I want step by step handwritten answer for each of the parts from (a) to (f) including their subparts in the part a and b. Appreciate your efforts in advance.

Consider the following linear program min:5x1x2+12x3s.t.:3x1+2x2+x3=105x1+3x2+x4=16x1,x2,x3,x40 An optimal solution to this problem is given by x=[2,2,0,0]T and the corresponding optimal simplex tableau is given by (a) Let us introduce a new variable x5 with A5=[3,3]T and c5=4. Answer the following: (i) Will the current solution remain optimal after updating the model by adding x5 ? Support your answer algebraically. (ii) If the answer is NO, then, what would be the new solution to the problem? In this case, to find a solution, I am asking that you update the current optimal simplex tableau, and carry on additional simplex iteration(s) to answer this question. (b) Let us introduce a new constraint x1+2x25. Answer the following: (i) Will the current solution remain optimal after updating the model by adding the new constraint? Support your answer algebraically. (ii) If the answer is NO, then, what would be the new solution to the problem? In this case, to find a solution, I am asking that you update the current optimal simplex tableau, and carry on additional simplex iteration(s) to answer this question. (c) Find the range of change of b2 so that B={1,2} remains an optimal basis sequence. How is the cost affected? Support your answer algebraically. (d) Find the range of change of c3 and c4 respectively under which so that B={1,2} remains an optimal basis sequence. Support your answer algebraically. (e) Find the range of change of c2 under which so that B={1,2} remains an optimal basis sequence. Support your answer algebraically. (f) Suppose that A3 is changed from [1,0]T to [2,1]T. Would the optimal solution be affected? Support your answer algebraically