Consider the following problem. maxZ=4x1+5x2+x3+2x4subjectto:2x1+x2+x3+3x4x1+3x2+x3+2x4x1,x2,x3,x4430 The optimal basis contains variables x1 and x2. Answer the following questions (standard notations are used). (a) Construct the complete optimal dictionary (without solving the problem, using the dictionary representation in matrix form). (b) Obtain the range of objective coefficient c3 for which the current solution is optimal. (c) Obtain the range of objective coefficient c2 for which the current solution is optimal. (d) Obtain the range of RHS b1 for which the current basis is optimal. (e) Obtain the range of RHS b2 for which the current basis is optimal. (f) Obtain the range of constraint coefficient a13 for which the current solution is still optimal. (g) Add a new variable x7 with c7=15 and a7=[1,1]T to the original problem. Is the current solution still optimal? If not, optimize and obtain the new optimal solution. (h) Add a new constraint x1+x2x32 to the original problem. Is the current solution still optimal? If not, optimize and obtain the new optimal solution. What is the value of the new dual variable corresponding to this new constraint. (i) Suppose b1 and b2 can vary simultaneously and independently. Obtain the region within which (b1,b2) can vary such that the current basis is still optimal. Compare your answers to the ones you obtained in (4) and (5), what do you observe