+2x =17 (2) (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 (d) If coefficient of x2 is changed to a,2=2. Determine whether the previous optimal a2」 solution is till optimal (e)If a new variable Xnew has been introduced into the model, Xnew coefficient is a6=3. Determine whether the previous optimal solution is till optimal 2 Solution: the final tableau Basic variable Eq 4 5 Right side 2 1/53/5 -1/3 1/3 5/3 (2) -1/5 /5 (a) The optimal solution is x1=5/3, x2=0, X3=3, maxz=17 min S=25y, + 20y2 6y1+3y2≥3 (b)The dual problem is 3y, +4y2 21 ≥0 (c) The optimal solution for the dual problem from the final set of equations is yl=1/5y2=3/5,minS=17 (d) The coefficient of x2 in row 0 is o,=CRB-A-C 3=--<0.so the 55八3 previous optimal solution is not till optimal (e) The coefficient of new variable Xnew iso=CRB-A6-C6(5 34 3 5 2 5 1 (2) 3 5 3 1 3 1 3 1 (1) 17 5 3 5 1 (0) 2 2 3 4 5 1 2 4 5 2 4 5 + − + = − + − = + + + = x x x x x x x x Z x x x (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. (d) If coefficient of x2 is changed to ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 3 2 3 22 12 2 a a c . Determine whether the previous optimal solution is till optimal. (e) If a new variable Xnew has been introduced into the model, Xnew coefficient is ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 2 3 2 26 16 6 a a c . Determine whether the previous optimal solution is till optimal. Solution: the final tableau is Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 Right side Z (0) 1 0 2 0 1/5 3/5 17 X1 (1) 0 1 -1/3 0 1/3 -1/3 5/3 X3 (2) 0 0 1 1 -1/5 2/5 3 (a) The optimal solution is x1=5/3, x2=0, x3=3, maxZ=17 (b) The dual problem is ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ + ≥ + ≥ + ≥ = + , 0 5 5 4 3 4 1 6 3 3 . . min 25 20 1 2 1 2 1 2 1 2 1 2 y y y y y y y y st S y y (c) The optimal solution for the dual problem from the final set of equations is y1=1/5,y2=3/5,minS=17 (d) The coefficient of x2 in row 0 is 0 5 4 3 3 2 5 3 5 1 2 2 1 2 − = − < ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − = − σ CB B A C , so the previous optimal solution is not till optimal (e) The coefficient of new variable Xnew is 0 5 1 2 2 3 5 3 5 1 6 6 1 6 − = > ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − = − σ CB B A C , so