6. Solve the following problem using the dual simplex method, and present the simplex tableau for each ofthe iterations.maxx1,x2∈Rn-x1+ 2x2s.t.5x1+ 4x2≥20x1+ 5x2= 10x1, x2≥0.7. Given the following problem:maxx1,x2,x3∈Rn-5x1+ 5x2+ 13x3s.t.-1x1+x2+ 3x3≤2012x1+ 4x2+ 10x3≤90x1, x2, x3≥0.(1)(a) Solve the above problem (1) using the primal simplex method.(b) Letx2andx5be basic variables at an optimal solutionx*of (1). Determine the following:i The optimality and feasibility of the optimal basis of (1) if the right-hand side is changed to[b1b2]0= [10 100]0.ii The optimality of (x*0) if a new variablex6is introduced with coefficientsc6= 10,a16= 13, anda26= 5.2
iii The optimality and feasibility of the optimal basis of (1) if the coefficients ofx1are changed toc1=-2,a11= 0, anda12= 5.8. Consider the following problem.maxZ=3x1+x2+4x3s.t.6x1+3x2+5x3≤253x1+4x2+5x3≤20x1≥0,x2≥0,x3≥0.The corresponding final set of equations yielding the optimal solution is(0)Z+2x2++15x4+35x5= 17(1)x1-13x2++13x4-13x5=53(2)x2+x3-15x4+25x5= 3.(a) Identify the optimal solution from this set of equations.(b) Construct the dual problem.(c) Identify the optimal solution for the dual problem from the final set of equations. Verify this solutionby solving the dual problem graphically.