Homework #2 2602327 Quantitative Business Analysis NX , :S , Xz : 8 1) Graphically solve the following problem isoprofit ; 401--841-5×2 Maximize profit = 8X 1 + 5X2 ✗ 1=10 subject to ✗ 2:10 X1 + X2 ≤ 10 X1 ≤ 6 - M=4 ✗ ② slope :& 2=2 2X1 + 4X2 ≥ 8 ✗ 1=2 Xz :3 3X1 -2X2 ≥0 slope : -1 , ✗ 1=8 xi " X1 + 2X2 ≤ 8 c- - 61-2×2=8 ⑤ 2×2=2 X1, X2 ≥ 0 a) What is the optimal solution? 16,11 , profit :S } b) What is the range of optimality for this problem? ' Xz :| 81611-5117 c) What is the range of the change in right-hand side for all constraints? d) What is value of the dual price for all constraints? e) What would be happened if the coefficient of X1 of the objective function change from 8 to 2 f) What would be happened if the coefficient of X2 of the objective function change from 5 to 9 g) Change the right-hand side of constraint 1 to 11 (instead of 10) and resolve the problem. How much did the profit increase or decrease as a result of this? h) Change the right-hand side of constraint 2 to 7 (instead of 6) and resolve the problem. How much did the profit increase or decrease as a result of this? Looking at the graph, what would happen if the rigth-hand side value to go to 4? i) Change the right-hand side of constraint 5 to 10 (instead of 8) and resolve the problem. How much did the profit increase or decrease as a result of this? (b) - E- Cx , so E tz - Cxz let let cx , :S - e- • I ¥ - Cxz NY § t , -164 , Cxzt ( ✗ 270 - - t - cx t • ca - of Range Cx - t N , Cx 16 , - Cx change 1- ✗ in ) tower limit at ✗ upper limit , = point 12,37 2 : upper limit = - N point 16,11 : 16 upper limit at 18,01=8 lower limit at 61-2101--6 = at point 16,47 point 16,01 61-2147=14 constraint lower limit lower limit 5th constraint constraint grd RHS 3161 -2117 N = N - point 21611-4111 = 16,11 16 , E -1g - 2.5 Cx , >12.5 E N Upper limit at 2=7 upper limit 2nd Cx 4th constraint lower limit at point 16,1 , - , ! 2.SE Cx , EN 1st constraint X ¥ - 5 Of Cx , £1b (C) :S , d) ✗ sub 7 : , ✗ 2×2--8 t , 5 sth constraint Dual price of t ⑤ - t sub 2×2--8 it : constraint ② - into 2 2nd price of Dual :b , - 1-2×2--9 , ⑤ - into 2 ② s 61-2×2=9 ; ✗ ✗ it , i. Profit : 8171 dual price i. - - Dual price of : The : profit of hl It : it It ' . . . Profit . : constraints ① solution because is 8167 t 5 dual price (y) :X ? , :b 55.5 = t 2. s : they are non - binding constraints . changed optimality out of the range of is 2 = because :O , optimal . . changed will be increased the ③ ④ , changed won't be It will not affect g) 7 58 s The profit will be decreased . : , tss will be f) The decision . sly ) / The decision e) + : X = a by profit because g 19-57×1 because = is in the range of optimality . 4 constraint ① is a non - binding constraint and the change is in the RHS is binding Total is profit binding Total profit constraint and the will be increased constraint and the will be increased change by the range the range . As a result As a result , we can use dual price to calculate we can use dual price to calculate (7-6115.5)=5.5 change by is within is within (10-8112.5) = 5 . , range ✗ 2 n " g t i. f - I - . " ←