max:=3x+2x,+5 x2+x3≤430 °n+2 x1+4x2≤420 x1,x2,x3≥0 Let x4, Xs, and x6 denote the slack variables for the respective constraints. The final simplex tableau is as folle Basic variable Eq Coefficient of Right side X3 (2)03/20 0 1/2 0001 230 2 (a) Identify the optimal solution from this table (b)Identify the optimal solution for the dual problem (c) Introduce a new constraint x,+2x,+3x, <480, Determine whether the previous optimal solution is till optimal (d)If we want the previous optimal solution to be always optimal, how much does the right-side of constraint 1 b, increase? (e)If a new variable Xnew has been introduced into the model, Xnew coefficient is 324 Determine whether the previous optimal solution is till optimal Solution: (a) The optimal solution is x1=0, x2=100, x3=230, maxZ=1350 (b) The optimal solution for the dual problem is y 1=1, y2=2, y3=0, min W=1350 (c) Put the optimal solution x1=0, x2=100, x3=230 into the new constraint x,2x2+3x3<480,0+2X 100+3 X230=890>480, so the previous optimal solution is not till optimal (d)If we want the previous optimal solution to be al ways optimal, then 2-1/40Tb B-b=01/20460≥0,230≤b1≤440, bi can increase 10 units 21 420 (e) If a new variable Xnew has been introduced into the model =CB4+-C,=0202|-9=-2 <0, so the previous optimal solution is ot till optimal4 ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ + ≤ + ≤ + + ≤ = + + 0 4 420 3 2 460 2 430 3 2 5 1 2 3 1 2 1 3 1 2 3 1 2 3 x x x x x x x x x x x x x , , s.t max z Let x4,x5, and x6 denote the slack variables for the respective constraints. The final simplex tableau is as follows: Coefficient of : Basic variable Eq. Z X1 X2 X3 X4 X5 X6 Right side Z (0) 1 4 0 0 1 2 0 X2 (1) 0 -1/4 1 0 1/2 -1/4 0 100 X3 (2) 0 3/2 0 1 0 1/2 0 230 X6 (3) 0 2 0 0 -2 1 1 20 (a) Identify the optimal solution from this table. (b) Identify the optimal solution for the dual problem. (c) Introduce a new constraint 480 x1 + 2x2 + 3x3 ≤ , Determine whether the previous optimal solution is till optimal. (d) If we want the previous optimal solution to be always optimal, how much does the right-side of constraint 1 b1 increase? . (e) If a new variable Xnew has been introduced into the model, Xnew coefficient is ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ 4 2 3 9 37 27 17 7 a a a c . Determine whether the previous optimal solution is till optimal. Solution: (a) The optimal solution is x1=0,x2=100,x3=230, maxZ=1350. (b) The optimal solution for the dual problem is y1=1,y2=2,y3=0,minW=1350 (c) Put the optimal solution x1=0,x2=100,x3=230 into the new constraint x1 + 2x2 + 3x3 ≤ 480 , 0+2×100+3×230=890>480, so the previous optimal solution is not till optimal. (d) If we want the previous optimal solution to be always optimal, then 0 420 460 2 1 1 0 1/ 2 0 1/ 2 1/ 4 0 1 1 ≥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − = − b B b , 230 440 ≤ b1 ≤ , b1 can increase 10 units. (e) If a new variable Xnew has been introduced into the model, ( ) 9 2 0 4 2 3 7 7 1 2 0 1 7 − = − < ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = − = − σ CB B A C , so the previous optimal solution is not till optimal