the defining equations of the corresponding corner point. (iii) Construct the Basis matrix B for this basic solution. 

(c) Consider the following simplex tableau associated with some basic solution of Problem P. Using B-1 (which is shown in the appropriate place of the tableau) and Fundamental Insight formulas, fill-in the remaining entries of the tableau associated with this basic solution. Coefficient of: Basic Variable Eq. Z x1 x2 x3 x4 x5 Right Side Z (0) 1 0 1 x1 (1) 0 0 1/3 -2/3 (2) 0 1 1/3 1/3 

(d) After we apply the simplex method, the optimal final tableau of P is as follows. (The following questions of this problem are based on this optimal simplex tableau). Coefficient of: Basic Variable Eq. Z x1 x2 x3 x4 x5 Right Side Z (0) 1 0 3 3 2 0 16 x1 (1) 0 1 2 1 1 0 8 x5 (2) 0 0 3 -1 1 1 12 Write the optimal solution of P, i.e. values of all variables (original and slack) and z-value and the optimal solution of its dual (dual variables, surplus and dual objective value). 

(e) Are all resources used at the optimal solution? If not, determine the unused amounts of the resources. (f) If the price of resource 1 is $3, should the company buy additional resources to increase profit? 

(g) Find the allowable range for the first resource (b1) so that the optimal basis remains unchanged. 

(h) Find the allowable range of c1 (unit profit for activity #1 - current value is 2) so that the optimal solution remains unchanged. What is the lowest value of c1 below which activity #1 is not profitable? 

(i) Find the allowable range for c2. Using the allowable ranges for c1 and c2 and the 100% Rule, find if an increase by 1 in c1 and a simultaneous increase in c2 by 2 will maintain optimality in the current optimal solution. If yes, find the new Z-value.